Abstract
This work is devoted to the analysis of the linear Boltzmann equation on the torus, in the presence of a force deriving from a potential. The collision operator is allowed to be degenerate in the following two senses: (1) the associated collision kernel may vanish in a large subset of the phase space; (2) we do not assume that it is bounded below by a Maxwellian at infinity in velocity. We study how the association of transport and collision phenomena can lead to convergence to equilibrium, using concepts and ideas from control theory. We prove two main classes of results. On the one hand, we show that convergence towards an equilibrium is equivalent to an almost everywhere geometric control condition. The equilibria (which are not necessarily Maxwellians with our general assumptions on the collision kernel) are described in terms of the equivalence classes of an appropriate equivalence relation involving transport and collisions. On the other hand, we characterize the exponential convergence to equilibrium in terms of the Lebeau constant, which involves some averages of the collision frequency along the flow of the transport. We also explain how to handle the case of linear Boltzmann equations posed on the phase space associated to a compact Riemannian manifold without boundary.
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Introduction
This paper is concerned with the study of the linear Boltzmann equation
for \(x \in {{\mathbb {T}}}^d, \, v \in {{\mathbb {R}}}^d\), \(d \in {{\mathbb {N}}}^*\), where \({{\mathbb {T}}}^d := {{\mathbb {R}}}^d/{{\mathbb {Z}}}^d\) is the flat torus. The linear Boltzmann equation is a classical model of statistical physics, allowing to describe the interaction between particles and a fixed background [11–13]. Among many possible applications, we mention the modeling of semi-conductors, cometary flows, or neutron transport. We refer the reader interested by further physical considerations or by a discussion of the validity of (1.1) in these contexts to [11, Chapter IV, §3] or [12, Chapter I, §5]. We also point out that this equation can be derived in various settings: see for instance [21] in the context of quantum scattering, or [9] in the context of a gas of interacting particles.
In (1.1), the unknown function \(f = f(t,x,v)\) is the so-called distribution function; the quantity \(f(t,x,v) \, dv dx\) can be understood as the (non-negative) density at time t of particles whose position is close to x and velocity close to v. The function V is a potential which drives the dynamics of particles; we shall assume throughout this work that V is smooth, more precisely that \( V \in W^{2,\infty }({{\mathbb {T}}}^d)\). We normalize the potential so that
This may always be assumed by changing the potential V (which only appears in (1.1) through \(\nabla _x V\)) into \(V + \log \left( \int _{{{\mathbb {T}}}^d}e^{-V} dx \right) \). The linear Boltzmann equation (1.1) is a typical example of a so-called hypocoercive equation, in the sense of Villani [42]. It is made of a conservative part, namely the kinetic transport operator \(v \cdot \nabla _x - \nabla _x V \cdot \nabla _v\) associated to the hamiltonian \(H(x,v)= \frac{1}{2} |v|^2 + V(x)\), and a degenerate dissipative part which is the collision operator (i.e. the right hand-side of (1.1)). According to the hypocoercivity mechanism of [42], only the interaction between the two parts can lead to convergence to some global equilibrium.
The function k is the so-called collision kernel, which describes the interaction between the particles and the background. In the following, we shall denote by \({\mathscr {C}}(f)\) the collision operator, which can be split as
where
are respectively the gain and the loss term. A first property of this operator is that, due to symmetry reasons, the formal identity holds:
This, together with the fact that the vector field \(v \cdot \nabla _x - \nabla _x V \cdot \nabla _v\) is divergence free, implies that the mass is conserved: any solution f of (1.1) satisfies
We shall now list the assumptions we make on the collision kernel k.
A1. The collision kernel k belongs to the class \(C^0({{{\mathbb {T}}}^d} \times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d)\) and is nonnegative.
A2. Introducing the Maxwellian distribution
we assume that \({\mathcal {M}}\) cancels the collision operator, that is
A3. Assume that
It will sometimes be convenient to work with the function
With this notation, Assumptions A2 and A3 may be rephrased in a more symmetric way as
Note that with assumption A2, the function \((x,v)\mapsto {\mathcal {M}}(v)e^{-V(x)}=\frac{1}{(2\pi )^{d/2} }e^{-H(x,v)}\), which we shall call the Maxwellian equilibrium, cancels both the transport operator and the collision operator and thus is a stationary solution of (1.1).
Assumption A3 is a mild growth condition that is in particular satisfied in the standard cases where \(\tilde{k}\) is bounded or has a polynomial growth in the variables v and \(v'\). One of its interests is that the gain operator \({\mathscr {C}}^+\) is then a bounded operator in the functional spaces under interest (see Section 4.1). In particular, this assumption will allow to prove well-posedness on appropriate weighted Lebesgue spaces and to justify the associated dissipation identity.
Assumption A3 is satisfied by kernels \(k(x,v,v') = \tilde{k}(x,v,v') {\mathcal {M}}(v')\) for instance as soon as we have a bound of the form
Before going any further, let us present usual classes of examples of collision kernels covered by Assumptions A1–A3 and addressed in the present article.
E1. “Symmetric” collision kernels. Let k be a collision kernel verifying A1 and A3. We moreover require \(\tilde{k}\) to be symmetric with respect to v and \(v'\), i.e. \(\tilde{k}(x,v,v') = \tilde{k}(x,v',v)\) for all \((x,v,v') \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d\). Notice that for these kernels, A2 is automatically satisfied.
A classical example of such a kernel is the following.
E1’. Linear relaxation kernel. Taking \({k}(x,v,v')=\sigma (x){\mathcal {M}}(v')\), with \(\sigma \ge 0,\) not identically vanishing and \(\sigma \in C^0({{{\mathbb {T}}}^d})\) provides the simplest example of kernel in the class E1. This corresponds to the following equation (often called linearized BGK):
This example also belongs to the following class.
E2. “Factorized” collision kernels Let k be a collision kernel verifying A1–A3. We require k to be of the form
with \(\sigma \in C^0({{{\mathbb {T}}}^d})\), \(\sigma \ge 0,\) not identically vanishing and \(k^* \in C^0({{{\mathbb {T}}}^d}\times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d)\), satisfying for some \(\lambda >0\), for all \(x \in {{\mathbb {T}}}^d\), \(v,v' \in {{\mathbb {R}}}^d\),
The sub-class of E2 which is the most studied in the literature (see e.g. [20]) consists in the following non-degenerate case.
E2’. Non-degenerate collision kernels. Let k be a collision kernel verifying A1–A3. The classical non-degeneracy condition consists in assuming that there exists \(\lambda >0\) such that for all \(x \in {{\mathbb {T}}}^d\), \(v,v' \in {{\mathbb {R}}}^d\)
Later in the paper (see Section 6), we will introduce other classes of collision kernels, that are interesting for our purposes.
Under assumptions A1-A3, the linear Boltzmann equation (1.1) is well-posed in appropriate Lebesgue spaces and some weighted \(L^2\) norm of its solutions, that is the quantity
is dissipated (i.e. decreasing with respect to time, see Lemma 4.1).
This work aims at describing the large time behavior of solutions of (1.1), under assumptions A1–A3. This large framework aims at encompassing collision kernels k which are degenerate in the following two senses:
-
the collision kernel k may vanish in a large subset of the phase space \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\);
-
we do not assume that \(\tilde{k}\) is bounded below by a fixed positive constant at infinity in velocity.
However, still in the spirit of Villani’s hypocercivity, one may hope that the transport term in (1.1) compensates for this strong degeneracy. Our goal is to find geometric criteria (on the hamiltonian H and the collision kernel k) to:
-
P1 characterize convergence to a global equilibrium,
-
P1’ characterize exponential convergence to this global equilibrium.
The study of these questions naturally leads to another problem:
-
P2 describe the structure and the localization properties of the spectrum of the underlying linear Boltzmann operator.
In recent works [4–6, 16], Bernard, Desvillettes and Salvarani investigated P1 and P1’ in a framework close to that of E2. In particular, in [5], the authors have shown that in the case where \(V=0\), \((x, v) \in {{\mathbb {T}}}^d\times {{\mathbb {S}}}^{d-1}\), and \(k^*(x,v,v') = k^*(v,v')\) (where \(k^*\) is defined in E2), the exponential convergence to equilibrium (in the Lebesgue space \(L^1\)) was equivalent to a geometric control condition (similar to that of Bardos-Lebeau-Rauch-Taylor in control theory [3, 37]).
Previous works on this topic, for the non-degenerate class of collision kernels E2’ include [33, 38–41] (spectral approach), [10, 17, 20, 35, 42] (hypocoercivity methods), [28] (Lie techniques), and references therein. There are also several related works which concern the non-linear Boltzmann equation, but we do not mention them since that equation is not studied in this paper.
In this article, we introduce another point of view on these questions (in particular different from [4–6, 16]), by implementing in this context different methods coming from control theory. We borrow several ideas from the seminal paper of Lebeau [32], which concerns the decay rates for the damped wave equation.
The goal of this paper is to give necessary and sufficient geometric conditions ensuring P1 and P1’. We also show that the methods we develop here are sufficiently robust to handle a general Riemannian setting.
In the companion paper [27], we shall explain how to adapt our methods for the case of specular reflection in bounded domains. The related question P2 will be studied as well in [26]. These results were announced in [25].
We now give a detailed overview of the main results of the present work.
Overview of the Paper
In this Section, we give an overview of the results contained in this paper. We state our results on the torus, i.e. when the phase space is \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), but they can all be generalized to Riemannian manifolds (see Sections 9 and 10).
Some Definitions
In this section, we introduce the notions needed to characterize convergence and exponential convergence to equilibrium. Given a collision kernel k satisfying Assumptions A1–A3, we first introduce the set \(\omega \) where the collisions are effective.
Definition 2.1
Define the open set of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\)
Note that because of A1–A2, we also have
Let us recall the definition of the hamiltonian flow associated to H, and associated characteristic curves (or characteristics) in the present setting.
Definition 2.2
The hamiltonian flow \((\phi _t)_{t\in {{\mathbb {R}}}}\) associated to \(H(x,v) = \frac{|v|^2}{2} + V(x)\) is the one parameter group of diffeomorphisms on \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) defined by \(\phi _t(x,v) := (X_t (x,v), \, \Xi _t(x,v))\) with \((x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) and
The characteristic curve stemming from \((x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) is the curve \(\{\phi _t(x,v), t \in {{\mathbb {R}}}^+\}\).
Recall that throughout the paper, we assume that \(V \in W^{2,\infty } ({{\mathbb {T}}}^d)\), so that the Cauchy-Lipshitz theorem ensures the local existence and uniqueness of the solutions of (2.3). Global existence follows from the fact that H is preserved along any characteristic curve. Note in particular that each energy level \(\{H = R\}\) is compact (V being continuous on \({{\mathbb {T}}}^d\), it is bounded from below). Hence, each characteristic curve is contained in a compact set of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\).
The notions needed to understand the interaction between collisions and transport are of two different nature. We start by expressing purely geometric definitions. Then, we formulate structural-geometric definitions. We finally introduce the weighted Lebesgue spaces used in this paper, as well as a definition of a unique continuation type property.
Geometric Definitions
We start by introducing the following definitions:
-
The Geometric Control Condition of [3, 37], in Definition 2.3,
-
The Lebeau constants of [32], \(C^-(\infty )\) and \(C^+(\infty )\), in Definition 2.4,
-
The almost everywhere in infinite time Geometric Control Condition, in Definition 2.5.
Let us first recall the Geometric Control Condition, which is a classical notion in the context of control theory. It is due to Rauch-Taylor [37] and Bardos-Lebeau-Rauch [3].
Definition 2.3
Let U be an open subset of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) and \(T>0\). We say that (U, T) satisfies the Geometric Control Condition (GCC) with respect to the hamiltonian \(H(x,v) = \frac{|v|^2}{2} + V(x)\) if for any \((x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), there exists \(t \in [0,T]\) such that \(\phi _t(x,v) = (X_t(x,v),\Xi _t(x,v)) \in U\).
We shall say that U satisfies the Geometric Control Condition with respect to the hamiltonian \(H(x,v) = \frac{|v|^2}{2} + V(x)\) if there exists \(T>0\) such that the couple (U, T) does.
We now define two important constants in view of the study of the large time behavior of the linear Boltzmann equation, which involve averages of the damping function (usually called collision frequency in kinetic theory) \(b(x,v):= \int _{{{\mathbb {R}}}^d} k(x,v,v') \, dv'\) along the flow \(\phi _t\).
Definition 2.4
Define the Lebeau constants ([32]) in \({{\mathbb {R}}}^+ \cup \{+\infty \}\) by
where \(\phi _t\) denotes the hamiltonian flow of Definition 2.2.
It is not clear at first sight that \(C^-(\infty )\) and \(C^+(\infty )\) are well defined: see [32] and the beginning of Section 7 for a short explanation. It turns out that only \(C^-(\infty )\) will be useful in this paper (but \(C^+(\infty )\) will be interesting in the companion paper [26]).
Finally, we introduce a weaker version of the Geometric Control Condition, which will also plays an important role in this work.
Definition 2.5
Let U be an open subset of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\). We say that U satisfies the almost everywhere infinite time (a.e.i.t.) Geometric Control Condition with respect to the hamiltonian \(H(x,v) = \frac{|v|^2}{2} + V(x)\) if for almost every \((x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), there exists \(s\ge 0\) such that the characteristics \((X_t(x,v), \, \Xi _t(x,v))_{t \ge 0}\) associated to H satisfy \((X_{t=s},\Xi _{t=s}) \in U\).
Using this terminology, the usual Geometric Control Condition of Definition 2.3 could be called “everywhere finite time” GCC.
Remark 2.1
We have the following characterization of the different geometric properties introduced here.
-
The couple (U, T) satisfies GCC if and only if \(\bigcup _{s \in (0,T)}\phi _{-s}(U)= {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\).
-
The set U satisfies the a.e.i.t. GCC if and only if there exists \({\mathcal {N}}\subset {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) with zero Lebesgue measure such that \(\bigcup _{s \in {{\mathbb {R}}}^+}\phi _{-s}(U)\cup {\mathcal {N}}= {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\). Note that this implies in particular that \({\mathcal {N}}\) is a closed subset of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) satisfying \(\phi _s({\mathcal {N}}) \subset {\mathcal {N}}\) for all \(s\ge 0\).
Structural-Geometric Definitions
In the sequel, the above geometric definitions are used with \(U = \omega \). They hence involve joint properties of the flow \(\phi _t\) together with the open set \(\omega \), i.e. of the damping function \(b =\int _{{{\mathbb {R}}}^d} k( \cdot , \cdot , v')\, dv'\). As such, they do not take into account the fine structure of the Boltzmann operator, and in particular the non-local property with respect to the velocity variable of the gain operator \(f \mapsto \big ( (x,v) \mapsto \int _{{{\mathbb {R}}}^d} k( x , v', v) f(x,v')\, dv' \big )\).
The next definitions aim at describing how the information may travel between the different connected components of \(\omega \). Let us first define two basic binary relations on the open sets of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\).
Definition 2.6
Let \(U_1\) and \(U_2\) be two open subsets of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\). We say that \(U_1 \, {\mathcal {R}} _{\phi } \, U_2\) if there exist \(s \in {{\mathbb {R}}}\) such that \(\phi _s(U_1) \cap U_2 \ne \emptyset \).
Definition 2.7
Let \(U_1\) and \(U_2\) be two open subsets of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\). We say that \(U_1 \, {\mathcal {R}} _k \, U_2\) if there exist \((x,v_1,v_2) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d\) with \((x,v_1) \in U_1\), \((x,v_2) \in U_2\) such that \(k(x,v_1,v_2)>0\) or \(k(x,v_2,v_1)>0\).
Both relations are symmetric and \(\, {\mathcal {R}} _{\phi }\) is moreover reflexive. When restricted to open sets intersecting \(\omega \), the relation \(\, {\mathcal {R}} _k\) also becomes reflexive. The relation \(\, {\mathcal {R}} _{\phi }\) expresses the fact that the open sets are “connected through” the flow \(\phi _s\), whereas the relation \(\, {\mathcal {R}} _{k}\) means that the open sets are “connected through” a collision.
We also define another convenient x-dependent binary relation.
Definition 2.8
Let \(x \in {{\mathbb {T}}}^d\) and \(O_1\) and \(O_2\) be two open subsets of \({{\mathbb {R}}}^d\). We say that \(O_1 \, {\mathcal {R}} _k^x \, O_2\) if there exists \((v_1,v_2) \in {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d\) with \(v_1 \in O_1\), \(v_2\in O_2\) such that \(k(x,v_1,v_2)>0\) or \(k(x,v_2,v_1)>0\).
Given now an open subset U of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), we define \(U(x) = \{v \in {{\mathbb {R}}}^d, (x,v) \in U\}\). With this notation, notice that \(U_1 \, {\mathcal {R}} _k \, U_2\) if and only if there exists \(x \in {{\mathbb {T}}}^d\) such that \(U_1(x) \, {\mathcal {R}} _k^x \, U_2(x)\).
Given U an open set of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), we denote by \({\mathcal {C}}{\mathcal {C}}(U)\) the set of connected components of U. Note that from the separability of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), it follows that for any open set \(U \subset {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), the cardinality of the set \({\mathcal {C}}{\mathcal {C}}(U)\) is at most countable.
In the sequel, the main open sets U we are interested in are \(\omega \) and \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\). We now define the key equivalence relation on \({\mathcal {C}}{\mathcal {C}}(\omega )\).
Definition 2.9
Given \(\omega _1\) and \(\omega _2\) two connected components of \(\omega \), we say that \(\omega _1 \Bumpeq \omega _2\) if there are \(N \in {{\mathbb {N}}}^*\) and N connected components \((\omega ^{(i)})_{1\le i \le N}\) of \(\omega \) such that
-
we have \(\omega _1 \, {\mathcal {R}} _{\phi } \, \omega ^{(1)}\) or \(\omega _1 \, {\mathcal {R}} _k \, \omega ^{(1)}\),
-
for all \(1\le i \le N-1\), we have \(\omega ^{(i)} \, {\mathcal {R}} _{\phi } \, \omega ^{(i+1)}\) or \(\omega ^{(i)} \, {\mathcal {R}} _k \, \omega ^{(i+1)}\),
-
we have \(\omega ^{(N)} \, {\mathcal {R}} _{\phi } \, \omega _2\) or \(\omega ^{(N)} \, {\mathcal {R}} _k \, \omega _2\).
The relation \(\Bumpeq \) is an equivalence relation on the set \({\mathcal {C}}{\mathcal {C}}(\omega )\) of connected components of \(\omega \). For \(\omega _1\in {\mathcal {C}}{\mathcal {C}}(\omega )\), we denote its equivalence class for \(\Bumpeq \) by \([\omega _1]\).
This definition means that the two connected components \(\omega _1\) and \(\omega _2\) are linked by \(\, {\mathcal {R}} _{\phi }\) or \(\, {\mathcal {R}} _k\) through a chain of connected components of \(\omega \).
Remark 2.2
We will introduce later in Section 3.2 a related equivalence relation on \({\mathcal {C}}{\mathcal {C}}\left( \bigcup _{t \ge 0} \phi _{-t} \omega \right) \).
Weighted Lebesgue Spaces and a Unique Continuation Type Property
Let us now introduce the weighted Lebesgue spaces that will be used throughout this paper.
Definition 2.10
(Weighted \(L^p\) spaces). We define the Banach spaces \({\mathcal {L}}^p({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\) (for \(p \in [1,+\infty )\)) and \({\mathcal {L}}^\infty ({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\) by
The space \({\mathcal {L}}^2\) is a (real) Hilbert space endowed with the inner product
We finally define a Unique Continuation Property for (1.1).
Definition 2.11
We say that the set \(\omega \) satisfies the Unique Continuation Property if the only solution \(f \in C^0_t({\mathcal {L}}^2)\) to
is \(f= \left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f \, dv \,dx\right) {e^{-V}} {\mathcal {M}}\).
It is actually possible to reformulate in a more explicit form the second equation in (2.6), involving the value of f on connected components of \(\omega \) (see Remark 4.1).
Convergence to Equilibrium
Recall that the main goal of this paper is to provide necessary and sufficient geometric conditions to ensure P1 and P1’. Our results can be formulated as follows.
We first give a general characterization of convergence to some equilibrium.
Theorem 2.1
The following statements are equivalent.
-
(1)
The set \(\omega \) satisfies the a.e.i.t. GCC with respect to H.
-
(2)
For all \(f_0 \in {\mathcal {L}}^2\), there exists a stationary solution \(Pf_0\) of (1.1) such that
$$\begin{aligned} \Vert f(t)- Pf_0 \Vert _{{\mathcal {L}}^2} \rightarrow _{t \rightarrow + \infty } 0, \end{aligned}$$where f(t) is the solution of (1.1) with initial datum \(f_0\).
Theorem 2.1 is actually a weak version of Theorem 5.1, which is our main result in this direction. If (1) or (2) holds, we can actually describe precisely the stationary solution \(Pf_0\). Note that this description is a key step of the proof of Theorem 5.1 (and thus, that of Theorem 2.1). It involves the equivalence classes of another equivalence relation, which is related to \(\Bumpeq \) (see Definition 3.1 and Lemma 3.1): we refer to the statement of Theorem 5.1. In particular, in several cases, the stationary solution \(Pf_0\) is not the Maxwellian equilibrium
As a matter of fact, we will see that the dimension of the vector space of stationary solutions is equal to the number of equivalence classes for \(\Bumpeq \). An explicit example of linear Boltzmann equation with several equivalence classes for \(\Bumpeq \) (and thus, for which \(Pf_0\) is not given by (2.7)) is exhibited in Section 6.2.
Among all possible stationary solutions of the linear Boltzmann equation, the Maxwellian equilibrium (2.7) of course particularly stands out. In the next theorem (which is actually also a particular case of Theorem 5.1), we characterize the situation for which the stationary solution ultimately reached is precisely the projection to the Maxwellian.
Theorem 2.2
The following statements are equivalent.
-
(i.)
The set \(\omega \) satisfies the Unique Continuation Property.
-
(ii.)
The set \(\omega \) satisfies the a.e.i.t. GCC and there exists one and only one equivalence class for \(\Bumpeq \).
-
(iii.)
For all \(f_0 \in {\mathcal {L}}^2({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\), denote by f(t) the unique solution to (1.1) with initial datum \(f_0\). We have
$$\begin{aligned} \left\| f(t)- \left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f_0 \, dv \,dx \right) {e^{-V(x)}} {\mathcal {M}}\right\| _{{\mathcal {L}}^2} \rightarrow _{t \rightarrow +\infty } 0. \end{aligned}$$(2.8)
As already mentioned in the introduction, our proofs are inspired by ideas which originate from control theory. For the sake of brevity, we shall not give a detailed explanation of the proof of these results in this introduction. Nevertheless, we would like to comment on an important aspect of the proof of (i.) implies (iii.) in Theorem 2.2. Our approach is based on the fact that the square of the \({\mathcal {L}}^2\) norm of a solution f(t) of (1.1), which we shall sometimes refer to as the energy, is damped via an explicit dissipation identity, see Lemma 4.1:
with \({\mathscr {D}}(f) \ge 0\), which we shall call the dissipation term.
The idea of the proof is to assume by contradiction that there exists an initial condition \(g_0\) in \({\mathcal {L}}^2\), with zero mean, such that the associated solution g(t) to (1.1) does not decay to 0. This yields the existence of \({\varepsilon }>0\) and of a sequence of times \((t_n)_{n \ge 0}\) going to \(+\infty \) such that \(\Vert g(t_n)\Vert _{{\mathcal {L}}^2} \ge {\varepsilon }\).
We then study the sequence of shifted functions \(h_n(t):= g(t+t_n)\). This is the core of our analysis, which basically consists in a uniqueness-compactness argument. We study the weak limit of \(h_n\) and show, using the identity (2.9) and the unique continuation property, that it is necessarily trivial. Then, we consider the associated sequence of defect measures and prove that it is also necessarily trivial, yielding a contradiction.
A difficulty in the analysis comes from the fact that in general, the dissipation term does not control neither the \({\mathcal {L}}^2\) distance to the projection on the set of stationary solutions, nor the \({\mathcal {L}}^2\) norm of the collision operator. However, what holds true is the weak coercivity property
see Lemma 4.5. This turns out to be sufficient for our needs. Denoting by
the linear Boltzmann operator, where \(T f = (v\cdot {\nabla }_x - {\nabla }_x V\cdot {\nabla }_v) f\) and \({\mathscr {C}}\) is the collision operator, the property (2.10) together with the skew-adjointness of T then implies that \({{\mathrm{Ker}}}(A) = {{\mathrm{Ker}}}(T) \cap {{\mathrm{Ker}}}({\mathscr {C}})\). This precise structure, together with the equivalence relation \(\Bumpeq \), allows to identify \({{\mathrm{Ker}}}(A)\), i.e. the space of stationary solutions of (1.1).
Besides, when studying defect measures, the analysis relies on another peculiar structure of (1.1), which is, loosely speaking, made of a propagative and dissipative part (transport and the loss term) and a relatively compact part (the gain term). That the gain term is relatively compact is proved via averaging lemmas (see Appendix 2 and the references therein).
Exponential Convergence to Equilibrium
For what concerns exponential convergence, we need to introduce a technical assumption, which implies an additional mild growth condition on the kernel k at infinity, with respect to A3:
A3’. Assume that there exists a continuous function \(\varphi (x,v):= \Theta \circ H(x,v)\) with \(\Theta : {{\mathbb {R}}}\rightarrow [ 1, +\infty )\), such that for all \((x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), we have
and
As for A3, this assumption is satisfied in the standard case where \(\tilde{k}\) has a polynomial growth in the variables v and \(v'\). Once A3 is assumed, it can be rephrased as
The assumption is satisfied by kernels \(k(x,v,v') = \tilde{k}(x,v,v') {\mathcal {M}}(v')\) for instance as soon as we have a bound of the form
with \(\varphi = \Theta \circ H\) and \(\Theta (t) = C e^{2 \max \{ {\varepsilon }_1,0\}t}\), C large enough. Furthermore, for some \(C_0, C_1>0\), we have \(C_0 e^{\max \{ {\varepsilon }_1,0\} |v|^2}\le \varphi (x,v) \le C_1 e^{\max \{ {\varepsilon }_1,0\} |v|^2}\). Of course, in the case \({\varepsilon }_1 + {\varepsilon }_2 = \frac{1}{4}\), this can be refined with a polynomial correction.
We have the following criterion, assuming that A3’ is satisfied in addition to A1–A3.
Theorem 2.3
(Exponential convergence to equilibrium). Assume that the collision kernel satisfies A3’. The following statements are equivalent:
-
(a.)
\(C^-(\infty ) > 0\).
-
(b.)
There exist \(C>0, \gamma >0\) such that for any \(f_0 \in {\mathcal {L}}^2({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\), the unique solution to (1.1) with initial datum \(f_0\) satisfies for all \(t \ge 0\),
$$\begin{aligned}&\left\| f(t)-\left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f_0 \, dv \,dx \right) {e^{-V(x)}} {\mathcal {M}}\right\| _{{\mathcal {L}}^2}\nonumber \\&\quad \le C e^{-\gamma t} \left\| f_0-\left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f_0 \, dv \,dx\right) {e^{-V(x)}} {\mathcal {M}}\right\| _{{\mathcal {L}}^2}. \end{aligned}$$(2.12) -
(c.)
There exists \(C>0, \gamma >0\) such that for any \(f_0 \in {\mathcal {L}}^2({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\), there exists a stationary solution \(Pf_0\) of (1.1) such that the unique solution to (1.1) with initial datum \(f_0\) satisfies for all \(t \ge 0\),
$$\begin{aligned} \left\| f(t)-Pf_0\right\| _{{\mathcal {L}}^2} \le C e^{-\gamma t} \left\| f_0-Pf_0\right\| _{{\mathcal {L}}^2}. \end{aligned}$$(2.13)
Remark 2.3
If we do not assume that A3’ is satisfied, then we still have that (a.) implies (b.) and (c.).
If \(C^-(\infty )>0\), note in particular that the Geometric Control Condition of Definition 2.3 is satisfied.
One interesting consequence of Theorem 2.3 is a rigidity property of the Maxwellian equilibrium with respect to exponential convergence: loosely speaking, given a linear Boltzmann equation with a collision kernel satisfying A1-A2-A3-A3’, if uniform convergence to (some) equilibrium holds, then the stationary solution ultimately reached is necessarily the projection of the initial datum on the Maxwellian equilibrium.
The assumption A3’ is used in the proof of \((b.) \Longrightarrow (a.)\) to solve an issue due to the combined effect of the nonlocality of the operator \({\mathscr {C}}^+\) (which, from small velocities, can instantaneously create arbitrary large ones), together with the unboundedness of the multiplication operator \({\mathscr {C}}^-\) (still for large velocities).
As an immediate consequence of Theorem 2.3, we deduce the following result.
Corollary 2.1
Assume that there is \(x \in {{\mathbb {T}}}^d\) such that \({\nabla }_x V(x)=0\) and \(\int k(x,0,v') \, dv' =0\). Then \(C^-(\infty )=0\) and there is no uniform exponential rate of convergence to equilibrium.
In particular we get in the free transport case:
Corollary 2.2
Assume that \(V=0\) and that \(p_x(\omega ) \ne {{\mathbb {T}}}^d\), where
denotes the projection on the space of positions. Then there is no uniform exponential rate of convergence to equilibrium.
The proof of Theorem 2.3 is as well inspired by ideas coming from control theory. The proof of \((a.)\implies (b.)\) relies on the following facts.
-
By (2.9), the exponential decay (i.e. (ii.) in Theorem 2.3) can be rephrased as a certain observability inequality relating the dissipation and the energy at time 0, see Lemma 11.1: there exist \(K,T>0\) such that for all \(f_0 \in {\mathcal {L}}^2({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d)\) with zero mean, we have
$$\begin{aligned} K\int _0^T {\mathscr {D}}(f) \, dt \ge \Vert f_0 \Vert _{{\mathcal {L}}^2}^2, \end{aligned}$$where f is the solution of (1.1) with initial datum \(f_0\).
-
This inequality is proved using a contradiction argument, following Lebeau [32], which also consists in a uniqueness-compactness argument. The analysis follows the same lines as those of (i.) implies (iii.) in the proof of Theorem 2.2. In particular, it also relies on the weak coercivity property (2.10). The main difference is that we need to use here the fact that the Lebeau constant is positive in order to show that the sequence of defect measures becomes trivial at the limit, yielding a contradiction.
For what concerns \((b.)\implies (a.)\), the idea is to contradict the observability condition: we construct a sequence of initial conditions in \({\mathcal {L}}^2({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d)\) for (1.1), which concentrate to a trapped ray (whose existence is guaranteed by the cancellation of the Lebeau constant). Loosely speaking, this corresponds to a geometric optics type construction. In order to justify this procedure, we need that the collision kernel satisfies A3’. Finally, we mention that the proof of \((c.)\implies (a.)\) is similar but relies on an additional argument based on the precise version of Theorem 2.1.
Remarks 2.1
-
(1)
As for the damped wave equation [3, 31, 32, 37], the study of asymptotic decay rates relies on “phase space” analysis. However, as opposed to the wave equation, the Boltzmann equation is directly set on the phase space. As a consequence, the study of associated propagation and damping phenomena only uses “local” analysis, whereas that of the wave equation (see [3, 31, 32, 37]) requires the use of microlocal analysis.
-
(2)
One technical difficulty here is to handle the lack of compactness of the phase space \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) in the variable v. It is also possible to consider the equations on a compact phase space. In this case, all our proofs apply, sometimes with significant simplifications. We refer to Section 10.
Organization of the Paper
The main part of this paper is dedicated to the proof of Theorems 2.1, 2.2 and 2.3. Before this, we start with two preliminary sections, namely Sections 3 and 4. In Section 3, we provide several comments on the results, the assumptions, and illustrate some geometric definitions; we also introduce another key equivalence relation used in the main proofs. In Section 4, we then give some preliminaries in the analysis; in Section 4.1, we start by proving the well-posedness of (1.1) and the associated dissipation identity, while Section 4.2 is dedicated to a detailed study of the kernel of the collision operator, which leads to the weak coercivity property (2.10). Section 5 is mainly devoted to the proofs of Theorems 2.2 and 2.1; in Section 5.1, we start by proving Theorem 2.2. Then, in Section 5.2, we state and prove Theorem 5.1, which is the precise version of Theorem 2.1. Section 6 is dedicated to the application of these results to some particular classes of collision kernels. In Section 7, we prove Theorem 2.3. In Section 8, we briefly revisit the recent work of Bernard and Salvarani [4] in our framework, in order to give some abstract lower bounds on the convergence rate when \(C^-(\infty )=0\). Finally, we adapt our analysis in order to handle other geometric situations. In Section 9, we deal with the case of a general compact Riemannian manifold (without boundary): we first explain how to express the linear Boltzmann equation in this setting and generalize Theorems 5.1 and 2.3 to this context. Also, in Section 10, we explain very shortly how to adapt all these results to the case of compact phase spaces.
This paper ends with five appendices. In Appendix 1, we give the equivalence between exponential decay and the observability inequality, used crucially in the proof of Theorem 2.3. In Appendix 2 we give a reminder about classical averaging Lemmas and adapt them to our purposes. In Appendix 3, we provide reformulations of some geometric properties. In Appendix 4, we provide the proof of Proposition 3.2, which relates the two equivalence relations, which are key notions in our analysis. In Appendix 5, to stress the robustness of our methods, we explain how the results of this paper concerning large time behavior can be adapted to other Boltzmann-like equations (e.g. relativistic Boltzmann equation or general linearized BGK equation).
Note added in proof After completion of this paper, we learnt about the work of Mokhtar-Karroubi [34] who studied question P1’ in the case where \(V=0\), \((x, v) \in {{\mathbb {T}}}^d\times U\), and U is a bounded open set of \({{\mathbb {R}}}^d\); he also introduced some quantities in his analysis which are the same as the Lebeau constants.
Remarks and Examples
In this section, we provide several comments on the different geometric definitions introduced in Sections 2.1.1 and 2.1.2.
About a.e.i.t. GCC in the Torus
A first natural question is to understand the a.e.i.t. GCC in the usual situation of free transport (i.e. \(V=0\)), when \(\omega \), i.e. the set where collisions are effective, is of the simple form \(\omega _x \times {{\mathbb {R}}}^d\). We prove that this condition is satisfied for any nonempty \(\omega _x \subset {{\mathbb {T}}}^d\). We also prove that this situation is very particular, and unstable with respect to small perturbations of the potential.
Proposition 3.1
Suppose that \(V = 0\) and that \( \omega = \omega _x \times {{\mathbb {R}}}^d\), where \(\omega _x\) is a non-empty open subset of \({{\mathbb {T}}}^d\). Then \((i.)-(ii.)-(iii.)\) in Theorem 2.2 hold.
Such a result is in particular relevant for the study of the linearized BGK equation (class E1’). Proposition 3.1 shows that there is convergence to the Maxwellian equilibrium (2.7) as soon as \(\sigma \) does not vanish identically.
On the other hand, \(\omega \) being fixed, we give an example of dynamics (i.e. exhibit a potential V) for which the a.e.i.t. GCC fails. More precisely, we prove that this property is very unstable with respect to small perturbations of the potential: for \(\omega = \omega _x \times {{\mathbb {R}}}^d \ne {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) there exist arbitrary small potentials (in any \(C^k\)-norm) such that \(\omega \) does not satisfy a.e.i.t. GCC for the associated Hamiltonian. This illustrates the fact that the dynamics associated to free transport in the torus is not generic.
Proposition 3.2
Assume that \(\overline{p_x(\omega )} \ne {{\mathbb {T}}}^d\), where \(p_x(\omega )\) denotes the projection of \(\omega \) on \({{\mathbb {T}}}^d\) defined in (2.14). Then there exists a potential \(V \in C^\infty ({{\mathbb {T}}}^d)\) such that for any \({\varepsilon }>0\), \(\omega \) does not satisfy a.e.i.t. GCC for the Hamiltonian \(H_{\varepsilon }(x,v) = \frac{|v|^2}{2}+ {\varepsilon }V(x)\).
The proof of Proposition 3.1 and Proposition 3.2 are given respectively in Section 6.3 and Appendix 4.
The Equivalence Relation on \({\mathcal {C}}{\mathcal {C}}\left( \bigcup _{t \ge 0} \phi _{-t} (\omega )\right) \)
We define here another key equivalence relation \(\sim \) on the set of connected components of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\). We then explain the link between the two equivalence relations \(\sim \) on \({\mathcal {C}}{\mathcal {C}}\left( \bigcup _{t \ge 0} \phi _{-t} (\omega )\right) \) and \(\Bumpeq \) on \({\mathcal {C}}{\mathcal {C}}(\omega )\).
Definition 3.1
Given \(\Omega _1, \Omega _2\) two connected components of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\), we say that \(\Omega _1 \sim \Omega _2\) if there are \(N \in {{\mathbb {N}}}\) and N connected components \((\Omega ^{i})_{1\le i \le N}\) of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) such that
-
we have \(\Omega _1 \, {\mathcal {R}} _k \, \Omega ^{(1)}\),
-
for all \(1\le i \le N-1\), we have \(\Omega ^{(i)} \, {\mathcal {R}} _k \, \Omega ^{(i+1)}\),
-
we have \(\Omega ^{(N)} \, {\mathcal {R}} _k \, \Omega _2\).
The relation \(\sim \) is an equivalence relation on the set of \({\mathcal {C}}{\mathcal {C}}\left( \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega ) \right) \) of connected components of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\). For \(\Omega _1 \in {\mathcal {C}}{\mathcal {C}}\left( \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega ) \right) \), we denote its equivalence class for \(\sim \) by \([\Omega _1]\).
The following lemma gives the link between the two equivalence relations. We define the function
The application \(\Psi \) maps \(\omega _0 \in {\mathcal {C}}{\mathcal {C}}(\omega )\) to the connected component \(\Omega _0\) of \(\bigcup _{s \in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) containing \(\omega _0\).
Lemma 3.1
Given \(\omega _1, \omega _2 \in {\mathcal {C}}{\mathcal {C}}(\omega )\), we have \(\omega _1 \Bumpeq \omega _2\) if and only if \(\Psi (\omega _1) \sim \Psi (\omega _2)\). As a consequence, \(\Psi \) goes to the quotient defining a bijection \(\tilde{\Psi }\) between the equivalence classes of \(\Bumpeq \) and \(\sim \):
In particular, the numbers of equivalence classes for \(\Bumpeq \) in \({\mathcal {C}}{\mathcal {C}}(\omega )\) and for \(\sim \) in \({\mathcal {C}}{\mathcal {C}}\Big (\bigcup _{s \in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\Big )\) are equal.
The proof of Lemma 3.1 is given in Appendix 3.1. As a consequence of this lemma, all the results of this paper (together with their proofs) can be formulated with \(\Bumpeq \) or with \(\sim \) equivalently.
Comparing \(C^-(\infty )>0\) and GCC
The statement that \(C^-(\infty )>0\) is in general stronger than the fact that \(\omega \) satisfies the Geometric Control Condition with respect to the hamiltonian \(H(x,v) = \frac{|v|^2}{2} + V(x)\). This is due to the the non-compactness of the phase space \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\).
Assume for instance that \(V = 0\) so that \(\phi _t(x,v) = ( x+tv ,v )\): the characteristic curves are straight lines and the velocity component v is preserved by the flow. Take \(k(x,v',v)>0\) on the whole \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d\). In this situation, \(\omega ={{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) and so, it satisfies automatically GCC in any positive time. Assume further that k does not depend on the space variable, i.e. \(k(x,v',v) = k(v',v)\) and that there exists a sequence \((v_n)\) such that \(\int _{{{\mathbb {R}}}^d} k(v_n,v') dv' \rightarrow 0\). Then, we have \(\left( \int _{{{\mathbb {R}}}^d} k( \cdot ,v') dv' \right) \circ \phi _t (x,v) = \int _{{{\mathbb {R}}}^d} k( v ,v') dv'\) (as the flow preserves v) and hence, for any \(n \in {{\mathbb {N}}}\), we have \(C^-(\infty ) \le \int _{{{\mathbb {R}}}^d} k(v_n,v') dv'\). This yields \(C^-(\infty )= 0\) although \(\omega \) satisfies GCC. As an explicit example, one can take \(k(x, v', v ) = {\mathcal {M}}(v') {\mathcal {M}}(v)^2\).
Note finally that if \(\int _{{{\mathbb {R}}}^d} k( x,v ,v') dv'\) is uniformly bounded from below at infinity (i.e. there exists \(C,R>0\) such that \(\int _{{{\mathbb {R}}}^d} k( x,v ,v') dv' \ge C\) for all \((x,v) \in {{\mathbb {T}}}^d \times B(0,R)^c\)), then \(C^-(\infty )>0\) and GCC become equivalent.
The next paragraph shows that our assumption is indeed more general.
Example of Exponential Convergence Without a Bound from Below at Infinity
Here, we produce a simple example of dynamics and collision kernel such that \(C^-(\infty )>0\), but neither \(\tilde{k}\) nor \(\int _{{{\mathbb {R}}}^d} k( x,v ,v') dv'\) are uniformly bounded from below at infinity.
For this, assume \((x,v) \in {{\mathbb {T}}}\times {{\mathbb {R}}}\) (we could construct similar examples in higher dimensions as well) and take \(V=0\), so that \(\phi _t(x,v) = ( x+tv ,v )\). We identify \({{\mathbb {T}}}\) to \([-1/2,1/2)\) with periodic boundary conditions. Define \(\alpha \in C^0({{\mathbb {T}}}; {{\mathbb {R}}}^+)\) with support contained in \((-1/3,1/3)\) and satisfying \(\alpha = 1\) on \([-1/4,1/4]\) and \(\psi \in C^0({{\mathbb {R}}}; {{\mathbb {R}}}^+)\) such that \(\psi (v) \rightarrow _{|v| \rightarrow +\infty } 0\), \(\psi >0\) and \(\psi = 1\) on \([-2,2]\). Consider the collision kernel in the class E1
We first readily check that \(\tilde{k}>0\) on \({{\mathbb {T}}}\times {{\mathbb {R}}}\times {{\mathbb {R}}}\) and hence \(\omega ={{\mathbb {T}}}\times {{\mathbb {R}}}\). We also remark that for any \(R>0\),
Nevertheless, we can prove that \(C^-(\infty ) \ge C^-(1)>0\), and thus, by Theorem 2.3, there is exponential convergence to the Maxwellian equilibrium. We set \(\beta := \int _{{{\mathbb {R}}}} \psi (v') {\mathcal {M}}(v') \, dv'>0\) and take \((x,v) \in {{\mathbb {T}}}\times {{\mathbb {R}}}\).
-
If \(v\in [-2,2]\), then we have \(k(\phi _t(x,v),v') = k(x+tv, v,v') \ge \psi (v) \psi (v'){\mathcal {M}}(v')\) so that
$$\begin{aligned} \int _0^1 \int _{{{\mathbb {R}}}} k(\phi _t(x,v),v') \, dv' \, dt \ge \beta \int _0^1 \psi (v) \,dt = \beta >0. \end{aligned}$$ -
If \(v\notin [-2,2]\), then, denoting by \(\lfloor v \rfloor \) the integer part of v, we have
$$\begin{aligned} \int _0^1 \int _{{{\mathbb {R}}}} k(\phi _t(x,v),v') \, dv' \, dt&\ge \int _0^1 \alpha (x+tv) \, dt \ge \int _0^{\frac{\lfloor v \rfloor }{|v|}} \alpha (x+tv) \, dt\\&\ge \frac{1}{2} \frac{\lfloor v \rfloor }{|v|} \ge \frac{1}{4}. \end{aligned}$$
This proves that \(C^-(1)>0\) and thus \(C^-(\infty )>0\).
Preliminary Results
Well-Posedness and Dissipation
For readability, we shall sometimes denote
The following dissipation identity holds for sufficiently smooth and decaying solutions to (1.1).
Lemma 4.1
Let k be collision kernel satisfying A1–A3. Let f be a smooth and sufficiently decaying (at infinity in the variable v) solution to (1.1). The following identity holds, for all \(t\in {{\mathbb {R}}}^+\):
where \({\mathscr {D}}(f) = - 2\langle {\mathscr {C}}(f), f \rangle _{{\mathcal {L}}^2}\) satisfies
The term \({\mathscr {D}}(f)\) will often be referred to as the dissipation term in the following. The proof is rather classical and follows [15].
Proof of Lemma 4.1
Multiply (1.1) by \(f \, \frac{e^V}{{\mathcal {M}}(v)}\) and integrate with respect to x and v. This yields
On the one hand, the contribution of the transport term vanishes
since \(\left( v \cdot \nabla _x - \nabla _x V \cdot \nabla _v \right) \frac{e^V}{{\mathcal {M}}(v)} =0\). On the other hand, following [15], we have for any \(x \in {{\mathbb {T}}}^d\) the identity
Symmetrizing the first term in the right hand-side of (4.3) yields
Concerning the second term in the right hand-side of (4.3), we use (1.4) to obtain
Combining the last two identities, we can now collect together the terms with \(k(x,v',v)\) (resp. \(k(x,v, v')\)) and rewrite the right hand-side of (4.3) as a sum of two squares. Namely, this provides
This yields (4.1) and concludes the proof of the Lemma. \(\square \)
We have the following useful lemma for the dissipation functional \({\mathscr {D}}\).
Lemma 4.2
Assume that \((h_n)\) is a sequence of measurable functions such that
and such that \(h_n \rightharpoonup h\) in the sense of distributions. Then, we have
Proof of Lemma 4.2
We denote
Then, introducing
we observe that \(\Vert \tilde{h}_n \Vert _{L^2(d\lambda )}^2= \int _0^T{\mathscr {D}}(h_n) \, dt\) and thus, by (4.4), we deduce that \(\tilde{h}_n(t,x,v,v')\) is uniformly bounded in \(L^2(d\lambda )\). Consequently, up to extracting a subsequence, \(\tilde{h}_n\) weakly converges in \(L^2(d\lambda )\) to some function \(\tilde{h}\). By uniqueness of the limit in the sense of distributions, we have
Then, by (4.4) and weak lower semi-continuity, we deduce that for any \(T>0\), we have
\(\square \)
We can now state the main well-posedness result (which uses all assumptions A1-A2-A3).
Proposition 4.1
(Well-posedness of the linear Boltzmann equation). Assume that \(f_0 \in {\mathcal {L}}^2\). Then there exists a unique solution \(f\in C^0({{\mathbb {R}}}^+;{\mathcal {L}}^2)\) of (1.1) satisfying \(f|_{t = 0} =f_0\). Moreover, the solution f satisfies
and we have
where \({\mathscr {D}}(f)\) is defined in (4.2). If moreover \(f_0 \ge 0\) a.e., then for all \(t \in {{\mathbb {R}}}\) we have \(f(t, \cdot ,\cdot )\ge 0\) a.e. (Maximum principle).
Note that in the case where \(\int _{ {{\mathbb {R}}}^d} k(x,v,v') dv'\) is not a bounded function (in the variable \(v \in {{\mathbb {R}}}^d\)), then this proposition in particular states a gain of integrability of solutions of (1.1).
Let us start with a preliminary description of the operators (on the space \({\mathcal {L}}^2\)) that are involved. We write
The domain of \(A_0\) is given by
The following lemma describes the very first properties enjoyed by functions of \(D(A_0)\).
Lemma 4.3
For all \(f \in D(A_0)\), we have
and
Proof of Lemma 4.3
Let \(\Psi \in C^\infty _c({{\mathbb {R}}}, [0,1])\), with \(\Psi \equiv 1\) on \([-1,1]\) and \(\Psi \equiv 0\) on \((-\infty ,2)\cup (2,+\infty )\). Let \(f \in D(A_0)\). We set \(\Psi _n (x,v) := \Psi \left( \frac{H(x,v)}{n} \right) \), where we recall that H is the hamiltonian. Hence, \(\Psi _n\) is compactly supported. Note also that we have
We define the compactly supported functions \(f_n :=\Psi _n f \). Using a dominated convergence argument, we have
noticing that \(A_0 f_n = \Psi _n A_0 f\). We deduce that
Furthermore, since \(f \in D(A_0)\), we have \((v\cdot {\nabla }_x - {\nabla }_x V\cdot {\nabla }_v) f \in L^2_{loc}\) and thus
We can hence compute (using a classical density argument)
Therefore, we deduce
and thus by a weak lower semi-continuity argument, that
We can therefore finally pass to the limit in the identity
to infer that
that is to say
This concludes the proof of the lemma.
Lemma 4.4
The operator K is bounded on \({\mathcal {L}}^2\) and \({D}(A) = {D}(A_0)\). The Boltzmann operator A generates a strongly continuous semigroup \(e^{-tA}\) on \({\mathcal {L}}^2\) and we have the Duhamel type formula
Note that this only uses A1 and A3.
Proof of Lemma 4.4
On the one hand, we have the explicit formula, for \(t \ge 0\),
where \(\phi _s(x,v) = (X_s(x,v),\Xi _s(x,v))\) denotes the hamiltonian flow of Definition 2.2. Using \(k \ge 0\) and remarking that the change of variables \(\phi _{-t}(x,v) \rightarrow (x,v)\) preserves \(\frac{e^V}{{\mathcal {M}}}\) and has unit Jacobian, we obtain
Moreover, for \(u \in C^\infty _c({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\), with \(S = \mathrm{supp}\,(u)\), we have
where \(K=S\cup \phi _t(S)\). The regularity of the flow \(\phi _t\) yields \(\phi _{-t}(x,v) = (x,v) + O(t)\) where O(t) is uniform on the compact set S as \(t \rightarrow 0^+\). Therefore, using the smoothness of u, a Taylor expansion proves that \(\lim _{t \rightarrow 0^+} \Vert e^{-tA_0} u - u \Vert _{{\mathcal {L}}^2}=0 \).
Next, take any \(u \in {\mathcal {L}}^2\) and fix \({\varepsilon }>0\). There is \(u^{\varepsilon }\in C^\infty _c({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\) such that \(\Vert u- u^{\varepsilon }\Vert _{{\mathcal {L}}^2}< {\varepsilon }/3\), and according to (4.7), \(\Vert e^{-tA_0}(u- u^{\varepsilon }) \Vert _{{\mathcal {L}}^2}\le \Vert u- u^{\varepsilon }\Vert _{{\mathcal {L}}^2}< {\varepsilon }/3\). We also have \(\Vert e^{-tA_0} u^{\varepsilon }- u^{\varepsilon }\Vert _{{\mathcal {L}}^2}^2 \rightarrow 0\) as \(t \rightarrow 0^+\) and thus, there exists \(\delta >0\) such that \(\Vert e^{-tA_0} u^{\varepsilon }- u^{\varepsilon }\Vert _{{\mathcal {L}}^2} <{\varepsilon }/3\) for \(0<t<\delta \). Finally, we have for \(0<t<\delta \)
This means that \(\lim _{t \rightarrow 0^+} \Vert e^{-tA_0} u - u \Vert _{{\mathcal {L}}^2} = 0\) so that the operator \(A_0\) generates a strongly continuous semigroup of contraction on \({\mathcal {L}}^2\), explicited by (4.6).
On the other hand, using the assumption A3, the operator K is bounded on \({\mathcal {L}}^2\). Indeed, by the Cauchy-Schwarz inequality, we have
The operator K is hence bounded in \({\mathcal {L}}^2\), with
Therefore \(D(A)=D(A_0)\) and according to [36, Chapter 3, Theorem 1.1], A therefore generates a strongly continuous semigroup \(e^{-tA}\) on \({\mathcal {L}}^2\). The Duhamel formula also follows from the boundedness of K. \(\square \)
Proof of Proposition 4.1
Lemma 4.4 implies the existence of a unique solution \(f(t) \in C^0({{\mathbb {R}}}^+; {\mathcal {L}}^2)\) to (1.1) with initial condition \(f_0\). For what concerns the maximum principle, we can argue as follows. Assume that \(f_0 \ge 0\) a.e. We may denote by \(C^0(0,T; {\mathcal {L}}^2)_+\) the subset of \(C^0(0,T; {\mathcal {L}}^2)\) of almost everywhere nonegative functions, and define the map
We have the contraction estimate
For \(T>0\) small enough, J is contracting in the Banach space \(C^0(0,T; {\mathcal {L}}^2)_+\), and therefore admits a unique fixed point in \(C^0(0,T; {\mathcal {L}}^2)_+\), which is f by uniqueness of the solution to (1.1). Thus, \(f \ge 0\) a.e. on [0, T], and then on \({{\mathbb {R}}}^+\) by the time translation invariance of (1.1).
Let us finally prove the dissipation identity (4.5). We rely on Lemma 4.1 and an approximation argument.
Note first that if \(f_0 \in D(A)\), it follows from [36, Chapter 1, Theorem 2.4 c)] that the associated solution f satisfies \(f \in C^1({{\mathbb {R}}}^+ ; {\mathcal {L}}^2) \cap C^0 ({{\mathbb {R}}}^+ ; D(A))\), and, in particular, for all \(t \ge 0\), \(f (t) \in D(A)\). This entails that for all \(t \ge 0\), we have
For \(f \in C^0 ({{\mathbb {R}}}^+ ; D(A))\), the quantity \({\mathscr {D}}(f(t))\) is well defined for every t, remarking that (1.4) gives \(\int _{{{\mathbb {R}}}^d} k(x,v', v) \frac{{\mathcal {M}}(v')}{{\mathcal {M}}(v)} \, dv' = \int _{{{\mathbb {R}}}^d} k(x,v , v') \, dv'\).
Moreover, for \(f \in C^1({{\mathbb {R}}}^+ ; {\mathcal {L}}^2) \cap C^0 ({{\mathbb {R}}}^+ ; D(A))\) all computations of the proof of Lemma 4.1 can be performed, so that for any \(f_0 \in D(A)\), we have
Then, note that \(C_c^\infty ({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d) \subset D(A) \subset {\mathcal {L}}^2\), so that D(A) is dense in \({\mathcal {L}}^2\). Let thus \((f_0^n)_{n>0}\) be a sequence such that \(f_0^n \in D(A)\) and \(\Vert f_0^n -f_0 \Vert _{{\mathcal {L}}^2} \rightarrow 0\) as \(n \rightarrow +\infty \). Denote by \(f^n(t) \in C^1({{\mathbb {R}}}^+ ; {\mathcal {L}}^2) \cap C^0 ({{\mathbb {R}}}^+ ; D(A))\) the solution of (1.1) with initial condition \(f_0^n\) obtained from Lemma 4.4. We have in particular
Since \(f_0^n \in D(A)\), the dissipation identity (4.9) holds for \(f^n\). From Lemma 4.2, we then deduce that for all \(t \ge 0\),
Using this information, we obtain from the definition of \(\mathscr {D}(f)\) that
We now want to prove the dissipation identity for any solution \(f \in C^0({{\mathbb {R}}}^+ ; {\mathcal {L}}^2)\). For this, introduce \(\rho \in C^\infty _c({{\mathbb {R}}})\) such that \(\rho \ge 0\), \(\int _{{\mathbb {R}}}\rho = 1\) so that \(\rho _\delta = \frac{1}{\delta } \rho (\frac{t}{\delta })\) is an approximation of identity. We define \(\overline{f}(t)\) such that \(\overline{f}(t) = f(t)\) for \(t\ge 0\) and \(\overline{f}(t) = 0\) for \(t<0\), and set \(f^\delta = \rho _\delta * \overline{f}\) where the convolution is only in the variable \(t \in {{\mathbb {R}}}\). Since \(f \in C^0({{\mathbb {R}}}^+ ; {\mathcal {L}}^2)\), we have (in particular) \(f^\delta \in C^1({{\mathbb {R}}}^+ ; {\mathcal {L}}^2)\), and in the sense of distributions, we have
Since \(f^\delta \in C^1({{\mathbb {R}}}^+ ; {\mathcal {L}}^2)\), this equation yields \(A f^\delta = - {\partial }_t f^\delta \in C^0({{\mathbb {R}}}^+_* ; {\mathcal {L}}^2)\), so that \(f^\delta \in C^0({{\mathbb {R}}}^+_* ;D(A))\). As a consequence, the following dissipation identity holds for \(f^\delta \):
Next, the usual proofs for approximations of identity show that
-
\(\Vert f^\delta (t) - f(t)\Vert _{{\mathcal {L}}^2} \rightarrow 0\) for all \(t>0\);
-
If \(f \in L^2(t' , t ; {\mathscr {L}}^2)\), with \({\mathscr {L}}^2 = L^2({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d , \Psi (x,v) dx\, dv)\) where \(\Psi \ge 0\) is a measurable real valued function, then \( f^\delta \in L^2(t' , t ; {\mathscr {L}}^2)\) and we have \( \Vert f^\delta - f\Vert _{L^2(t' , t ; {\mathscr {L}}^2)} \rightarrow 0\).
The first item allows to pass to the limit \(\delta \rightarrow 0^+\) in the right hand-side of (4.10).
Moreover, we have proved that the solution \(f \in C^0({{\mathbb {R}}}^+ ; {\mathcal {L}}^2)\) is such that the two quantities \(\int _{t'}^t {\mathscr {D}}(f) \, ds\) and \(\int _{t'}^t \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} \left( \int _{{{\mathbb {R}}}^d} k(x,v , v') dv'\right) \frac{|f(s,x,v)|^2 e^{V(x)}}{{\mathcal {M}}(v)} \, dv \, dx \, ds\) are finite. In particular, we have
and the same identity holds with f replaced by \(f^\delta \in C^0({{\mathbb {R}}}^+_* ;D(A))\). We estimate separately the two different terms appearing in \( \int _{t'}^t {\mathscr {D}}(f^\delta ) \, ds - \int _{t'}^t {\mathscr {D}}(f) \, ds\). First, we have
Using the second item above with \({\mathscr {L}}^2 = {\mathcal {L}}^2\) (i.e. \(\Psi = \frac{e^V}{{\mathcal {M}}}\)) together with the fact that \(\Vert f^\delta \Vert _{L^2(t' , t ;{\mathcal {L}}^2)} \le C \Vert f_0 \Vert _{{\mathcal {L}}^2}\), this implies
as \(\delta \rightarrow 0^+\). Second, taking \({\mathscr {L}}^2 = L^2({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d , \Psi (x,v) dx\, dv)\) with \(\Psi = \left( \int _{{{\mathbb {R}}}^d} k(x,v , v') dv'\right) \frac{ e^{V(x)}}{{\mathcal {M}}(v)}\) in the second item above, we obtain \( \Vert f^\delta - f\Vert _{L^2(t' , t ; {\mathscr {L}}^2)} \rightarrow 0\) and in particular
as \(\delta \rightarrow 0^+\). This finally yields \( \int _{t'}^t {\mathscr {D}}(f^\delta ) \, ds \rightarrow \int _{t'}^t {\mathscr {D}}(f) \, ds\), and we can take the limit \(\delta \rightarrow 0^+\) in (4.10), providing the dissipation identity (4.5) for \(0<t'\le t\). Both terms in this identity are continuous as \(t' \rightarrow 0^+\) so that it remains valid for \(t'=0\), which concludes the proof of the proposition. \(\square \)
A useful consequence of the maximum principle, of the linearity of the equation, and of Assumption A2 is the following statement. If \(f_0 \in {\mathcal {L}}^2 \cap {\mathcal {L}}^\infty \), then the unique solution of (1.1) starting from \(f|_{t = 0} =f_0\), satisfies
Weak Coercivity
In this section, we describe some properties of the collision kernel \({\mathscr {C}}\) and associated dissipation D. In several proofs of the paper, we shall need to exploit some local coercivity properties of the dissipation. In particular, we would like to have the weak coercivity property
(and thus, \({\mathscr {D}}(f)=0\) is equivalent to \({\mathscr {C}}(f)=0\)). A difficulty comes from the fact that in general the dissipation term does not control neither the \({\mathcal {L}}^2\) distance to the projection on the set of stationary solutions, nor the \({\mathcal {L}}^2\) norm of the collision operator.
The main result is the following lemma.
Lemma 4.5
Let k be a collision kernel satisfying A1–A3. Let \(T \in (0, +\infty ]\) and denote \(\omega = \cup _{i \in I} \omega _i\) the partition of \(\omega \) in connected components. Then, the following three properties are equivalent
-
(1)
\(f \in C^0(0,T; {\mathcal {L}}^2)\) satisfies \({{\mathscr {C}}}(f(t))=0\) for all \(t \in [0,T]\).
-
(2)
\(f \in C^0(0,T; {\mathcal {L}}^2)\) satisfies \({\mathscr {D}}(f(t))=0\) for all \(t \in [0,T]\).
-
(3)
-
for all \(i \in I\), we have \(f(t,x,v)=\rho _i (t,x){\mathcal {M}}(v)\) on \([0,T] \times \omega _i\);
-
for \(i,j \in I\) and \(x \in {{\mathbb {T}}}^d\), we have: \(\omega _i(x) \, {\mathcal {R}} _k^x \, \omega _j(x) \Longrightarrow \rho _i(t,x) = \rho _j(t,x)\) for all \(t \in [0,T]\).
-
This lemma only states properties of the collision kernel. As such, it is not concerned with the time dependence, that we shall drop in the proof.
Proof of Lemma 4.5
By definition, we have \({\mathscr {D}}(f)= - 2 \langle {\mathscr {C}}(f) , f\rangle _{{\mathcal {L}}^2}\) so that \((1) \Longrightarrow (2)\).
Then, from \({\mathscr {D}}(f)=0\), Equation (4.2) implies that
Let \((x,v) \in \omega \). Thus, there exists \(v' \in {{\mathbb {R}}}^d\) such that \((x,v,v') \in S\). By continuity of k, there exists a neighborhood U of (x, v) such that for all \((y,w) \in U\), we have \((y,w,v') \in S\). Thus, for all \((y,w) \in U\), we have
that is to say that locally, \((y,w) \mapsto \frac{f(y,w)}{{\mathcal {M}}(w)}\) is function of y only. Therefore, for all \(i \in I\), there is a function \(\rho _i\) such that \(\frac{f(x,v)}{{\mathcal {M}}(v)} = \rho _i (x)\) on \( \omega _i\).
Furthermore, take \(i,j \in I\) and \(x \in {{\mathbb {T}}}^d\) such that \(\omega _i(x)\, {\mathcal {R}} _k^x \, \omega _j(x)\). There exists \(v_i,v_j \in {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d\) such that \((x,v_i) \in \omega _i\), \((x,v_j) \in \omega _j\), and \((x,v_i,v_j) \in S\). It then follows from (4.13) and \(\frac{f(x,v_i)}{{\mathcal {M}}(v_i)} = \rho _i (x)\), \(\frac{f(x,v_j)}{{\mathcal {M}}(v_j)} = \rho _j (x)\) that \(\rho _i(x) = \rho _j(x)\). This concludes the proof of \((2)\Longrightarrow (3)\).
Finally, let us check that a function satisfying the two assumptions of (3) cancels the collision operator, i.e. that for all x, v, we have \({\mathscr {C}}(f)(x,v)=0\).
Let \((x,v) \in \omega \) (if \((x,v) \notin \omega \), then \({\mathscr {C}}(f)(x,v)=0\)). Let \(i \in I\) such that \((x,v) \in \omega _i\). We have
where \(J_i\) is the largest subset of I such that for all \(j \in J_i\), there exists \(v' \in {{\mathbb {R}}}^d\) such that \((x,v') \in \omega _j\) and \(\tilde{k}(x,v',v)>0\). According to the second property satisfied by f, for all \(j \in J_i\),
and thus we deduce
The last line comes from the fact that k satisfies A2. This concludes the proof of \((3) \Longrightarrow (1)\). \(\square \)
Remark 4.1
Another benefit of Lemma 4.5 is that it allows to rephrase the Unique Continuation Property, in a slightly more explicit way.
Denote \(\omega = \cup _{i \in I} \omega _i\) the partition of \(\omega \) in connected components. The set \(\omega \) satisfies the Unique Continuation Property if and only if the following holds. The only solution \(f \in C^0({{\mathbb {R}}}; {\mathcal {L}}^2)\) to
satisfying the following properties
-
for all \(i \in I\), \(f(t,x,v)=\rho _i (t,x){\mathcal {M}}(v)\) on \([0,T] \times \omega _i\);
-
for \(i,j \in I\) and \(x \in {{\mathbb {T}}}^d\), \(\omega _i(x) \, {\mathcal {R}} _k^x \, \omega _j(x) \Longrightarrow \rho _i(t,x) = \rho _j(t,x)\) for all \(t \in [0,T]\),
is \(f= \left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f \, dv \,dx\right) {e^{-V(x)}} {\mathcal {M}}(v)\).
Remark 4.2
If \(\tilde{k}\) is in \(L^\infty \) (but only in this case), we can actually prove a stronger result, namely that the dissipation controls the norm of a “symmetrized” collision operator. To state and prove such a result, we introduce the symmetrized collision kernel
Note in particular that \( \overline{k}(x,v',v) \in L^\infty ({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d)\) if \(\tilde{k} \in L^\infty ({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d)\).
We also introduce the associated collision operator
Note that we have \( {\mathcal {C}} (f) = {\mathscr {C}}(f)\) if and only if \(\overline{k}(x,v',v)=\tilde{k}(x,v',v)\), i.e. if \(\tilde{k}(x,v',v)\) is symmetric with respect to v and \(v'\) (this corresponds to the class E1).
Lemma 4.6
Let k be a collision kernel satisfying A1–A2, and such that \(\tilde{k} \in L^\infty \). For any \(f \in {\mathcal {L}}^2\), we have
Proof of Lemma 4.6
According to the definition of the dissipation (4.2) and the symmetry of \(\overline{k}\), we have
and hence
By Jensen’s inequality it follows that
where we used again the symmetry of \(\overline{k}\). This concludes the proof of the lemma. \(\square \)
Characterization of Convergence to Equilibrium
In this Section, we shall first give a proof of Theorem 2.2; then we will provide a proof of Theorem 2.1, which will be a consequence of our main result in this direction, namely Theorem 5.1.
We start with a technical lemma concerning the evolution under the flow of the connected components of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\).
Lemma 5.1
Set \(\tilde{\Omega }= \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) and denote by \((\Omega _i)_{i\in I}\) the partition of \(\tilde{\Omega }\) in connected components, and \({\mathcal {A}}= {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \setminus \tilde{\Omega }\). Then we have for all \(t \ge 0\), for all \(i \in I\),
If moreover \(\omega \) satisfies a.e.i.t. GCC (i.e. \({\mathcal {A}}\) has zero Lebesgue measure), then for all \(i \in I\), for all \(t \in {{\mathbb {R}}}\),
Proof of Lemma 5.1
First, we just remark that for all \(t \ge 0\), we have \(\phi _{-t}(\tilde{\Omega }) = \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-t-s}(\omega ) \subset \tilde{\Omega }\). Taking the complement of this inclusion yields for \(t \ge 0\), \(\phi _{t}({\mathcal {A}}) \subset {\mathcal {A}}\).
Let us now fix \(i \in I\) and prove that
Take \((x,v) \in \Omega _i\) and \(t> 0\). If \(\phi _{-t}(x, v) \notin \Omega _i\), then there exists \(t_0 \in (0,t]\) such that \(\phi _{-t_0}(x, v) \notin \tilde{\Omega }\) since \(\Omega _i\) is a connected component of this set. This is in contradiction with \(\phi _{-t_0}(\tilde{\Omega }) \subset \tilde{\Omega }\). This implies (5.3).
Finally, if \({\mathcal {A}}\) has zero Lebesgue measure, then \(|\phi _{-t}({\mathcal {A}})| = 0\) as well. As a consequence, the identity
yields \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d = \bigcup _{i \in I} \phi _{-t} (\Omega _i )\) up to a set of measure zero. Since for all \(i \in I\) and \(t\ge 0\), \(\phi _{-t} (\Omega _i ) \subset \Omega _i\), we obtain (still for \(t \ge 0\)) \(\phi _{-t} (\Omega _i ) = \Omega _i\) up to a set of measure zero, from which the conclusion of the lemma follows. \(\square \)
Remark 5.1
Note that if \(V=0\) and \(\omega \) satisfies the following property: \((x,v) \in \omega \Leftrightarrow (x,-v) \in \omega \), then the inclusions in (5.1) become equalities. Hence, all sets considered in (5.1) are then invariant by \(\phi _t\) for all \(t\in {{\mathbb {R}}}\).
Proof of Theorem 2.2
We shall prove that \((i.) \implies (iii.)\), that \((iii.) \implies (ii.)\) and finally that \((ii.) \implies (i.)\).
\((i.) \implies (iii.)\) We prove that the Unique Continuation Property (of Definition 2.11) implies the decay.
We first prove the expected convergence for data enjoying more regularity, i.e. we prove
Since (1.1) is linear,
is a solution to (1.1) with initial datum
satisfying \(\int g(0) \, dv \, dx =0\). Therefore proving (5.4) is equivalent to proving that \(\Vert g(t)\Vert _{{\mathcal {L}}^2} \rightarrow 0\).
We argue by contradiction. Assume that there exists an initial datum \(g_{0}\) in \({\mathcal {L}}^2 \cap {\mathcal {L}}^\infty \) (with \(\int g_0 \, dv \,dx = 0 \)), \({\varepsilon }>0\) and an increasing sequence \((t_n)_{n \in {{\mathbb {N}}}}\) such that:
From this sequence, we may extract a subsequence (still denoted \((t_n)\)) satisfying
According to the Maximum Principle of Proposition 4.1, we have
We introduce the shifted function
By the time translation invariance of (1.1), \(h_n\) is still a solution to (1.1), with initial datum \(h_n(0)= g(t_n)\). Using (5.5), up to some extraction, we can assume that there is \(\alpha \in [{{\varepsilon }}, \left\| g_0\right\| _{{\mathcal {L}}^2}]\) such that
Note also that by conservation of the mass, for all \(n \in {{\mathbb {N}}}\) and all \(t \ge 0\),
Using the dissipation identity (4.5) for g, we have:
that is (using the time translation invariance):
This, together with (5.6) and (5.8), implies that for any \(T>0\),
Now, up to another extraction, since for any \(n \in {{\mathbb {N}}}\),
we can assume that \(h_n \rightharpoonup h\) weakly in \(L^2_{t, loc}{\mathcal {L}}^2\). Let us now prove that \(h=0\). First, since \(h_n\) is a solution to (1.1), by linearity, h also satisfies (1.1). Then, according to Lemma 4.2, we have
Thus, by weak coercivity (see Lemma 4.5), we infer that \({{\mathscr {C}}}(h)=0\) on [0, T], for any \(T>0\), and therefore h satisfies the kinetic transport equation (2.6). Using the Unique Continuation Property (see Definition 2.11), we deduce that
Since \(h_{n} \rightharpoonup h\) weakly in \(L^2_t {\mathcal {L}}^2\), using (5.9), we obtain in particular that for any \(T>0\)
Since \(\int _0^{T} \left( \int h \, dv dx \right) dt = T \left( \int h(0) \, dv dx \right) \), we deduce that \(\int h \, dv dx=0\) so that \(h=0\).
Let us now consider the sequence of defect measures \(\nu _n := |h_n|^2\), which, according to (5.11) and (5.7) satisfies, for all \(n \in {{\mathbb {N}}}\),
for \(C_0 = \max _{(x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} e^{-V(x)}{\mathcal {M}}(v)\). We have that, up to another subsequence \(\nu _n \rightharpoonup \nu \) weakly-\(\star \) in \(L^{\infty }_{t,loc} {\mathcal {L}}^\infty \). Let us compute the equation satisfied by \(\nu \): to this purpose, we consider (1.1) satisfied by \(h_n\) and multiply it by \(h_n\). We obtain:
Using the averaging lemma of Corollary 12.2 and the fact that \(h_n\) weakly converges to 0, we deduce that
strongly in \(L^2_{t,loc}{\mathcal {L}}^2\). On the other hand, according to (5.11), the sequence \((h_n)\) is uniformly bounded in \(L^2_{t,loc}{\mathcal {L}}^2\). We hence obtain
The second term, by definition of \(\nu \), weakly converges to \(- 2 \left( \int _{{{\mathbb {R}}}^d} k(x,v , v')\, dv' \right) \nu \) in the sense of distributions, so that \(\nu \) satisfies the equation
Moreover, writing
and using (5.10) together with (5.14), we deduce that
Thus, (5.15) combined with (5.17) entails that \(\nu \) satisfies the kinetic transport equation
which also shows that \(\nu \in C^0_t({\mathcal {L}}^{2})\). This, combined with the fact that \(\nu =0\) on \({{\mathbb {R}}}^+ \times \omega \) (again coming from (5.17)) and the Unique Continuation Property, implies
According to (5.17), this means that \(\nu =0\).
We now prove that there is no loss of mass at infinity. Let \(R>0\). We have:
which is exponentially converging to zero as \(R\rightarrow \infty \). This yields
Therefore, on the one hand, up to a subsequence, we can assume that \(\nu _n(0)\frac{e^{V}}{{\mathcal {M}}} \rightharpoonup \nu (0)\frac{e^{V}}{{\mathcal {M}}}\) tightly in \( {\mathcal {M}}^+_{x,v}\) and thus
On the other hand, using (5.8), we have
This yields a contradiction, and concludes the proof of (5.4).
We finally deduce (2.8) by an approximation argument. Let \(f_0 \in {\mathcal {L}}^2\) and f(t) be the solution associated to \(f_0\). Let \({\varepsilon }>0\). There exists \(f_{0,{\varepsilon }} \in {\mathcal {L}}^2 \cap {\mathcal {L}}^\infty \) such that \(\Vert f_0 - f_{0,{\varepsilon }}\Vert _{{\mathcal {L}}^2} \le {\varepsilon }\). Let \(f_{\varepsilon }(t)\) be the solution associated to \(f_{0,{\varepsilon }}\). Since \(f- f_{\varepsilon }\) is a solution of (1.1) with initial datum \(f_0 -f_{0,{\varepsilon }}\) we also have for any \(t\ge 0\), \(\Vert f(t) - f_{\varepsilon }(t)\Vert _{{\mathcal {L}}^2} \le {\varepsilon }\).
By (5.4), there exists \(t_0 \ge 0\) such that for all \(t\ge t_0\),
Thus, it follows that for all \(t\ge t_0\),
Besides, we have
The last two inequalities together yield, for all \(t \ge t_0\),
which concludes the proof of (2.8).
\((iii) \Rightarrow (ii.)\) Assume that (ii.) does not hold. Either \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \setminus \bigcup _{s \in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) has positive Lebesgue measure, or the equivalence relation \(\Bumpeq \) has two or more equivalence classes (or equivalently, by Lemma 3.1, the binary relation \(\sim \) has two or more equivalence classes).
Suppose first that \({\mathcal {A}}:= {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \setminus \bigcup _{s \in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) has positive Lebesgue measure. We set
Note that \(f_0\) satisfies \(\int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d}f_0(x,v) dx \, dv > 0\) as \({\mathcal {A}}\) is of positive Lebesgue measure. We consider \(f(t, \cdot )\) the solution to (5.20) with initial datum \(f_0\), given by
Moreover, for all \(t\ge 0\), we have \(\phi _{t}({\mathcal {A}}) \cap \omega = \emptyset \) since \({\mathcal {A}}\cap \phi _{-t}(\omega ) = \emptyset \). Therefore, \(f =0\) on \({{\mathbb {R}}}^+\times \omega \) so that, according to the characterization of \(\omega \) in (2.2), \({\mathscr {C}}(f) = 0\) and f is also a solution of (1.1). Moreover, this implies that
Thus, (iii.) does not hold.
Suppose now that \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \setminus \bigcup _{s \in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) has zero Lebesgue measure and that the equivalence relation \(\sim \) has (at least) two distinct equivalence classes, say \([\Omega _1]\) and \([\Omega _2]\).
We define now a function f(x, v) as follows
Using (5.2) in Lemma 5.1, we deduce that for all \(t \ge 0\),
so that f is a stationary solution of the Vlasov equation \(\partial _t f + v \cdot \nabla _x f - \nabla _x V \cdot \nabla _v f= 0\).
There remains to prove that f cancels the collision operator. Denote \(\omega = \cup _{i \in I} \omega _i\) the partition of \(\omega \) in connected components.
By Lemma 4.5, f cancels the collision operator if and only if f satisfies the following two properties.
-
(1)
For all \(i \in I\),
$$\begin{aligned} f=\rho _i (x){\mathcal {M}}(v) \quad \text {on } \omega _i, \end{aligned}$$ -
(2)
For \(i,j \in I\), and \(x \in {{\mathbb {T}}}^d\), \(\omega _i(x) \, {\mathcal {R}} _k^x \, \omega _j(x) \Longrightarrow \rho _i(x) = \rho _j(x)\).
We check now that the function f defined in (5.19) satisfies these two properties.
Let \(i \in I\). If for all \(\Omega ' \in [\Omega _1]\), \(\omega _i \cap \Omega ' =\emptyset \), then \(f=0\) on \(\omega _i\) (and hence satisfies (1) with \(\rho _i=0\)). If there is \(\Omega ' \in [\Omega _1]\) such that \(\omega _i \cap \Omega ' \ne \emptyset \), then since \(\Omega '\) is a connected component of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\), we have \(\omega _i \subset \Omega '\). Thus we have \(f=e^{-V}{\mathcal {M}}\) on \(\omega _i\) (and hence f satisfies (1) with \(\rho _i=e^{-V}\)). We deduce that for all \(i \in I\), f is of the form \(\rho _i (x){\mathcal {M}}(v)\) on \(\omega _i\).
Take now \(i,j \in I\) and \(x\in {{\mathbb {T}}}^d\) such that \(\omega _i(x) \, {\mathcal {R}} _k^x \, \omega _j(x)\). Let \(\Omega ^{(i)}\) (resp. \(\Omega ^{(j)}\)) be the connected component of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) which contains \(\omega _i\) (resp. \(\omega _j\)). By definition of the relations \(\, {\mathcal {R}} _k\) and \(\, {\mathcal {R}} _k^x\), this directly yields \(\Omega ^{(i)} \, {\mathcal {R}} _k \, \Omega ^{(j)}\), and a fortiori we deduce that \(\Omega ^{(i)}\sim \Omega ^{(j)}\): in other words, these are in the same equivalence class for \(\sim \). According to the definition of f, this implies \(\rho _i(x) = \rho _j(x)\) (which is equal to \( e^{-V(x)}\) if \(\Omega ^{(i,j)} \in [\Omega _1]\) and to 0 if not).
Therefore, the function f satisfies the two properties and, by Lemma 4.5, cancels the collision operator.
However, we have
and \(\cup _{\Omega ' \in [\Omega _2]} \Omega ' \) has a positive Lebesgue measure. Consequently, the measure of \({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \setminus \cup _{\Omega ' \in [\Omega _1]} \Omega '\) is positive, so that f is a stationary solution of (1.1) which is not a uniform Maxwellian. As a consequence, (iii.) does not hold.
\((ii.) \Rightarrow (i.)\) Assume that (ii.) holds. Let \(f \in C^0_t({\mathcal {L}}^2)\) be a solution to
As usual, without loss of generality, we can assume that \(\int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f \, dv \,dx=0\). The goal is to show that \(f=0\).
Since f cancels the collision operator, by Lemma 4.5, the restriction of f to \(\omega \) is necessarily of the form
where \(\omega = \bigcup _{i \in I} \omega _i\) is the partition of \(\omega \) in connected components. Furthermore, for \(i,j \in I\), if there is \(x \in {{\mathbb {T}}}^d\) such that \(\omega _i(x) \, {\mathcal {R}} _k^x \, \omega _j(x)\), then \(\rho _i(t,x) = \rho _j(t,x)\).
Consider now \(\tilde{\omega }\) a connected component of \(\omega \). For \((t,x, v) \in {{\mathbb {R}}}^+ \times \tilde{\omega }\), remark that the function \(g(t,x) := \frac{e^V}{{\mathcal {M}}(v)} \, f\) does not depend on the variable v according to (5.22). We have, in the sense of distributions in \({{\mathbb {R}}}^+ \times \tilde{\omega }\),
Since f satisfies (5.20), this implies that g satisfies the free transport equation
in the sense of distributions in \({{\mathbb {R}}}^+ \times \tilde{\omega }\).
Let \((x, v) \in \tilde{\omega }\). Since \(\tilde{\omega }\) is open, there exists \(\delta >0\) such that \(B(x,\delta )\times B(v,\delta ) \subset \tilde{\omega }\) and \(\eta >0\) such that for all \(t \in (-\eta ,\eta )\), for all \((x', v')\in B(x,\delta )\times B(v,\delta )\), we have \((x'+tv' ,v') \in \tilde{\omega }\). Integrating (5.23) along characteristics we obtain
Setting \(U_x := \{ x - \frac{\eta }{2} v + \frac{\eta }{2} v', v' \in B(v, \delta )\}\), we remark that \(U_x\) is an open set containing x. Moreover, for all \(y \in U_x\), we have \(y = x - \frac{\eta }{2} v + \frac{\eta }{2} v'\) for some \(v' \in B(v, \delta )\) so that, using (5.24), we have
Hence, \(g(0, \cdot )\) is constant on \(U_x\), and therefore constant on \(\tilde{\omega }\) (since \(\tilde{\omega }\) is connected).
Using the time translation invariance of (5.23), we also have that for all \(t\ge 0\), \(g(t, \cdot )\) is locally constant on \(\omega \). As a consequence, for all \(t \ge 0\), \(\frac{e^V}{{\mathcal {M}}} \, f(t,\cdot )\) is locally constant on \(\omega \), which means that \(\rho _i(t,x) = \kappa _i (t)\), i.e.
Since f satisfies the transport equation (5.20) on \({{\mathbb {R}}}^+ \times \omega _i\), we infer that \(\kappa _i\) is constant, so that
Using again the transport equation (5.20), we deduce that \(\frac{e^V}{{\mathcal {M}}} \, f(t,x,v) = \frac{e^V}{{\mathcal {M}}} \, f(0,\cdot ) \circ \phi _{-t}(x,v)\). Since there is only one equivalence class for \(\Bumpeq \), we first deduce that \(f= \kappa e^{-V(x)} {\mathcal {M}}(v) \) on \(\omega \) and then that \(f= \kappa e^{-V(x)} {\mathcal {M}}(v) \) on \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\). Since \(\omega \) satisfies a.e.i.t. GCC, this is a full measure set and we deduce that \(f =\kappa e^{-V} {\mathcal {M}}\). Since \(\int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f \, dv \,dx=0\), necessarily there holds \(f=0\).
This concludes the proof of Theorem 2.2.
Remark 5.2
Note that the proof of \((i.) \implies (iii.)\) relies on the maximum principle for the linear Boltzmann equation (1.1) (i.e. the \({\mathcal {L}}^\infty \) bound). This was in particular useful to prevent loss of mass at infinity for the sequence of solutions under study. This will turn out to be also very useful to overcome another issue in the proof of the analogous theorem in the case of a bounded domain of \({{\mathbb {R}}}^d\) with specular reflection [27].
If \(\omega \) satisfies the Geometric Control Condition of Defintion 2.3, we can actually show a slightly stronger property than the Unique Continuation Property. The above proof of \((ii.) \Longrightarrow (i.)\) in Theorem 2.2 together with the fact that there is a unique equivalence class for \(\sim \) under GCC (indeed, \(\bigcup _{s \in (0,T)}\phi _{-s}(U)= {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) is connected) yields the following proposition.
Proposition 5.1
Assume that \((\omega ,T)\) satisfies the Geometric Control Condition. If \(f \in C^0_t({\mathcal {L}}^2)\) is a solution to
where I is an interval of time of length larger than T, then \(f= \left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f \, dv \,dx\right) {e^{-V(x)}} {\mathcal {M}}(v)\).
This will be useful for the proof of Theorem 2.3.
Proof of Theorem 2.1
We start by describing the vector space of stationary solutions of the linear Boltzmann equation (1.1), when the associated set \(\omega \) satisfies the a.e.i.t. GCC.
For the sake of readability, we set here
We denote \(\tilde{\Omega }= \cup _{i \in I} \Omega _i\) the partition of \(\tilde{\Omega }\) in connected components. We write \(([\Omega _j])_{j\in J}\) the equivalence classes for the equivalence relation \(\sim \). We denote for all \(j \in J\)
We have the following description of the vector space of stationary solutions of (1.1).
Proposition 5.2
Assume that \(\omega \) satisfies the a.e.i.t. GCC. Then a Hilbert basis of the subspace of stationary solutions to the linear Boltzmann equation (1.1) (or, equivalently of \({{\mathrm{Ker}}}(A)\), where A is the linear Boltzmann operator defined in (2.11)) is given by the family \((f_j)_{j \in J}\), with
In particular, the cardinality of the set of equivalence classes for \(\sim \) is equal to the dimension of the vector space of stationary solutions to the linear Boltzmann equation (1.1), i.e.
We can introduce a generalized Unique Continuation Property, as follows.
Definition 5.1
We say that the set \(\omega \) satisfies the generalized Unique Continuation Property if the only solutions \(f \in C^0_t({\mathcal {L}}^2)\) to
are of the form \(f=\sum _{j\in J} \langle f, f_j \rangle _{{\mathcal {L}}^2} \, f_j =\sum _{j \in J} \frac{1}{\Vert \mathbbm {1}_{U_j} e^{-V} {\mathcal {M}}\Vert _{{\mathcal {L}}^2}} \left( \int _{U_j} f \, dv dx\right) f_j\), where \((U_j)_{j \in J}\) is defined in (5.25) and \((f_j)_{j \in J}\) in (5.26).
We can now state the precise version of Theorem 2.1:
Theorem 5.1
We keep the notations of Proposition 5.2. The following statements are equivalent.
-
(i.)
The set \(\omega \) satisfies the generalized Unique Continuation Property (see Definition 5.1).
-
(ii.)
The set \(\omega \) satisfies the a.e.i.t. GCC.
-
(iii.)
For all \(f_0 \in {\mathcal {L}}^2({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\), denoting by f(t) the unique solution to (1.1) with initial datum \(f_0\), we have
$$\begin{aligned} \left\| f(t)-Pf_0 \right\| _{{\mathcal {L}}^2} \rightarrow _{t \rightarrow +\infty } 0, \end{aligned}$$(5.28)where
$$\begin{aligned} P f_0 (x,v) = \sum _{j\in J} \frac{1}{\Vert \mathbbm {1}_{U_j} e^{-V} {\mathcal {M}}\Vert _{{\mathcal {L}}^2}} \left( \int _{U_j} f_0 \, dv dx\right) f_j , \end{aligned}$$(5.29)with \((U_j)_{j \in J}\) defined in (5.25) and \((f_j)_{j \in J}\) defined in (5.26).
-
(iv.)
For all \(f_0 \in {\mathcal {L}}^2({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\), there exists a stationary solution \(Pf_0\) of (1.1) such that we have
$$\begin{aligned} \left\| f(t)-Pf_0 \right\| _{{\mathcal {L}}^2} \rightarrow _{t \rightarrow +\infty } 0, \end{aligned}$$(5.30)where f(t) is the unique solution to (1.1) with initial datum \(f_0\).
Note that Theorem 2.2 is a particular case of Theorem 5.1, when there is only one equivalence class for \(\sim \) (or equivalently for \(\Bumpeq \)).
This section is devoted to the proof of Theorem 5.1 and is organized as follows: in Paragraph 5.2.1, we prove Proposition 5.2. Then, in Paragraph 5.2.2, we prove that (iv.) implies (ii.). Finally, in Paragraph 5.2.3, we show that (i.)–(ii.)–(iii.) are equivalent. Since the implication \((iii) \Longrightarrow (iv.)\) is straightforward, this will conclude the proof of Theorem 5.1.
Proof of Proposition 5.2
We start by checking that for all \(i \in J\), \(f_j\) is a stationary solution of (1.1). From Lemma 5.1, we know that for any connected component \(\Omega '\) of \(\tilde{\Omega }\) and any \(t\ge 0\), \(\phi _{-t} (\Omega ') = \Omega '\) up to a set of zero measure. Thus for all \(t\ge 0\), \(\phi _{-t} (U_j) = U_j\) up to a set of zero measure. The function \(f_j\) hence cancels the kinetic transport part.
We now check that \(f_j\) cancels the collision operator, i.e. \({\mathscr {C}}(\mathbbm {1}_{U_i} e^{-V}{\mathcal {M}})=0\). We use for this Lemma 4.5.
Denote \(\omega = \cup _{i \in I} \omega _i\) the partition of \(\omega \) in connected components. Let \(i \in I\). If \(\omega _i \cap U_j =\emptyset \), then \(f_j=0\) on \(\omega _i\). If \(\omega _i \cap U_j \ne \emptyset \), then there exists \(\Omega ' \in [\Omega _j]\) such that \(\omega _i \cap \Omega ' \ne \emptyset \). Since \(\Omega '\) is a connected component of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\), we have \(\omega _i \subset \Omega '\) and thus \(f_j=\frac{ e^{-V(x)}{\mathcal {M}}(v)}{\Vert \mathbbm {1}_{U_j} e^{-V} {\mathcal {M}}\Vert _{{\mathcal {L}}^2}} = \rho _j(x) {\mathcal {M}}(v)\) on \(\omega _i\), with \(\rho _j(x)=\frac{ e^{-V(x)}}{\Vert \mathbbm {1}_{U_j} e^{-V} {\mathcal {M}}\Vert _{{\mathcal {L}}^2}}\).
Assume now that there exist \(k,l \in I\) and \(x\in {{\mathbb {T}}}^d\) such that \(\omega _k(x) \, {\mathcal {R}} _k^x \, \omega _l(x)\). Denote by \(\Omega ' \in [\Omega _j]\), the connected component of \(\tilde{\Omega }\) such that \(\omega _l \subset \Omega '\), and \(\Omega ''\) the connected component of \(\tilde{\Omega }\) such that \(\omega _k \subset \Omega ''\). Note then that \(\omega _k(x) \, {\mathcal {R}} _k^x \, \omega _l(x)\) implies \(\Omega ' \sim \Omega ''\). By definition of \(f_j\), this implies that \(\rho _k(x)= \rho _l(x)\). Therefore, by Lemma 4.5, we infer that the function \(f_j\) cancels the collision operator. We deduce that \(f_j\) is a stationary solution of (1.1).
Furthermore, since the supports of the \((f_j)_{j \in J}\) are disjoint, \((f_j)_{j \in J}\) is an orthonormal family of \({\mathcal {L}}^2\).
Finally, let \(\varphi \) be a stationary solution of (1.1). Then \(\varphi \) satisfies
Taking the \({\mathcal {L}}^2\) scalar product with \(\varphi \), we deduce that \({\mathscr {D}}(\varphi )= \langle {\mathscr {C}}(\varphi ), \varphi \rangle _{{\mathcal {L}}^2}=0\), so that by Lemma 4.5, \({{\mathscr {C}}}(\varphi )=0\). Then, with the same analysis as the proof of \((ii.) \implies (i.)\) in Theorem 2.2, we deduce that \(\frac{e^{V}}{{\mathcal {M}}}\varphi \) is constant on each \(U_j\). Using the fact that \(\omega \) satisfies a.e.i.t. GCC, we deduce that we can write
that is
and this concludes the proof.
Necessity of the a.e.i.t. Geometric Control Condition
We prove here that (iv.) implies (ii.) in Theorem 5.1.
We argue by contradiction. Assume that the a.e.i.t. Geometric Control Condition does not hold. Then \({\mathcal {A}}:= {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \setminus \bigcup _{s \in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) has positive Lebesgue measure.
We set
with \(\Psi \) to be determined later on. We define
which satisfies, by construction,
Note that for all \(t\ge 0\), we have \(\phi _{t}({\mathcal {A}}) \cap \omega = \emptyset \) since \({\mathcal {A}}\cap \phi _{-t}(\omega ) = \emptyset \), which yields \({\mathscr {C}}(f(t)) = 0\) for all \(t \ge 0\) and thus f is also a solution of (1.1). We now fix \(\Psi \) in order to ensure that f(t) is not stationary.
-
If \((v \cdot {\nabla }_x - {\nabla }_x V\cdot {\nabla }_v)(\mathbbm {1}_{{\mathcal {A}}}) \ne 0\), then we take \(\Psi =1\).
-
If \((v \cdot {\nabla }_x - {\nabla }_x V\cdot {\nabla }_v )(\mathbbm {1}_{{\mathcal {A}}}) = 0\), we take \(\Psi \) to be a Morse function on \({{\mathbb {T}}}^d\), so that, in particular, \(\Psi \) is smooth and \(\nabla \Psi (x) \ne 0\) for almost every \(x \in {{\mathbb {T}}}^d\). Note that with such a function \(\Psi \), we have \(f_0 \in {\mathcal {L}}^2\). We compute
$$\begin{aligned} \left[ v \cdot {\nabla }_x \Psi - {\nabla }_x V \cdot {\nabla }_v \Psi \right] \mathbbm {1}_{{\mathcal {A}}} = (v \cdot {\nabla }_x \Psi (x) )\mathbbm {1}_{{\mathcal {A}}}(x,v). \end{aligned}$$Therefore for almost all \((x,v) \in {\mathcal {A}}\), this is not null, which shows that f(t) is not stationary.
Finally if there existed a stationary solution \(f_\infty \) of (1.1) such that
then since for all \(t\ge 0\), f(t) is supported in \({\mathcal {A}}\), we also have \(f_\infty \) supported in \({\mathcal {A}}\). Thus \(f_{\infty }\) cancels the collision operator, i.e. \({\mathscr {C}}(f_\infty )=0\). We deduce that \(f(t)- f_\infty \) satisfies
This yields for all \(t\ge 0\),
Moreover, the solution f defined in (5.31) is a non-stationary solution of (5.32) according to the definition of \(\Psi \). In conclusion, we have \( f_0 \ne f_\infty \). This yields \(\Vert f_0 -f_\infty \Vert _{{\mathcal {L}}^2} >0\), which, together with (5.34) contradicts (5.33). This concludes the proof of \((iv.) \Longrightarrow (ii.)\) in Theorem 5.1.
End of the Proof of Theorem 5.1
We have the following key lemma.
Lemma 5.2
Let f be a solution in \(C^0_t({\mathcal {L}}^2)\) of (1.1). Then for all \(j \in J\),
Proof of Lemma 5.2
Let f be a solution in \(C^0_t({\mathcal {L}}^2)\) of (1.1); denote by \(f_0\) its initial datum. Let \(j \in J\). We take the \({\mathcal {L}}^2\) scalar product with \(f_j\) in (1.1) to obtain
where \({\mathscr {C}}^*\) is the collision operator of collision kernel defined by
with \(\tilde{k}(x,v,v') = \frac{k(x,v,v')}{{\mathcal {M}}(v')}\). This follows from Property A2 satisfied by k.
We then use the following two facts.
-
(1)
We have \((v\cdot {\nabla }_x -{\nabla }_x V\cdot {\nabla }_v) f_j =0\) (see the proof of Proposition 5.2).
-
(2)
We have \({\mathscr {C}}^*(f_j)=0\). This follows from Property A2, the fact that \({\mathscr {C}}(f_j)= 0\) (see again the proof of Proposition 5.2) and Lemma 4.5.
We conclude that \(\frac{d}{dt} \langle f, f_j\rangle _{{\mathcal {L}}^2}=0\). \(\square \)
We therefore infer that if f(t) satisfies (1.1) with an initial datum \(f_0\), then for all \(t\ge 0\),
Equipped with this result, we can prove the equivalence between (i.)–(ii.)–(iii.) exactly as for Theorem 2.2, with only minor adaptations. The details are left to the reader.
Application to Particular Classes of Collision Kernels
In this section, we introduce different classes of collision kernels to illustrate the main results of the previous sections. We then draw consequences of the additional assumptions made in these examples.
E3. Let k be a collision kernel verifying A1–A3. Let \(\omega \) be the set where collisions are effective, defined in (2.1). We moreover require that for all \((x,v), (x,v') \in \omega \), there exist \(N \in {{\mathbb {N}}}^*\) and a “chain” \(v_1,\cdots , v_N \in {{\mathbb {R}}}^d\) such that the following hold.
-
For all i, \(1\le i \le N\), \((x,v_i) \in \omega \).
-
The points (x, v) and \((x,v_1)\) belong to the same connected component of \(\omega \).
-
The points \((x,v')\) and \((x,v_N)\) belong to the same connected component of \(\omega \).
-
For all i, \(1\le i \le N-1\), we have
$$\begin{aligned} k(x,v_i,v_{i+1}) > 0 \text { or } k(x,v_{i+1},v_i) > 0. \end{aligned}$$
As a subclass of E3, we have
E3’. Let k be a collision kernel verifying A1–A3. We require that for all \(y \in p_x(\omega )\) (where \(p_x(\omega )\) is the projection of \(\omega \) on \({{\mathbb {T}}}^d\)), the set \(p_x^{-1}( \{y\})\) is included in one single connected component of \(\omega \).
A trivial subclass of E3’ is the case where \(\omega \) is connected. Another subclass of E3’ is given in the following example.
E3”. Let k be a collision kernel verifying A1–A3. We require that
where \(\omega _x\) is an open subset of \({{\mathbb {T}}}^d\).
Remark that E2 is a subclass of E3”.
In what follows, we explain the interest of these classes of collision kernels regarding the geometric definitions introduced before.
The Case of Collision Kernels in The Class E3
The interest of E3 lies in the simple description of the kernel of the associated collision operator \({\mathscr {C}}\).
Using Lemma 4.5 and the “chain” in the definition of a collision kernel in E3, we deduce the following result.
Lemma 6.1
Let k be a collision kernel in the class E3. Let \(T \in (0, +\infty ]\) and assume that \(f \in L^2(0,T; {\mathcal {L}}^2)\) satisfies \({{\mathscr {C}}}(f(t))=0\) for almost every \(t \in [0,T]\). Then there is a function \(\rho \in L^2(0,T ;L^2({{\mathbb {T}}}^d))\) such that
Reciprocally, any function f satisfying this property satisfies \({\mathscr {C}}(f) =0\).
In other words, the kernel of the associated collision operator is equal to the set of functions which are Maxwellians on \(\omega \):
We recall that this property, which is usual in the non degenerate case \(\omega = {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), is not true in general for collision kernels satisfying merely A1, A2 and A3.
This allows us to reformulate in a very simple way the Unique Continuation Property for collision kernels in E3.
Lemma 6.2
Let k be a collision kernel in the class E3. Then the set \(\omega \) satisfies the Unique Continuation Property if and only if the only solution \(f \in C^0_t({\mathcal {L}}^2)\) to
is \(f= \left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f \, dv \,dx\right) {e^{-V(x)}} {\mathcal {M}}(v)\).
Conversely, using again Lemma 4.5, we have the following result.
Lemma 6.3
Let k be a collision kernel satisfying A1–A3. If any function \(f \in {\mathcal {L}}^2\) cancelling the collision operator has its restriction to \(\omega \) satisfying
for some \(\rho \in L^2({{\mathbb {T}}}^d)\), then k necessarily belongs to the class E3.
Therefore, E3 is the largest class of collision kernels such that the kernel of the associated collision operator is equal to the set of functions which are Maxwellians on \(\omega \), i.e. for which (6.1) holds.
The Case of Collision Kernels in the Class E3’
To explain the interest of E3’, let us introduce now another geometric condition:
- (iv.):
-
The set \(\omega \) satisfies the a.e.i.t. GCC and \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) is connected.
This condition is compared to other geometric conditions in Appendix 3.2. It has to be confronted to the geometric condition of Theorem 2.2 (rephrased using Lemma 3.1):
- (ii.):
-
The set \(\omega \) satisfies the a.e.i.t. GCC and there is only one equivalence class for \(\sim \).
In what follows, we shall adopt the notations of Theorem 2.2. It is clear that (iv.) implies (ii.), as (iv.) means that there is a single equivalence class for an equivalence relation defined as in Definition 2.9 with \(\, {\mathcal {R}} _\phi \) only. However, it is false in general that items (i.)–(iii.) in Theorem 2.2 and (iv.) above are equivalent; see Proposition 6.1 below for an example of collision kernel such that (i.)–(iii.) are satisfied, but not (iv.).
Proposition 6.1
For \(V=0\), there exists a collision kernel k in the class E1 and E3, for which (ii) holds, but not (iv).
Proof of Proposition 6.1
Consider \((x,v) \in {{\mathbb {T}}}\times {{\mathbb {R}}}\) (a similar example can be constructed in higher dimension as well). We take a function \(\varphi \in C^0({{\mathbb {R}}})\), such that \(\varphi (0)=0\) and \(\varphi (v) >0\) for all \(v \in {{\mathbb {R}}}\setminus \{0\}\). Define
By construction, A1 and A3 are satisfied and we notice that \(\tilde{k}\) is symmetric (so that k is in E1). We can also readily check that k is in E3 (with \(N=2\)).
We have
and so
from which we deduce that a.e.i.t. GCC is satisfied, but \(\cup _{t\ge 0} \phi _{-t}(\omega )\) is not connected.
On the other hand, the set \(\omega \) satisfies the unique continuation property. Indeed, let f satisfying
and \({\mathscr {C}}(f) =0\). Using Lemma 6.1 and the definition of k, we deduce that \(f= \rho (t,x) {\mathcal {M}}(v)\) on \({{\mathbb {R}}}^+ \times {{\mathbb {T}}}\times {{\mathbb {R}}}\setminus \{0\}\), and thus almost everywhere in \({{\mathbb {R}}}^+ \times {{\mathbb {T}}}\times {{\mathbb {R}}}\).
As f satisfies the transport equation (6.2), this implies that \(f = C {\mathcal {M}}(v)\) for some \(C>0\) and we conclude that the unique continuation property holds.
Therefore, by Theorem 2.2, we deduce that (ii.) holds. \(\square \)
However, when restricting to collision kernels in the class E3’, Conditions (ii.) and (iv.) become equivalent.
Proposition 6.2
Let k be a collision kernel in the class E3’. Then (i.)–(iv.) are equivalent.
Proof of Proposition 6.2
We consider k a collision kernel in the class E3’. Assume that (ii.) holds. The aim is to prove that (iv.) holds. By contradiction, assume that there are at least two connected components \(\Omega _1\), \(\Omega _2\) of \(\bigcup _{t\ge 0} \phi _{-t}(\omega )\).
By (ii.), \(\Omega _1\) and \(\Omega _2\) belong to the same equivalence class for \(\sim \). Thus, there exist \(x,v_1,v_2\) with \((x,v_1) \in \Omega _1\), \((x,v_2) \in \Omega _2\) and
But since k is in the class E3’, the set \(p_x^{-1}( \{x\})\) is included in one connected component of \(\omega \). Thus we cannot have \((x,v_1) \in \Omega _1\) and \((x,v_2) \in \Omega _2\). This is a contradiction and this concludes the proof. \(\square \)
More generally, we observe that for collision kernels in E3’, the equivalence classes for \(\sim \) are exactly the connected components of \(\bigcup _{t\ge 0} \phi _{-t}(\omega )\). Thus Theorem 5.1 can be reformulated as follows.
Corollary 6.1
Let k be a collision kernel in the class E3’. The following statements are equivalent.
-
(1)
The set \(\omega \) satisfies the generalized Unique Continuation Property.
-
(2)
The set \(\omega \) satisfies the a.e.i.t. GCC.
-
(3)
Let \((\Omega _i)_{i \in I}\) be the connected components of \(\cup _{t\ge 0} \phi _{-t}(\omega )\). For all \(f_0 \in {\mathcal {L}}^2({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\), denote by f(t) the unique solution to (1.1) with initial datum \(f_0\). We have
$$\begin{aligned} \left\| f(t)-P f_0\right\| _{{\mathcal {L}}^2} \rightarrow _{t \rightarrow +\infty } 0, \end{aligned}$$(6.3)where
$$\begin{aligned} P f_0 = \sum _{i\in I} \frac{1}{\Vert \mathbbm {1}_{\Omega _i} e^{-V} {\mathcal {M}}\Vert _{{\mathcal {L}}^2}} \left( \int _{\Omega _i} f_0 \, dv dx\right) g_j, \end{aligned}$$with for all \(i \in I\),
$$\begin{aligned} g_i = \frac{\mathbbm {1}_{\Omega _i} e^{-V}{\mathcal {M}}}{\Vert \mathbbm {1}_{\Omega _i} e^{-V} {\mathcal {M}}\Vert _{{\mathcal {L}}^2}}. \end{aligned}$$
We close this section by exhibiting an example of collision kernel in E3’, for which Corollary 6.1, and thus Theorem 5.1, are relevant.
We restrict ourselves to the case \({{\mathbb {T}}}\times {{\mathbb {R}}}\) (this can be easily adapted to higher dimensions). We consider the free transport case, i.e. \(V=0\). We identify \({{\mathbb {T}}}\) to \([-1/2,1/2)\). Consider \(\alpha \in C^0({{\mathbb {T}}})\) supported in \([-1/2,0)\) and \(\beta \in C^0({{\mathbb {T}}})\) supported in [0, 1 / 2) that do not vanish identically.
Let \(\varphi \in L^\infty \cap C^0({{\mathbb {R}}})\) such that \(\varphi >0\) on \({{\mathbb {R}}}^-_*\) and \(\varphi =0\) on \({{\mathbb {R}}}^+\). Likewise, let \(\Psi \in L^\infty \cap C^0({{\mathbb {R}}})\) such that \(\Psi >0\) on \({{\mathbb {R}}}^+_*\) and \(\Psi =0\) on \({{\mathbb {R}}}^-\). We define the collision kernel
Note that \(\tilde{k}(x,v,v') = \alpha (x) \varphi (v) \varphi (v') + \beta (x)\Psi (v) \Psi (v')\) is symmetric in v and \(v'\), and belongs to \(L^\infty \). Thus k is in the class E1. Furthermore, we readily check k is in the class E3’.
Moreover, we have \(\omega = \{\{\alpha >0\} \times {{\mathbb {R}}}^-_*\} \cup \{\{\beta >0\} \times {{\mathbb {R}}}^+_*\}\), and
Thus \(\omega \) satisfies the a.e.i.t. GCC but \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) is not connected.
The basis of the subspace of stationary solutions of (1.1) is given by \((f_j)_{j=1,2}\) with
The Case of Collision Kernels in the Class E3”
We finally study collision kernels in the class E3”. Because of the remarkable properties of the geodesic flow on the torus \({{\mathbb {T}}}^d\), the following holds.
Lemma 6.4
Suppose that \(V = 0\) and that the collision kernel belongs to the class E3”. Then \(\omega \) satisfies a.e.i.t. GCC.
Proof of Lemma 6.4
Define \(T_v^t : \, x \mapsto x + t \, v\); then \((T_v^t \, x)_{t \ge 0}\) is dense in \({{\mathbb {T}}}^d\) for almost every \((x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) (with respect to the Lebesgue measure). This proves the lemma. \(\square \)
We deduce a proof of Proposition 3.1.
Proof of Proposition 3.1
Take \(\omega _x^0\) a connected component of \(\omega _x\). According to Lemma 6.4, \(\omega _x^0 \times {{\mathbb {R}}}^d\) satisfies a.e.i.t. GCC. The result then follows from Proposition 13.1, \((i.) \Rightarrow (iii.)\), and Theorem 2.2. \(\square \)
Characterization of Exponential Convergence
Let us first briefly recall why \(C^-(\infty )\) is well-defined (see [32]). We can first define for all \(T>0\),
which is a continuous nonnegative function since k is. We then remark that the function \(T \mapsto - T C^-(T)\) is subadditive. This entails that \(C^-(\infty ) = \lim _{T \rightarrow +\infty } C^-(T)\) exists.
In this section, we assume that k satisfies A3’ and provide the proof of Theorem 2.3. To this end, we first prove that (a.) and (b.) are equivalent, and finally that (c.) implies (a.). This will conclude the proof, noticing that (b.) implies (c.) is straightforward. One can also readily check that in the proof of (a.) implies (b.), the assumption A3’ is not used.
Proof of Theorem 2.3, (a.) \(\Longleftrightarrow \) (b.)
Since the equation (1.1) is linear, if f(t) satisfies (1.1) then
is still a solution to (1.1). Thus we can deal only with initial data which have zero average.
By conservation of the mass, the Boltzmann equation (1.1) is well-posed in the space
and we can use Lemma 11.1 for solutions in \({\mathcal {L}}^2_0\), which yields that (b.) is equivalent to
- (b’.):
-
There exists \(T>0\) and \(K>0\) such that for all \(f_0 \in {\mathcal {L}}^2_0\), the associated solution f to (1.1) satisfies
$$\begin{aligned} K \int _0^T {\mathscr {D}}(f(t)) \, dt \ge \Vert f_0\Vert _{{\mathcal {L}}^2}^2. \end{aligned}$$
We first prove that (a.) implies \((b'.)\), then that \((b'.)\) implies (a.).
\({(a.) \implies (b'.)}\) Assume that (a.) holds.
We argue by contradiction. Denying \((b'.)\) is equivalent to assuming for all \(T>0\) and all \(C>0\), the existence of \(g_0^{C,T} \in {\mathcal {L}}^2_0\), such that
where \(g^{C,T}(t)\) is the unique solution to (1.1) with initial datum \(g_0^{C,T}\). Taking \(T=n\) and \(C=n\), this yields for all \(n \in {{\mathbb {N}}}^*\), the existence of \(g_{0,n} \in {\mathcal {L}}^2_0\) such that
where \(g_n(t)\) is the unique solution to (1.1) with initial datum \(g_{0,n}\). Furthermore, by linearity of (1.1), we can normalize the initial data so that for all \(n \in {{\mathbb {N}}}^*\),
Recall that by (4.5), we have, for all \(t\ge 0\),
In particular, the sequence \((g_n)_{n \in {{\mathbb {N}}}^*}\) is uniformly bounded in \(L^\infty _t {\mathcal {L}}^2\); thus, up to some extraction, we can assume that \(g_n \rightharpoonup g\) weakly in \(L^2_{t,loc} {\mathcal {L}}^2\). Let us prove that \(g=0\). By linearity of (1.1), g still satisfies (1.1) since \(g_n\) does.
Note also that by conservation of the mass, for all \(n \in {{\mathbb {N}}}\) and all \(t \ge 0\), we have
Take \(T'>0\) such that \((\omega ,T')\) satisfies GCC. By (7.1), we have \(\int _0^{T'} {\mathscr {D}}(g_n(t)) \, dt \rightarrow 0\) and therefore by Lemma 4.2, we deduce
As a consequence, by weak coercivity (see Lemma 4.5), we infer that \({{\mathscr {C}}}(h)=0\) on \([0,T']\), and therefore h satisfies the kinetic transport equation (2.6). The Unique Continuation Property of Proposition 5.1 then implies that
Since \(g_{n} \rightharpoonup g\) weakly in \(L^2_{t,loc} {\mathcal {L}}^2\), using (7.4), we obtain in particular that
Since \(\int _0^{T'} \left( \int g \, dv dx \right) dt = T' \left( \int g(0) \, dv dx \right) \), we deduce that \(\int g \, dv dx=0\), so that \(g=0\). Therefore, this leads to \(g=0\).
Now, let us study the sequence of defect measures \(\nu _n := |g_n|^2\) and \(\nu _{0,n} := |g_{0,n}|^2\). Consider the equation (1.1) satisfied by \(g_n\) and multiply it by \(g_n\) to get:
By Duhamel’s formula, we infer
where, more explicitely, we have
As a consequence, we get for all \(t\ge 0\),
By definition of \(C^-(\infty )\), there exists \(T_0>0\) large enough such that for all \(t\ge T_0, C^-(t)\ge C^-(\infty )/2>0\). We infer, after the change of variables \(\phi _{-t}(x,v) \mapsto (x,v)\), which has unit Jacobian (recall also that the hamiltonian is left invariant by this transform), that for all \(t\ge T_0\),
and thus we can choose \(T_1 \ge T_0\) large enough such that (we recall that \( \Vert \nu _{0,n} \Vert _{{\mathcal {L}}^1} =1\))
which tackles the first term in (7.7).
We now study the second term in (7.7) on the time interval \((0,T_1)\). Since \(g_n \rightharpoonup 0\), by the averaging lemma of Corollary 12.2, we deduce that
Hence, by the weak/strong convergence principle,
We may now estimate the second term in (7.7) by
change variable \(t-s \mapsto s\) in the time integral, apply the Fubini theorem, and finally use the change of variables \(\phi _{s}(x,v) \mapsto (x,v)\), which has unit Jacobian, to obtain, for \(t=T_1\),
according to (7.8). Thus, coming back to (7.7), for n large enough, we finally obtain
But integrating with respect to time (7.3) and using (7.2)-(7.1), we also have
which is a contradiction with (7.9). This concludes the proof of \({(a.) \implies (b'.)}\).
\({(b'.) \implies (a.)}\) We show that if (a.) does not hold (i.e. \(C^-(\infty )=0\)), then \((b'.)\) does not either. Assume that (a.) does not hold. The goal is to show that for all \(T>0\), for all \({\varepsilon }>0\), there exists \(g_{0,{\varepsilon }} \in {\mathcal {L}}^2_0\) such that
where \(g_{\varepsilon }\) is the solution to (1.1) with initial datum \(g_{0,{\varepsilon }}\).
Fix \(T>0\) and \({\varepsilon }>0\). Since \(C^-(\infty )=0\), there exists \((x_0,v_0) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\), such that
Let \(\chi \) be a smooth compactly supported cutoff function defined from \({{\mathbb {R}}}^+\) to \({{\mathbb {R}}}\) such that \(\chi \equiv 1\) on [0, 1] and \(\chi \equiv 0\) on \([2,\infty )\) and such that \(\int _{{{\mathbb {R}}}^+} \chi (r) r^{d-1} \, dr =0\). Consider
Then notice that there is \(\alpha >0\) independent of n such that
The function \(g_{0,n} := \frac{n^{d}}{\alpha } \tilde{g}_{0,n}\) is thus normalized in \({\mathcal {L}}^2\). Note that by construction,
and thus \(g_{0,n} \in {\mathcal {L}}^2_0\).
We call \(g_n\) the solution to (1.1) with initial datum \(g_{0,n}\). By construction, we observe that \(g_{0,n} \rightharpoonup 0\) weakly in \({\mathcal {L}}^2\) and we deduce that \(g_n\rightharpoonup 0\) weakly in \(L^2_{t,loc} {\mathcal {L}}^2\). As in the previous proofs, by the averaging lemma of Corollary 12.2, this implies that
Now, consider \(\nu _n := |g_n|^2\). By construction, we have, in the sense of distributions,
where \(\delta \) denotes as usual the Dirac measure. As in (7.6), we have the Duhamel’s formula
where, more explicitely, we have
We now want to prove that \(\int _0^T {\mathscr {D}}(g_n(t)) dt \rightarrow 0\). To this end, we recall that
and study each term separately. Using (7.12), we directly obtain that
and it only remains to prove \(\int _0^T \langle {\mathscr {C}}^-(g_n), g_n \rangle _{{\mathcal {L}}^2}dt \rightarrow 0\). We have
with, according to (7.14),
We first have
According to (7.13) (together with the fact that \(\nu _{0,n}\) is uniformly compactly supported), we have, as \({n \rightarrow \infty }\) the following convergence
As a consequence of (7.11), there is \(N_0\) such that for \(n \ge N_0\), we have
We now want to study the term \(\int _0^T B_n(t) dt\). For this, we define the following new weighted \(L^2\) norm
where \(\varphi \) is the function given by Assumption A3’. We have the following \({\mathbb {L}}^2\) estimate for the Boltzmann equation (1.1).
Lemma 7.1
For any function \(f_0 \in {\mathcal {L}}^2\) with \(\Vert f_0 \Vert _{{\mathbb {L}}^2}<+\infty \), the solution f(t) to the Boltzmann equation (1.1) with initial datum \(f_0\) satisfies, for all \(t\ge 0\),
where
which is finite by A3’.
Proof of Lemma 7.1
The proof follows from an energy estimate for (1.1). We first multiply (1.1) by \(f \, \varphi ^2(x,v) \frac{e^{V}}{{\mathcal {M}}}\). Recalling that \(\varphi \) is a function of the hamiltonian, we can integrate and argue as in Lemma 4.1 to treat the terms coming from the collision operator and thus obtain:
Above we have used that the dissipation term \({\mathscr {D}}(f \varphi )\) is non-negative. Using the Cauchy-Schwarz inequality, we have, for fixed \(x \in {{\mathbb {T}}}^d\)
from which we deduce \(\frac{1}{2} \frac{d}{dt} \Vert f(t) \Vert _{{\mathbb {L}}^2}^2 \le \Gamma \Vert f(t) \Vert _{{\mathbb {L}}^2}^2\). This implies (7.17), and concludes the proof of the lemma. \(\square \)
By construction of the sequence \((g_{0,n})\), it is uniformly compactly supported and we hence observe that there exists \(C_0>0\) such that for all \(n \in {{\mathbb {N}}}\),
We thus use Lemma 7.1 to infer that there exists \(C_T>0\), such that for all \(n \in {{\mathbb {N}}}\), for all \(t \in [0,T]\),
To estimate \(\int _0^T B_n(t) dt\), we write
and use A3’. We notice that since \(\varphi \) is a function of the hamiltonian, we have
Therefore, using the change of variables \(\phi _{s-t}(x,v) \mapsto (x,v)\), the Fubini theorem, and the Cauchy-Schwarz inequality, we obtain
By (7.12) and (7.18), we deduce that \(\int _0^TB_n(t)\rightarrow 0\) as \(n \rightarrow + \infty \). This, combined with (7.15) and (7.16) implies that the function \(g_{0,{\varepsilon }} := g_{0,n}\) (with n large enough) satisfies (7.10), which concludes the proof of \({(b'.) \implies (a.)}\).
Rigidity with respect to exponential convergence of the Maxwellian
We prove here that (c.) implies (a.) in Theorem 2.3.
Assume that (c.) holds. By (1) implies (2) in Theorem 2.1, \(\omega \) satisfies the a.e.i.t. GCC. Therefore, by Theorem 5.1, this means that \(Pf_0\) is of the form defined in (5.29). We use these notations again.
Recall by Lemma 5.2 that given an equivalence class \([\Omega _j]\) for \(\sim \), denoting as usual \(U_j= \bigcup _{\Omega ' \in [\Omega _j]} \Omega '\), we have for all \(t\ge 0\),
where f(t) is the solution of (1.1) with initial condition \(f_0\).
Thus, the linear Boltzmann equation (1.1) is well-posed in the space
and we can use Lemma 11.1 for solutions in \({\mathcal {L}}^2_{00}\), which yields that the exponential convergence property is equivalent to
-
(c’.)
There exists \(T>0\) and \(K>0\) such that for all \(f_0 \in {\mathcal {L}}^2_{00}\), the associated solution f to (1.1) satisfies
$$\begin{aligned} K \int _0^T {\mathscr {D}}(f(t)) \, dt \ge \Vert f_0\Vert _{{\mathcal {L}}^2}^2. \end{aligned}$$
We can then make the same proof as \((b'.) \implies (a.)\) in Theorem 2.3 in order to infer that if \(C^-(\infty )=0\), then \((c')\) does not hold. We keep the notations of that proof. The only thing to check is that \(g_{0,n}\) defined there belongs to \({\mathcal {L}}^2_{00}\) for n large enough. Let \(j \in J\) such that \((x_0,v_0) \in U_j\). Then for n large enough, \(\mathrm{supp}\,g_{0,n} \subset U_j\). Thus for all \(i \ne j\), we have \( \int _{U_i} g_{0,n}\, dv \, dx =0\) and
by definition of \(g_{0,n}\). Thus \(g_{0,n} \in {\mathcal {L}}^2_{00}\) for n large enough, which concludes the proof of (c.) implies (a.) in Theorem 2.3.
Remarks on Lower Bounds for Convergence when \(C^-(\infty )=0\)
In the situation where \(\omega \) satisfies a.e.i.t. GCC but \(C^-(\infty )=0\), we know by Theorem 5.1 that for all data in \({\mathcal {L}}^2\) there is convergence to some \(Pf_0\) (defined in (5.29)). It is natural to wonder if there is a uniform decay rate for smoother data (in an appropriately defined way). If so, then the question of the convergence rate one can obtain becomes particularly interesting.
Let us provide here some a priori results in this direction. The following is nothing but a rephrasing in a general framework of a result of Bernard and Salvarani [4]. Note that they consider in their work free transport (\(V=0\)) and velocities on the sphere \({{\mathbb {S}}}^{d-1}\), but one can readily check that their methods are relevant for (1.1). In their computations, one should add the weight \(e^{V}/{\mathcal {M}}\) in the integrals.
Theorem 8.1
Denote \(\tau (x,v) := \inf \{ t \ge 0, \phi _{-t}(x,v) \in \omega )\}\). Assume that there is a function of time \(\varphi (t)\) such that
Then, there exists a non-negative initial datum \(f_0\) of \(C^\infty \) class and \(C>0\) such that for any \(t \ge 0\), denoting by f(t) the solution of the linear Boltzmann equation (1.1) with initial datum \(f_0\), we have
where \(Pf_0\) is defined in (5.29).
In particular, we obtain
Corollary 8.1
Assume that \(V=0\) and that \(\overline{p_x(\omega )} \ne {{\mathbb {T}}}^d\), where \(p_x\) denotes the projection on the space of positions. Then there exists a non-negative initial datum \(f_0\) of \(C^\infty \) class and \(C>0\) such that for any \(t \ge 0\), denoting by f(t) the solution of the linear Boltzmann equation with initial datum \(f_0\), we have
where \(Pf_0\) is defined in (5.29).
Proof of Corollary 8.1
Take \(x_0 \in {{\mathbb {T}}}^d \setminus \overline{p_x(\omega )}\). Let \(\delta := \text {dist}(x_0, \overline{p_x(\omega )})\). Consider \(U:=B(x_0,\delta /2)\times B(0,1)\); here \(\tau (x,v) := \inf \{ t \ge 0, x-tv \in p_x(\omega )\}\). Then the crucial point is the straightforward lower bound
and we can thus apply Theorem 8.1. \(\square \)
Combining with Bernard-Salvarani’s theorem which concerns the case with trapped trajectories [4], that we recall below, one may deduce that the “worst” lower bound in the free transport case is due to trapped trajectories, and not to low velocities.
Theorem 8.2
(Bernard-Salvarani [4]). Let k a collision kernel belonging to the class E3” and \(V=0\). Assume that there is \((x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) such that for all \(t\in {{\mathbb {R}}}^+\), \(x+tv \notin \omega _x\). Then there exists a non-negative initial datum \(f_0\) of \(C^\infty \) class and \(C>0\) such that for any \(t \ge 0\), denoting by f(t) the solution of the linear Boltzmann equation with initial datum \(f_0\), we have
It is natural to conjecture that in this case, the bound in \(1/\sqrt{t}\) is optimal (this is supported by numerical evidence, as shown by De Vuyst and Salvarani [14]).
The Case of a General Compact Riemannian Manifold
In this section, we show how the above results adapt to a general Riemannian setting. More precisely, we explain how they apply to the linear Boltzmann equation written on the phase space \(T^*M\) (it could be written on TM equivalently) for a compact Riemannian manifold M, of which \({{\mathbb {T}}}^d\) is a (very) particular case.
Let (M, g) be a smooth compact connected d-dimensional Riemannian manifold (without boundary). In local coordinates, the metric g is a symmetric positive definite matrix such that for all \(x \in M\) and \(u, w \in T_x M\), we have
where the Einstein summation notations are used. This provides a canonical identification between the tangent bundle TM and the cotangent bundle \(T^*M\) via the following formula. For any covector \(\eta \in T^*_x M\) there exists a unique vector \(u \in T_x M\) satisfying
In local coordinates, we have
We can define an inner product on \(T^*_x M\) using the above identification, denoted by \((\cdot ,\cdot )_{g^{-1}(x)}\). In local coordinates, we have
For all \(x \in M\) and all \(\eta \in T^*_x M\), we denote by \(|\eta |_x = (\eta ,\eta )_{g^{-1}(x)}^{\frac{1}{2}}\) the associated norm. Let \(d{{\mathrm{Vol}}}(x)\) be the canonical Riemannian measure on M. In local charts this reads
Up to a renormalization factor, we may assume that M has unit volume, i.e. Vol\((M) = 1.\)
The cotangent bundle \(T^*M\) is canonically endowed with a symplectic 2-form \(\omega \) (in local charts, \(\omega = \sum _{j=1}^d dx_j \wedge d\xi _j\)). Let \(\omega ^d\) be the canonical symplectic volume form on \(T^*M\) and by a slight abuse of notation \(d \omega ^d\) the associated normalized measure on \(T^*M\) (see for instance [29, p. 274]). In local coordinates, we have
The canonical projection \(\pi : T^*M \rightarrow M\) is measurable from \((T^*M , d\omega ^d)\) to \((M, d{{\mathrm{Vol}}})\). For \(f \in L^1 (T^*M , d\omega ^d)\), we define \(\pi _* f \in L^1 (M , d{{\mathrm{Vol}}})\) by
In local charts, we have
Note also that we have the following desintegration formula
where the measure \(dm_x\) on \(T^*_xM\) is given in local charts by
Let \(V \in W^{2, \infty }(M)\), normalized so that \(\int _{M} e^{-V(x)} \, d{{\mathrm{Vol}}}(x)=1\), and define on \(T^*M\) the hamiltonian
We define the associated Hamilton vector field \(X_H\), given in local coordinates by
Using the 2-form \(\omega \), we can also define the Poisson bracket \(\{\cdot , \cdot \}\), see again [29, p. 271]. We have \(X_H f = \{H, f\}\).
We denote by \(\Lambda = \{(x, \xi , \xi '), x \in M, (\xi , \xi ') \in T^*_x M \times T^*_x M\}\) the vector bundle over M whose fiber above x is \(T^*_x M \times T^*_x M\).
With these notations, the Boltzmann equation on \(T^*M\) can be written as follows, for \((t,x, \xi ) \in {{\mathbb {R}}}\times T^*M\),
We recover the key properties of the usual linear Boltzmann collision operator on flat spaces. We have, for all \(x \in M\),
Besides,
since \((X_H f) (x, \xi ) d\omega ^d\) is an exact form (since \(X_H\) is hamiltonian). As a consequence, the mass is conserved: any solution f of (9.1) satisfies
Consider now the (generalized) Maxwellian distribution:
Note that for all \(x \in M\), \(\int _{T^*_x M} \frac{1}{(2\pi )^{d/2}}e^{-\frac{|\xi |_x^2}{2}} \, dm_x (\xi ) = 1\). As usual, we make the following assumptions on the collision kernel k.
A1. The collision kernel \(k \in C^0(\Lambda )\), is nonnegative.
A2. We assume that the Maxwellian cancels the collision operator, that is:
A3. Assume that
We can define the characteristics in this Riemannian setting as follows.
Definition 9.1
The hamiltonian flow \((\phi _t)_{t\in {{\mathbb {R}}}}\) associated to H is the one parameter group of diffeomorphisms on \(T^*M\) defined, for \((x, \xi ) \in T^*M\), by \(s \mapsto \phi _s(x , \xi ) \in T^*M \), where
The characteristic curve stemming from \((x, \xi ) \in T^*M\) is the curve \(\{\phi _t(x,\xi ), t \in {{\mathbb {R}}}^+\}\).
Note also for any function h defined on \({{\mathbb {R}}}\), \(h \circ H\) is preserved along these integral curves, as
In particular, this holds for the function \(\frac{e^V}{{\mathcal {M}}} = \frac{1}{(2\pi )^{d/2}} e^{H}\).
With this definition of characteristics, we can then properly define
-
the set \(\omega \) where collisions are effective, as in Definition 2.1,
-
\(C^-(\infty )\), as in Definition 2.4,
-
a.e.i.t. GCC, as in Definition 2.5,
-
the Unique Continuation Property, as in Definition 2.11,
-
the generalized Unique Continuation Property, as in Definition 5.1,
-
the equivalence relations \(\sim \) and \(\Bumpeq \), as in Definitions 3.1 and 2.9,
-
the sets \(U_j\) as in (5.25).
Note that in this Riemannian setting, the classes of collision operators E1, E2, E3 still make sense, up to some obvious adaptations.
Let us now introduce the relevant weighted Lebesgue spaces.
Definition 9.2
(Weighted \(L^p\) spaces). We define the Banach spaces \({\mathcal {L}}^2\) and \({\mathcal {L}}^\infty \) by
The space \({\mathcal {L}}^2\) is a Hilbert space endowed with the inner product
As in the case of the torus, we have the following well-posedness result for the Boltzmann equation (9.1).
Proposition 9.1
(Well-posedness of the linear Boltzmann equation). Assume that \(f_0 \in {\mathcal {L}}^2\). Then there exists a unique \(f\in C^0({{\mathbb {R}}};{\mathcal {L}}^2)\) solution of (9.1) satisfying \(f|_{t = 0} =f_0\), and we have
where
If moreover \(f_0 \ge 0\) a.e., then for all \(t \in {{\mathbb {R}}}\) we have \(f(t, \cdot ,\cdot )\ge 0\) a.e. (Maximum principle).
More generally, all results of Section 4 (up to obvious adaptations) are still relevant.
The crucial point we have to check now concerns velocity averaging lemmas for kinetic transport equations on a Riemannian manifold.
Lemma 9.1
Let H be defined as above, and \(X_H\) the associated vector field. Let \(T>0\) and \(\Psi \in C^\infty _c(T^*M)\). There exists \(C>0\) a constant such that the following holds. For any \(f,h \in L^2((0,T)\times T^*M)\) satisfying
we have
i.e.
Remark 9.1
Assuming that V is smooth enough, we may obtain the optimal Sobolev regularity \(H^{1/2}\) (instead of \(H^{1/4}\)), see Remark 12.1.
Proof of Lemma 9.1
In local charts, we have
and f satisfies the kinetic equation
We use the change of variables \(\overline{f}(t,x,v^i) = {f}(t,x,g_{i,j}v^j)\) (we define as well \(\overline{h}\) and \(\overline{\Psi }\)), which satisfies the equation
where \(\Gamma ^i_{j,k}(x) = \frac{1}{2} g^{i ,\ell } (x)\left( {\partial }_{x_j}g_{k,\ell }(x) +{\partial }_{x_k}g_{j,\ell }(x) - {\partial }_{x_\ell }g_{j ,k}(x)\right) \) are the Christoffel symbols. Using a classical averaging lemma (see (12.1) in Theorem 12.1 in Appendix 2 with \(m=1\) and \(s=0\)), we deduce that
Going back to the original variables, we deduce that
which proves our claim. \(\square \)
Equipped with this tool (more generally the analogues of all averaging lemmas of Appendix 2 can be obtained as well), we have the following analogue of the general convergence result of Theorem 5.1 (which includes Theorems 2.2 and 2.1). The same proof applies with only minor adaptations. Recall that the sets \((U_j)_{j \in J}\) are defined in (5.25).
Theorem 9.1
The following statements are equivalent.
-
(i.)
The set \(\omega \) satisfies the generalized Unique Continuation Property.
-
(ii.)
The set \(\omega \) satisfies the a.e.i.t. GCC.
-
(iii.)
For all \(f_0 \in {\mathcal {L}}^2\), denote by f(t) the unique solution to (9.1) with initial datum \(f_0\). We have
$$\begin{aligned} \left\| f(t)-Pf_0 \right\| _{{\mathcal {L}}^2} \rightarrow _{t \rightarrow +\infty } 0, \end{aligned}$$(9.7)where
$$\begin{aligned} P f_0 (x,v) = \sum _{j\in J} \frac{1}{\Vert \mathbbm {1}_{U_j} e^{-V} {\mathcal {M}}\Vert _{{\mathcal {L}}^2}} \left( \int _{U_j} f_0 \, d\omega ^d \right) f_j, \end{aligned}$$(9.8)with \((U_j)_{j \in J}\) defined in (5.25) and \(f_j = \frac{ \mathbbm {1}_{U_j} e^{-V} {\mathcal {M}}}{\Vert \mathbbm {1}_{U_j} e^{-V} {\mathcal {M}}\Vert _{{\mathcal {L}}^2}}\).
-
(iv.)
For all \(f_0 \in {\mathcal {L}}^2\), denote by f(t) the unique solution to (9.1) with initial datum \(f_0\). We have
$$\begin{aligned} \left\| f(t)-Pf_0 \right\| _{{\mathcal {L}}^2} \rightarrow _{t \rightarrow +\infty } 0, \end{aligned}$$(9.9)where \(Pf_0\) is a stationary solution of (9.1).
We obtain as well the analogue of Theorem 2.3. As in the torus case, we make the additional technical assumption:
A3’. Assume that there exists a continuous function \(\varphi (x,\xi ):= \Theta \circ H(x,\xi )\), with \(\varphi \ge 1\), such that for all \((x,\xi ) \in T^*M\), we have
and
Theorem 9.2
(Exponential convergence to equilibrium). Assume that the collision kernel satisfies A3’. The following statements are equivalent:
-
(a.)
\(C^-(\infty ) > 0\).
-
(b.)
There exists \(C>0, \gamma >0\) such that for any \(f_0 \in {\mathcal {L}}^2\), the unique solution to (9.1) with initial datum \(f_0\) satisfies for all \(t\ge 0\)
$$\begin{aligned} \left\| f(t)-\left( \int _{T^*M} f_0 \, d\omega ^d \right) {e^{-V}} {\mathcal {M}}\right\| _{{\mathcal {L}}^2}\le C e^{-\gamma t}\left\| f_0-\left( \int _{T^*M} f_0 \, d\omega ^d \right) {e^{-V}} {\mathcal {M}}\right\| _{{\mathcal {L}}^2}.\nonumber \\ \end{aligned}$$(9.10) -
(c.)
There exists \(C>0, \gamma >0\) such that for any \(f_0 \in {\mathcal {L}}^2\), there exists \(Pf_0\) a stationary solution of (9.1) such that the unique solution to (9.1) with initial datum \(f_0\) satisfies for all \(t \ge 0\),
$$\begin{aligned} \left\| f(t)-Pf_0\right\| _{{\mathcal {L}}^2} \le C e^{-\gamma t} \left\| f_0-Pf_0\right\| _{{\mathcal {L}}^2}. \end{aligned}$$(9.11)
As a particular case of Theorem 9.1, we have the following corollary.
Corollary 9.1
Assume that \(V=0\) and \(\omega =T^*\omega _x\), where \(\omega _x\) is a non-empty open subset of M. Suppose that the dynamics associated to \((\phi _t)_{t \ge 0}\) on
is ergodic. Then for all \(f_0 \in {\mathcal {L}}^2\), denoting by f(t) the unique solution to (9.1) with initial datum \(f_0\), we have
Note that if the dynamics of \((\phi _t)_{t \ge 0}\) is ergodic on \(S^* M\), then it is also ergodic on cosphere bundles of any positive radius (since for \(V=0\), the flow is homogeneous of degree one).
Classical examples of Riemannian manifolds satisfying this dynamical assumption are given by compact Riemannian manifolds with negative curvature.
The case of compact phase spaces
Instead of studying the linear Boltzmann equation on the “whole” phase space \(T^*M\), it is possible to consider this equation set on the “reduced” compact phase spaces
for \(R'>R>0\). Note that by continuity, the potential V is always bounded from below (and above), so that \(B_H^*M\), \( S_H^*M\) and \({\mathcal {R}}_H^*M\) are indeed compact.
We focus now on the case of \(S_H^*M\) (the other cases being handled similarly). We require that the collision kernel k is defined on
and make the following assumptions
A1. The collision kernel \(k \in C^0(\Gamma )\), is nonnegative.
A2. We assume that the Maxwellian cancels the collision operator, that is:
Note that in the compact framework, the former assumption A3 is useless.
Within these assumptions, the linear Boltzmann equation (1.1) is well-posed in appropriate weighted Lebesgue spaces based on \(L^2(S_H^*M)\), in particular because the hamiltonian is preserved by the dynamics. The case of \(S_H^*M\) is for instance relevant for the equations of radiative transfer or neutronics.
The analogues of Theorems 2.2, 2.3, 5.1 still hold in this framework, up to the appropriate modifications of the various geometric conditions. For the sake of conciseness, we do not write these results again. All proofs remain valid, with some simplifications, since the phase space is now compact. Note that the fact that \(C^-(\infty )>0\) is equivalent to GCC in this compact case.
References
Agoshkov, V.I.: Spaces of functions with differential-difference characteristics and the smoothness of solutions of the transport equation. Dokl. Akad. Nauk SSSR 276(6), 1289–1293 (1984)
Arsénio, D., Saint-Raymond, L.: Compactness in kinetic transport equations and hypoellipticity. J. Funct. Anal. 261(10), 3044–3098 (2011)
Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30, 1024–1065 (1992)
Bernard, É., Salvarani, F.: On the convergence to equilibrium for degenerate transport problems. Arch. Ration. Mech. Anal. 208(3), 977–984 (2013)
Bernard, É., Salvarani, F.: On the exponential decay to equilibrium of the degenerate linear Boltzmann equation. J. Funct. Anal. 265(9), 1934–1954 (2013)
Bernard, É., Salvarani, F.: Optimal estimate of the spectral gap for the degenerate Goldstein–Taylor model. J. Stat. Phys. 153(2), 363–375 (2013)
Berthelin, F., Junca, S.: Averaging lemmas with a force term in the transport equation. J. Math. Pures Appl. (9) 93(2), 113–131 (2010)
Bézard, M.: Régularité \(L^p\) précisée des moyennes dans les équations de transport. Bull. Soc. Math. France 122(1), 29–76 (1994)
Bodineau, T., Gallagher, I., Saint-Raymond, L.: The Brownian motion as the limit of a deterministic system of hard-spheres. arXiv preprint arXiv:1305.3397 (2013)
Cáceres, M.J., Carrillo, J.A., Goudon, T.: Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles. Commun. Part. Differ. Equ. 28(5–6), 969–989 (2003)
Cercignani, C.: The Boltzmann Equation and Its Applications. Applied Mathematical Sciences, vol. 67. Springer-Verlag, New York (1988)
Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 1. Springer-Verlag, Berlin (1990). Physical origins and classical methods, With the collaboration of Philippe Bénilan, Michel Cessenat, André Gervat, Alain Kavenoky and Hélène Lanchon, Translated from the French by Ian N. Sneddon, With a preface by Jean Teillac
Dautray, R., Lions, J.-L.: Mathematical Analysis and Numerical Methods for Science and Technology, vol. 6. Springer-Verlag, Berlin (1993). Evolution problems. II, With the collaboration of Claude Bardos, Michel Cessenat, Alain Kavenoky, Patrick Lascaux, Bertrand Mercier, Olivier Pironneau, Bruno Scheurer and Rémi Sentis, Translated from the French by Alan Craig
De Vuyst, F., Salvarani, F.: Numerical simulations of degenerate transport problems. Kinet. Relat. Models. 7(3), 463–476 (2014)
Degond, P., Goudon, T., Poupaud, F.: Diffusion limit for nonhomogeneous and non-micro-reversible processes. Indiana Univ. Math. J. 49(3), 1175–1198 (2000)
Desvillettes, L., Salvarani, F.: Asymptotic behavior of degenerate linear transport equations. Bull. Sci. Math. 133(8), 848–858 (2009)
Desvillettes, L., Villani, C.: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. Commun. Pure Appl. Math. 54(1), 1–42 (2001)
DiPerna, R.J., Lions, P.-L.: Global weak solutions of Vlasov–Maxwell systems. Commun. Pure Appl. Math. 42(6), 729–757 (1989)
DiPerna, R.J., Lions, P.-L., Meyer, Y.: \(L^p\) regularity of velocity averages. Ann. Inst. H. Poincaré Anal. Non Linéaire 8(3–4), 271–287 (1991)
Dolbeault, J., Mouhot, C., Schmeiser, C.: Hypocoercivity for linear kinetic equations conserving mass. arXiv preprint arXiv:1005.1495, to appear in Trans. AMS (2010)
Erdős, L., Yau, H.-T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger equation. Commun. Pure Appl. Math. 53(6), 667–735 (2000)
Gérard, P., Golse, F.: Averaging regularity results for PDEs under transversality assumptions. Commun. Pure Appl. Math. 45(1), 1–26 (1992)
Golse, F., Lions, P.-L., Perthame, B., Sentis, R.: Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76(1), 110–125 (1988)
Golse, F., Perthame, B., Sentis, R.: Un résultat de compacité pour les équations de transport et application au calcul de la limite de la valeur propre principale d’un opérateur de transport. C. R. Acad. Sci. Paris Sér. I Math. 301(7), 341–344 (1985)
Han-Kwan, D., Léautaud, M.: Trend to equilibrium and spectral localization properties for the linear Boltzmann equation. Séminaire Laurent S chwartz—Équations aux dérivées partielles et applications. Année 2013–2014. Sémin. Équ. Dériv. Partielles, VII, 15 École Polytech., Palaiseau (2014)
Han-Kwan, D., Léautaud, M.: Geometric analysis of the linear Boltzmann equation III. Spectral localization properties. In preparation
Han-Kwan, D., Léautaud, M.: Geometric analysis of the linear Boltzmann equation II. Trend to equilibrium in bounded domains with specular conditions. In preparation
Hérau, F.: Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. Asymptot. Anal. 46(3–4), 349–359 (2006)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274. Springer-Verlag, Berlin (1985). Pseudodifferential operators
Jabin, P.-E.: Averaging lemmas and dispersion estimates for kinetic equations. Riv. Mat. Univ. Parma 8(1), 71–138 (2009)
Koch, H., Tataru, D.: On the spectrum of hyperbolic semigroups. Commun. Part. Differ. Equ. 20(5–6), 901–937 (1995)
Lebeau, G.: Équation des ondes amorties. In: de Monvel, AB., Marchenko, V. (eds.) Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993). Mathematical Physics Studies, vol. 19, pp. 73–109. Kluwer Acad. Publ., Dordrecht (1996)
Mokhtar-Kharroubi, M.: Optimal spectral theory of the linear Boltzmann equation. J. Funct. Anal. 226(1), 21–47 (2005)
Mokhtar-Kharroubi, M.: On \(L^1\) exponential trend to equilibrium for conservative linear kinetic equations on the torus. J. Funct. Anal. 266(11), 6418–6455 (2014)
Mouhot, C., Neumann, L.: Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity 19(4), 969–998 (2006)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer-Verlag, New York (1983)
Rauch, J., Taylor, M.: Exponential decay of solutions to hyperbolic equations in bounded domains. Indiana Univ. Math. J. 24, 79–86 (1974)
Ukai, S.: On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Jpn. Acad. 50, 179–184 (1974)
Ukai, S., Point, N., Ghidouche, H.: Sur la solution globale du problème mixte de l’équation de Boltzmann nonlinéaire. J. Math. Pures Appl. 57(3), 203–229 (1978)
Vidav, I.: Existence and uniqueness of nonnegative eigenfunctions of the Boltzmann operator. J. Math. Anal. Appl. 22, 144–155 (1968)
Vidav, I.: Spectra of perturbed semigroups with applications to transport theory. J. Math. Anal. Appl. 30, 264–279 (1970)
Villani, C.: Hypocoercivity. Mem. Am. Math. Soc 202(95), iv+141 (2009)
Acknowledgments
We wish to thank Diogo Arsénio for several interesting and stimulating discussions related to this work.
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Appendices
Appendix 1: A Stabilization Criterion
We provide in this appendix a characterization of exponential decay for dissipative evolution equations. The following lemma is very classical and we reproduce it here for the convenience of the reader.
Lemma 11.1
Consider the evolution equation
assumed to be:
-
globally wellposed in some functional space X in the sense that for any \(f_0 \in X\), there is a unique \(f \in C^0_t(X)\) solution to (11.1),
-
invariant by translation in time, in the sense that if \(f \in C^0_t(X)\) is the solution of (11.1), then for all \(t_0\ge 0\), \(g(t) := f(t+t_0)\) is the unique solution of
$$\begin{aligned} \left\{ \begin{aligned} \partial _t g + L g = 0, \\ g_{|t=0} = f_{|t=t_0}. \end{aligned} \right. \end{aligned}$$(11.2)
Let \({\mathscr {E}}(f)\) and \({\mathscr {D}}(f)\) be two non-negative functionals defined for all \(f \in X\), and such that if f is a solution to (11.1),
Then, the following two properties are equivalent:
-
(1)
There exist \(C,\gamma >0\) such that for all \(f(0) \in X\), the associated solution f to (11.1) satisfies
$$\begin{aligned} \text { for all } t \ge 0, \quad {\mathscr {E}}(f(t))\le C e^{-\gamma t} {\mathscr {E}}(f(0)). \end{aligned}$$(11.4) -
(2)
There exists \(T>0\) and \(K>0\) such that for all \(f(0) \in X\), the associated solution f to (11.1) satisfies
$$\begin{aligned} K \int _0^T {\mathscr {D}}(f(t)) \, dt \ge {\mathscr {E}}(f(0)). \end{aligned}$$(11.5)
For the sake of completeness, we provide a short proof of this lemma.
Proof of Lemma 11.1
\((1) \Rightarrow (2)\) Assume that (1) holds. Let \(T_0>0\) such that \(C e^{-\gamma T_0} = \frac{1}{2}\). Then, using (11.3) betwen 0 and \(T_0\), we have:
so that, by (11.4),
and we can therefore take \(T =T_0\) and \(C=2\) in (11.5).
\((2) \Rightarrow (1)\) Assume that (2) holds. Here (and only here), we need the property of invariance by time translations for (11.1). By (11.3) and (11.5), we have
Note that the assumption \({\mathscr {E}}(f) \ge 0\) implies in particular that \(K\ge 1\). We may assume that \(K>1\). Indeed, for \(K=1\), we have \({\mathscr {E}}(f(t)) = 0\) for all \(t \ge T\) so that for any \(\gamma >0\), there exists \(C>0\) such that (1) holds. By invariance by translation in time of (11.1), one likewise obtains
Thus, by a straightforward induction, for any \(k \in {{\mathbb {N}}}\), we have the bound:
Defining \(\gamma _0 := \frac{- \log \left( 1-\frac{1}{K} \right) }{T}>0\) and \(C_0 := \left( 1-\frac{1}{K} \right) ^{-1} = e^{\gamma _0 T}>0\), we can now check that
Indeed, let \(t\ge 0\) and \(k \in {{\mathbb {N}}}\) such that \(t \in [kT, (k+1) T[\); since \({\mathscr {E}}(f(\cdot ))\) is decreasing (see (11.3)), we have
which concludes the proof. \(\square \)
Appendix 2: Velocity Averaging Lemmas
Velocity averaging lemmas play an important role in many proofs of this paper. In this appendix, we recall some classical results and also state the versions precisely adapted to our needs.
Kinetic transport equations are hyperbolic partial differential equations and as it can be seen from Duhamel’s formula, there is propagation of potential singularities at initial time and/or from a source in the equations. Thus there is no hope that the solution of a kinetic equation becomes more regular than the initial condition.
It was nevertheless observed by Golse, Perthame and Sentis [24] that the averages in velocity of the solution of a kinetic transport equation enjoy extra regularity/compactness properties (see also the independent paper of Agoshkov [1]). We refer to the by now classical paper of Golse, Lions, Perthame, Sentis [23], DiPerna, Lions [18], DiPerna, Lions, Meyer [19], Bézard [8] for quantitative estimates of this compactness property in various settings of increasing complexity.
We also refer to the review paper of Jabin [30] and to the recent work of Arsénio and Saint-Raymond [2].
We start by recalling classical averaging lemmas in the whole space \({{\mathbb {R}}}^d\). There are also versions of these lemmas for \(p\in (1,\infty )\), but we stick to the case \(p=2\), which is sufficient for our needs.
Theorem 12.1
(Kinetic averaging lemma [8, 18, 19, 23]). Let \(s\in [0,1)\) and \(m \in {{\mathbb {R}}}^+\). For any \(T>0\) and any bounded open sets \(\Omega _x ,\Omega _v \subset {{\mathbb {R}}}^d\), there exists a constant \(C>0\) such that for all \(\Psi \in C^{\infty }_c({\mathbb {R}}^d)\) supported in \(\Omega _v\) and all \(f,g \in L^2_{loc}({{\mathbb {R}}}\times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d)\) satisfying
we have
where \(\rho _{\Psi }(t,x):=\int _{{{\mathbb {R}}}^d} f(t,x,v)\Psi (v)dv\) and \(\alpha = {\frac{(1-s)}{2(1+m)}}\).
As a consequence, we obtain
Corollary 12.1
Let \(T>0\) and \((f_n)_{n\in {{\mathbb {N}}}}\) and \((g_n)_{n\in {{\mathbb {N}}}}\) be two sequences of \(L^2(0,T ; L^2_{loc}( {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d))\) such that the following holds
with \(s \in [0,1), m \ge 0\). Assume that for any bounded open sets \(\Omega _x ,\Omega _v \subset {{\mathbb {R}}}^d\), there exists \(C_1>0\), such that for all \(n \in {{\mathbb {N}}}\),
Then, for any \(\Psi \in C^{\infty }_c({\mathbb {R}}^d)\), the sequence \((\rho _{\Psi ,n})_{n\in {{\mathbb {N}}}}\) defined for \(n \in {{\mathbb {N}}}\) by
is relatively compact in \(L^2(0,T ; L^2_{loc}( {{\mathbb {R}}}^d))\).
Remark 12.1
In the main part of the paper, we apply this averaging lemma to the Boltzmann equation (1.1) by writing it under the form
To this end, we consider the case \(s=0, m=1\) in Theorem 12.1. This implies that the averages in v belong to the Sobolev space \(H^{1/4}\).
Nevertheless, assuming that the potential V is smooth enough, we can also use the approach of Gérard-Golse [22] or Berthelin-Junca [7] to obtain the optimal Sobolev space \(H^{1/2}\) for these averages (which is not needed in this paper).
Remark 12.2
We also have a version of these lemmas for kinetic transport equations set in general Riemannian manifolds, see Lemma 9.1.
We now state the result as needed in the main part of this work. Assuming an extra uniform integrability, we can deduce some compactness on moments of f without having to consider compactly supported test functions in v. This is the purpose of the next result, which is actually the version of averaging lemmas used most of the time in this work.
Corollary 12.2
Let \(\Omega _x\) be a bounded open set of \({{\mathbb {R}}}^d\), \(T>0\), and \((f_n)_{n\in {{\mathbb {N}}}}\), \((g_n)_{n\in {{\mathbb {N}}}}\) be two sequences of \(L^2(0,T ; L^2_{loc}( \Omega _x \times {{\mathbb {R}}}^d))\) satisfying \( \partial _t f_n + v \cdot \nabla _x f_n = (1- \Delta _{v})^{m/2} g_n, \) for some \( m \ge 0\). Suppose that there exists \(V \in L^\infty \) such that for any bounded open set \(\Omega _v \subset {{\mathbb {R}}}^d\), there exists \(C_0>0\) such that, for any \(n \in {{\mathbb {N}}}\),
Assume moreover that there is \(f \in L^\infty (0,T ;L^2(\Omega _x \times {{\mathbb {R}}}^d))\) such that \(f_n \rightharpoonup f\) weakly\(-\star \) in \(L^\infty (0,T ; {\mathcal {L}}^2(\Omega _x \times {{\mathbb {R}}}^d))\). Consider \(\rho _{n}(t,x):=\int _{{{\mathbb {R}}}^d} f_n(t,x,v) \,dv\). Then up to a subsequence, we have
and for any continuous kernel \(k(\cdot ,\cdot ,\cdot ): {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d \rightarrow {{\mathbb {R}}}\) satisfying A3, we have
Moreover, the same result still holds if we replace \(\Omega _x\) by \({{\mathbb {T}}}^d\).
Proof of Corollary 12.2
Note first that the result for \({{\mathbb {T}}}^d\) can be deduced by a finite covering of \({{\mathbb {T}}}^d\) by local charts of the form \(\Omega _x\). Hence, it suffices to prove the result with \(\Omega _x\). By Fatou’s lemma, the function f satisfies:
Since \(\rho _n\) does not depend on v, proving (12.4) is equivalent to show that:
Let \(\Psi \in C^{\infty }_c({{\mathbb {R}}})\) such that \(\Psi = 1\) in a neighborhood of 0, and define \(\Psi _R(v) = \Psi (\frac{|v|}{R})\), \(v\in {{\mathbb {R}}}^d\).
By Corollary 12.1, we can assume, up to a subsequence, that
Let \({\varepsilon }>0\). We can write the decomposition:
By Cauchy-Schwarz inequality, using (12.3), we have for all \(n \in {{\mathbb {N}}}\) and all \(t \in (0,T)\),
As a consequence, there exists \(R_0>0\) large enough such that for all \(R\ge R_0\) and all \(n \in {{\mathbb {N}}}\), we have
Likewise, we use (12.6) to get for all \(n \in {{\mathbb {N}}}\),
Using (12.8) (R is now fixed), there is \(N\ge 0\) such that for any \(n\ge N\),
from which we infer that
and this concludes the proof of (12.4).
For the proof of (12.5), let us first assume for a while that k is smooth (namely for all x, \(k(x, \cdot , \cdot )\) belongs to the \(C^\infty \) class). We first have to be careful about the integration in the velocity variable. The convergence in (12.5) results from the following two facts:
-
For all \(v \in {{\mathbb {R}}}^d\), we have the following convergence
$$\begin{aligned}&\int _{{{\mathbb {R}}}^d} k(x,v',v) f_n(t,x,v') \, dv' \\&\quad \rightarrow \int _{{{\mathbb {R}}}^d} k(x,v',v) f(t,x,v') \, dv' \quad \text {strongly in } L^2(0,T;L^2_{x}(\Omega _x)). \end{aligned}$$This follows from a truncation argument and Corollary 12.1, exactly as for \(\rho _n\). Keeping the same notations, the only difference is that we have to study
$$\begin{aligned} \left( \int _{{{\mathbb {R}}}^d} (1-\Psi _R)^{2}k^2(x,v',v) {{\mathcal {M}}(v')} \, dv' \right) , \end{aligned}$$which, using A3, is small for R large enough.
-
By the Cauchy-Schwarz inequality and the bound (12.3), we have
$$\begin{aligned}&\int _0^T \int _{\Omega _x} \left( \int _{{{\mathbb {R}}}^d} k(x,v',v) (f_n-f)(t,x,v') \, dv' \right) ^2 \frac{e^V}{{\mathcal {M}}(v)}\, dx \, dt \\&\le \int _0^T \left( \sup _{x \in \Omega _x} \int _{{{\mathbb {R}}}^d} k^2(x,v',v) \frac{{\mathcal {M}}(v')}{{\mathcal {M}}(v)} \, dv' \right) \\&\quad \int _{\Omega _x}\left( \int _{{{\mathbb {R}}}^d} \frac{|f_n-f|^2(t,x,v')}{{\mathcal {M}}(v')}\, dv' \right) e^V \, dx \, dt \\&\le C_0 T \sup _{x \in \Omega _x} \int _{{{\mathbb {R}}}^d} k^2(x,v',v) \frac{{\mathcal {M}}(v')}{{\mathcal {M}}(v)} \, dv' , \end{aligned}$$which is independent of n and in \(L^1(dv)\), since by A3, we have
$$\begin{aligned} \sup _{x \in \Omega _x} \int _{{{\mathbb {R}}}^d \times {{\mathbb {R}}}^d} k^2(x,v',v) \frac{{\mathcal {M}}(v')}{{\mathcal {M}}(v)} \, dv' \, dv < +\infty . \end{aligned}$$
Hence, by Lebesgue dominated convergence theorem, we deduce (12.5).
We now use an approximation argument to handle the general case, i.e. when k is only assumed to be continuous. Consider \((\phi _\delta )_{\delta >0}\) a family of mollifiers in \(C^\infty _c({{\mathbb {R}}}^d \times {{\mathbb {R}}}^{d})\) for the measure \( {\mathcal {M}}(v){\mathcal {M}}(v') \, dv' dv\). We set for all \(x,v,v'\)
We use the following classical properties of mollifiers:
-
for all x, \(k_\delta (x, \cdot ,\cdot )\) is in the \(C^\infty \) class;
-
we have for all \(\delta >0\)
$$\begin{aligned} \sup _{x \in \Omega _x} \Vert \tilde{k}- k_\delta \Vert _{L^2({\mathcal {M}}(v){\mathcal {M}}(v') \, dv' dv)} \rightarrow _{\delta \rightarrow 0} 0. \end{aligned}$$(12.10)
Let \({\varepsilon }>0\). We write the decomposition
with
We estimate \(A_2\) as follows, using (12.10)
Using (12.10), we fix \(\delta >0\) small enough so that for all \(n\in {{\mathbb {N}}}\),
and thus for all \(n\in {{\mathbb {N}}}\), we have
For \(A_1\), we use the above analysis in the smooth case to deduce that we can take N large enough to get for all \(n\ge N\),
Finally, we have proven that for any \({\varepsilon }>0\), there is N such that for all \(n\ge N\),
which concludes the proof of the convergence. \(\square \)
Appendix 3: Reformulation of Some Geometric Properties
In this appendix, we first provide a proof of Lemma 3.1, giving the link between the two equivalence relations \(\Bumpeq \) and \(\sim \). Then, we explore related geometric conditions with additional connectedness assumptions, that are sufficient for having a single equivalence class for \(\Bumpeq \) and \(\sim \).
Appendix 3.1: Proof of Lemma 3.1
We first define another convenient equivalence relation.
Definition 13.1
Given \(\omega _1\) and \(\omega _2\) two connected components of \(\omega \), we say that \(\omega _1 \bumpeq \omega _2\) if there are \(N \in {{\mathbb {N}}}\) and N connected components \((\omega ^{i})_{1\le i \le N}\) of \(\omega \) such that
-
we have \(\omega _1 \, {\mathcal {R}} _{\phi } \, \omega ^{(1)}\),
-
for all \(1\le i \le N-1\), we have \(\omega ^{(i)} \, {\mathcal {R}} _{\phi } \, \omega ^{(i+1)}\),
-
we have \(\omega ^{(N)} \, {\mathcal {R}} _{\phi } \, \omega _2\).
The relation \(\bumpeq \) is an equivalence relation on the set of connected components of \(\omega \). For \(\omega _1\) a connected component of \(\omega \), we denote its equivalence class for \(\bumpeq \) by \(\{\omega _1\}\).
The following result is useful for the proof of Lemma 3.1.
Lemma 13.2
Let \(\Omega _0\) be a connected component of \(\bigcup _{s \in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) and let \((\omega _\ell )_{\ell \in L}\) be the connected components of \(\omega \) such that for all \(\ell \in L\), there exists \(t\ge 0\) with \(\phi _{-t} (\omega _\ell ) \cap \Omega _0 \ne \emptyset \). Then, for all \(\ell ,\ell ' \in L\), we have \(\omega _\ell \bumpeq \omega _{\ell '}\).
Proof of Lemma 13.2
Assume that there exist at least two equivalence classes for \(\bumpeq \) among the \(\omega _\ell \), \(\ell \in L\). Let \(\ell _0 \in L\) and consider \(\{\omega _{\ell _0}\}\) the equivalence class of \(\omega _{\ell _0}\) for \(\bumpeq \). Defining
we have by construction that \(U_1, U_2 \) are two open non-empty subsets of \(\Omega _0\) and that \(U_1 \cup U_2 = \Omega _0\).
Let us check \(U_1 \cap U_2 = \emptyset \): otherwise it means that there exist two connected components \(\Omega _1\) and \(\Omega _2\) of \(\omega \) such that \(\Omega _1 \in \{\omega _{\ell _0}\}\), \(\Omega _2 \notin \{\omega _{\ell _0}\}\) and \(\Omega _1 \, {\mathcal {R}} _\phi \Omega _2\), which is excluded by definition of the equivalence class.
This is a contradiction with the fact that \(\Omega _0\) is connected. \(\square \)
We are now in position to prove Lemma 3.1.
Proof of Lemma 3.1
Let us first prove that \(\omega _1 \Bumpeq \omega _2 \Longrightarrow \Psi (\omega _1) \sim \Psi (\omega _2)\). It suffices to prove that
The conclusion then follows from an iterative use of this argument.
If \(\omega _1 \, {\mathcal {R}} _k \, \omega _2\), then \(\Psi (\omega _1) \, {\mathcal {R}} _k \Psi (\omega _2)\). This follows from the fact that \(\omega _j \subset \Psi (\omega _j)\) and the definition of \(\, {\mathcal {R}} _k\). Similarly, if \(\omega _1 \, {\mathcal {R}} _\phi \, \omega _2\), then \(\Psi (\omega _1) \, {\mathcal {R}} _\phi \, \Psi (\omega _2)\).
Let us now prove that \(\Psi (\omega _1) \sim \Psi (\omega _2) \Longrightarrow \omega _1 \Bumpeq \omega _2 \). According to Lemma 13.2, it is sufficient to prove that \(\Omega ^{(1)}, \Omega ^{(2)} \) being two given connected components of \(\bigcup _{t\ge 0} \phi _{-t} (\omega )\), we have
The conclusion then follows from an iterative use of this argument.
By definition of \(\, {\mathcal {R}} _k\), there exist \((x,v_1,v_2) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d \times {{\mathbb {R}}}^d\) with \((x,v_1) \in \Omega ^{(1)}\) and \((x,v_2) \in \Omega ^{(2)}\) such that \(k(x,v_1,v_2)>0\) or \(k(x,v_2,v_1)>0\).
Note in particular that this implies \((x,v_1) ,(x,v_2) \in \omega \). Denoting by \(\omega _1^*\) (resp. \(\omega _2^*\)) the connected component of \(\omega \) such that \((x,v_1) \in \omega _1^*\) (resp. \((x,v_2) \in \omega _2^*\)), we hence have \(\omega _1^* \, {\mathcal {R}} _k \, \omega _2^*\). The conclusion of (13.2) follows from the fact that \(\omega _j^* \subset \Omega ^{(j)}\), for \(j=1,2\) and the very definition of \(\, {\mathcal {R}} _k\). \(\square \)
Appendix 3.2: Almost Everywhere Geometric Control Conditions and Connectedness
In this section, we give sufficient conditions for \(\omega \) to satisfy the a.e.i.t. GCC with \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) connected. In this case, whatever the collision kernel k, there is a single equivlence class for \(\Bumpeq \) and \(\sim \), so that the dimension of the set of stationary solutions is one (and the equilibrium ultimately reached is the global Maxwellian).
Proposition 13.1
Let \(\omega \subset {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\) be an open subset. Consider the following geometric properties:
-
(i)
There exists \(\tilde{\omega } \subset \omega \), \(\tilde{\omega }\) connected satisfying the a.e.i.t. Geometric Control Condition;
-
(ii)
The set \(\omega \) satisfies the a.e.i.t. GCC and for any connected components \((\omega _1,\omega _2)\) of \(\omega \), there exists \((x_0,v_0) \in \omega _1\) and \(s \in {{\mathbb {R}}}\) such that \(\phi _s(x_0,v_0) \in \omega _2\);
-
(iii)
The set \(\omega \) satisfies the a.e.i.t. GCC and \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) is connected.
Then \((i) \implies (iii)\) and \((ii) \implies (iii)\).
Note in particular that \((i)-(ii)-(iii)\) hold as soon as \(\omega \) is connected and satisfies a.e.i.t. GCC.
Proof of Proposition 13.1
Before starting the proof, let us remark that is \(\omega \) is a connected open subset of \(\Omega \times {{\mathbb {R}}}^d\), then \( \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) is also a connected open subset. Indeed it is first an open subset of \(\Omega \times {{\mathbb {R}}}^d\), and it is equivalent to show that it is path-connected. Let \(y_1, y_2 \in \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\); there exists \(s_1,s_2 \ge 0\) and \(z_1, z_2 \in \omega \) such that \(y_1 = \phi _{-s_1} (z_1)\) and \(y_2 = \phi _{-s_2} (z_2)\). Since \(\omega \) is a connected open subset of \(\Omega \times {{\mathbb {R}}}^d\), it is also path-connected and one can find a continuous path in \(\omega \) between \(z_1\) and \(z_2\). Using the application \(\phi _{-s}\), we also get a continuous path between \(y_1\) and \(z_1\) in \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) (resp. between \(y_2\) and \(z_2\)).
Gluing these paths together, this yields a continuous path in \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) between \(y_1\) and \(y_2\).
\(\bullet \) \((i) \implies (iii)\). Since \(\tilde{\omega } \subset \omega \), we have \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\tilde{\omega }) \subset \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\). Denote by \((\Omega _i)_{i \in I}\) the connected components of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\). The sets \(\Omega _i\) are connected open sets so that the inclusion \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\tilde{\omega }) \subset \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) together with the connectedness of \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\tilde{\omega })\) yields the existence of \(i_0 \in I\) such that \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\tilde{\omega }) \subset \Omega _{i_0}\). Since \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\tilde{\omega })\) is of full measure, this is also the case for \(\Omega _{i_0}\). As \(\Omega _i\) is open, we obtain that \(\Omega _i = \emptyset \) for \(i \ne i_0\), so that \(\Omega _{i_0} = \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) is connected (and of full measure).
\(\bullet \) \((ii) \implies (iii)\). Let \(y_1, y_2 \in \bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\); there exists \(s_1,s_2 \ge 0\) and \(z_1, z_2 \in \omega \) such that \(y_1 = \phi _{-s_1} (z_1)\) and \(y_2 = \phi _{-s_2} (z_2)\). If \(z_1, z_2\) belong to the same connected component \(\tilde{\omega }\) of \(\omega \), then since \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\tilde{\omega })\) is connected, one can find a continuous path between \(z_1\) and \(z_2\).
If \(z_1, z_2\) belong to two different connected components \(\omega _1\) and \(\omega _2\), apply (i) to find, up to a permutation between the indices 1 and 2, \(u \in \omega _1\) and \(s \in {{\mathbb {R}}}^+\) such that \(\phi _{-s}(u) \in \omega _2\). Then one can find a continuous path between u and \(z_1\) in \(\omega _1\), and another between \(\phi _{-s}(u)\) and \(z_2\) in \(\omega _2\). We conclude as in the previous subcase by gluing the paths together. \(\square \)
Appendix 4: Proof of Proposition 3.2
In this section, we prove Proposition 3.2.
Proof of Proposition 3.2
Let \(x_0 \in {{\mathbb {T}}}^d \setminus \overline{p_x(\omega )} \ne \emptyset \) and take \(\eta >0\) such that \(B(x_0 , 2\eta ) \cap \overline{p_x(\omega )} = \emptyset \). Define the potential \(V (x) := \frac{|x-x_0|^2}{2} \Psi (x)\), where \(\Psi \) is a “corrector” to ensure \(V\in C^\infty ({{\mathbb {T}}}^d)\), and such that \(\Psi \equiv 1\) on \(B(x_0 , 2\eta )\) (reduce \(\eta \) if necessary). Denote \(V_{\varepsilon }= {\varepsilon }V\) and notice that \(\nabla V_{\varepsilon }(x) = {\varepsilon }(x-x_0)\) on \(B(x_0 , 2\eta )\). As a consequence, the hamiltonian flow \((\phi _t)_{t \in {{\mathbb {R}}}}\) associated to the vector field \(v \cdot \nabla _x - \nabla _x V_{\varepsilon }\cdot \nabla _v\) may be explicited in the set \(B(x_0,2\eta ) \times {{\mathbb {R}}}^d\): we have
as long as \(\phi _t(x,v) \in B(x_0,2\eta ) \times {{\mathbb {R}}}^d\). In particular, note that if \((x,v)\in B(x_0,\eta ) \times B(0, \sqrt{{\varepsilon }}\eta )\), then \(\phi _t(x,v)\) remains in \(B(x_0,2\eta ) \times B(0, 2 \sqrt{{\varepsilon }}\eta )\) for all \(t \in {{\mathbb {R}}}^+\). This reads \(\phi _t(B(x_0,\eta ) \times B(0,\sqrt{{\varepsilon }}\eta )) \subset B(x_0,2\eta ) \times B(0,2\sqrt{{\varepsilon }}\eta )\), i.e. in particular \(\phi _t(B(x_0,\eta ) \times B(0,\sqrt{{\varepsilon }}\eta )) \cap \omega = \emptyset \) for all \(t \in {{\mathbb {R}}}^+\). This proves that a.e.i.t. GCC is not satisfied. \(\square \)
Remark 14.1
Notice that in the previous proof, to handle small potential, we consider small speeds, i.e. \(v \in B(0,\sqrt{{\varepsilon }}\eta )\). In the opposite direction, if one fixes the speeds in a large Hamiltonian sphere \(v \in S_H(0,R)\) (note that with the particular potential used in the proof, on the set \(\{\Psi =1\}\) we have \(S_H(0,R) = S(0,R)\)) for some \(R>0\), then one can find a (large) potential (namely \(R^2/\eta ^2 V\) where V is that of the previous proof) such that a.e.i.t. GCC fails.
Appendix 5: Other Linear Boltzmann Type Equations
The goal of this appendix is to show that the methods developed here can be adapted to handle other types of Boltzmann-like equations.
Appendix 5.1: Generalization to a Wider Class of Kinetic Transport Equations
Here, we consider the equation
where \(a(v) = \nabla _v A(v)\) with \(A \in W^{2,\infty }_{loc} ({{\mathbb {R}}}^d)\) is such that \(A(v) \rightarrow +\infty \) as \(|v|\rightarrow + \infty \) and \(\int _{{{\mathbb {R}}}^d} e^{-A(v)} \, dv<+\infty \).
For simplicity, we assume that (15.1) is set on \({{\mathbb {T}}}^d\times {{\mathbb {R}}}^d\).
Assume that a(v) satisfies a non degeneracy property: there exists \(\gamma \in (0,2)\) and \(C>0\) such that, for all \(\xi \in {{\mathbb {S}}}^{d-1}\),
This assumption prevents from concentration of a(v) in some directions of \({{\mathbb {S}}}^{d-1}\).
The hamiltonian associated to the transport equation is then the following:
Define the global Maxwellian associated to a(v) by
with \(C_A = 1/ \left( \int _{{{\mathbb {R}}}^d} e^{-A(v)} \, dv \right) \).
In addition to the usual assumption A1 on the collision kernel k, we shall assume the following (which replace A2 –A3):
A2’. We assume that \({\mathcal {M}}_A\) cancels the collision operator, that is
A3’. We assume that
For \(a(v)=v\) (for which \(\gamma =1\) in (15.2)), we recover the framework which has been already treated before. One physically relevant case is
(for which we also have \(\gamma =1\) in (15.2)), which allows to model relativistic transport. Note that in this case, we have \(A_{rel}(v) :=\sqrt{1+|v|^2}\), and the related Maxwellian is then the so-called relativistic Maxwellian:
where \(C_{rel}\) is a normalizing constant, so that \(\int _{{{\mathbb {R}}}^d} {\mathcal {M}}_{rel}(v) \, dv =1\).
Our aim in this paragraph is to show that our methods are still relevant. The characteristics of the equation are defined in the following way:
Definition 15.1
Let \(V \in W^{2,\infty }_{loc} ({{\mathbb {T}}}^d)\). Let \((x_0,v_0) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\). The characteristics \(\phi _t(x_0,v_0) := (X_t (x_0,v_0), \, \Xi _t(x_0,v_0))\) associated to the hamiltonian \(H(x,v) = A(v) + V(x)\) are defined as the solutions to the system:
With this definition of characteristics, we can then properly define
-
the set \(\omega \) where collisions are effective, as in Definition 2.1,
-
the Unique Continuation Property, as in Definition 2.11
-
\(C^-(\infty )\), as in Definition 2.4,
-
a.e.i.t. GCC, as in Definition 2.5,
-
the equivalence relation \(\sim \), as in Definition 3.1.
The next thing to do concerns the local well-posedness of (15.1) in some relevant weighted spaces, which we introduce below.
Definition 15.2
(Weighted \(L^p\) spaces). We define the Banach spaces \({\mathcal {L}}^2_A\) and \({\mathcal {L}}^\infty _A\) by
The space \({\mathcal {L}}^2_A\) is a Hilbert space endowed with the inner product
As usual, we have
Proposition 15.3
(Well-posedness of the linear Boltzmann equation with modified transport). Assume that \(f_0 \in {\mathcal {L}}^2_A\). Then there exists a unique \(f\in C^0({{\mathbb {R}}};{\mathcal {L}}^2_A)\) solution of (15.1) satisfying \(f|_{t = 0} =f_0\), and we have
where
If moreover \(f_0 \ge 0\) a.e., then for all \(t \in {{\mathbb {R}}}\) we have \(f(t, \cdot ,\cdot )\ge 0\) a.e. (Maximum principle).
Then the analogues of Theorems 2.1, 2.2 and 2.3 hold in this setting (with some obvious modifications); for the sake of conciseness, we omit these statements.
Such results can be proved exactly as Theorems 2.1, 2.2 and 2.3. The crucial additional ingredient is the fact that averaging lemmas for the operator \(a(v) \cdot {\nabla }_x\) still hold, precisely when a satisfies the non degeneracy condition (15.2), see [23]. Note that in this case, the gain of regularity on averages depends on the index \(\gamma \) in (15.2), but in any case, this is always sufficient to obtain compactness.
Appendix 5.2: Generalization to linearized BGK operators
Once again, for simplicity, we assume that \((x,v) \in {{\mathbb {T}}}^d \times {{\mathbb {R}}}^d\). Let \(V \in W^{2,\infty }_{loc}({{\mathbb {T}}}^d)\). Let \(\varphi : {{\mathbb {R}}}\rightarrow {{\mathbb {R}}}^+_*\) be a function in \(L^\infty ({{\mathbb {R}}})\) such that
Denote \(F(x,v) = \varphi \left( \frac{|v|^2}{2} + V(x) \right) \) and \(\rho _F(x) = \int F(x,v) \, dv\).
Let \(\sigma \in C^0({{\mathbb {T}}}^d)\) be a non-negative function. We study in this paragraph the following degenerate linearized BGK equation:
with an initial condition \(f_0\) at time 0. The natural equilibrium is given by
The main feature of this equilibrium is that there is no separation of variables contrary to the Maxwellian case.
Our aim in this paragraph is again to show that the methods developed in this paper are still relevant here. For what concerns well-posedness, we introduce the relevant weighted \(L^p\) spaces and have the usual result.
Definition 15.4
(Weighted \(L^p\) spaces). We define the Banach spaces \({\mathcal {L}}^2_{bgk}\) and \({\mathcal {L}}^\infty _{bgk}\) by
The space \({\mathcal {L}}^2_{bgk}\) is a Hilbert space endowed with the inner product
Proposition 15.5
(Well-posedness of the linearized BGK equation). Assume that \(f_0 \in {\mathcal {L}}^2_{bgk}\). Then there exists a unique \(f\in C^0({{\mathbb {R}}};{\mathcal {L}}^2_{bgk})\) solution of (15.11) satisfying \(f|_{t = 0} =f_0\), and we have
where
If moreover \(f_0 \ge 0\) a.e., then for all \(t \in {{\mathbb {R}}}\) we have \(f(t, \cdot ,\cdot )\ge 0\) a.e. (Maximum principle).
With the same geometric definitions of Section 2, we have the following results. Note that the set \(\omega \) where the collisions are effective is equal to \(\omega _x \times {{\mathbb {R}}}^d\), where
Theorem 15.6
(Convergence to equilibrium). The following statements are equivalent.
-
(i.)
The set \(\omega \) satisfies the Unique Continuation Property.
-
(ii.)
The set \(\omega \) satisfies the a.e.i.t. GCC and \(\bigcup _{s\in {{\mathbb {R}}}^+}\phi _{-s}(\omega )\) is connected.
-
(iii.)
For all \(f_0 \in {\mathcal {L}}^2_{bgk}\), denote by f(t) the unique solution to (15.11) with initial datum \(f_0\). We have
$$\begin{aligned} \left\| f(t)-\left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f_0 \, dv \,dx\right) \frac{F(x,v)}{\int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} F(x,v) \, dv \,dx}\right\| _{{\mathcal {L}}^2_{bgk}} \rightarrow _{t \rightarrow +\infty } 0,\nonumber \\ \end{aligned}$$(15.13)
Theorem 15.7
(Exponential convergence to equilibrium). The two following statements are equivalent:
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(i.)
\(C^-(\infty ) > 0\).
-
(ii.)
There exists \(C>0, \gamma >0\) such that for any \(f_0 \in {\mathcal {L}}^2({{\mathbb {T}}}^d \times {{\mathbb {R}}}^d)\), the unique solution to (15.11) with initial datum \(f_0\) satisfies
$$\begin{aligned}&\left\| f(t)-\left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f_0 \, dv \,dx \right) \frac{F(x,v)}{\int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} F(x,v) \, dv \,dx} \right\| _{{\mathcal {L}}^2} \nonumber \\&\quad \! \le \! C e^{-\gamma t} \left\| f_0-\left( \int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f_0 \, dv \,dx \right) \frac{F(x,v)}{\int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} F(x,v) \, dv \,dx}\right\| _{{\mathcal {L}}^2}.\qquad \qquad \end{aligned}$$(15.14)
We shall not dwell on the proofs of Theorems 15.6 and 15.7, since they are very similar to those of Theorems 2.2 and 2.3. Indeed, we note that the structure of the equation (15.11) is similar to that of (1.1), in the sense that the “degenerate dissipative” part is still made of a dissipative term plus a relatively compact term. This compactness, as usual, comes from averaging lemmas.
Let us just underline a crucial point in the proof (ii.) implies (i.) of Theorem 15.6. Here, we have to be careful of the fact that F(x, v) does not separate the x and v variables, contrary to the Maxwellian equilibrium of (1.1). Let us check that the proof we gave in the Boltzmann case is still relevant (see the proof of \((ii.) \implies (i.)\) of Theorem 2.2).
Let \(f \in C^0_t({\mathcal {L}}^2_{bgk})\) be a solution to
Assume that \(\int _{{{\mathbb {T}}}^d \times {{\mathbb {R}}}^d} f \, dv \,dx=0\). The goal is to show that \(f=0\).
To this purpose, as before, consider for \((t,x, v) \in {{\mathbb {R}}}^+ \times \omega \), \(g(t,x) := \frac{1}{F(x,v)} \, f\) (note that by (15.16), g does not depend on v). We have, for \((t,x, v) \in {{\mathbb {R}}}^+ \times \omega \):
Since f satisfies (15.15) and (15.16),
By definition of F, we have
from which we deduce that g satisfies the free transport equation on \(\omega \):
We then conclude as in the proof of \((ii.) \implies (i.)\) of Theorem 2.2, mutatis mutandis.
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Han-Kwan, D., Léautaud, M. Geometric Analysis of the Linear Boltzmann Equation I. Trend to Equilibrium. Ann. PDE 1, 3 (2015). https://doi.org/10.1007/s40818-015-0003-z
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DOI: https://doi.org/10.1007/s40818-015-0003-z
Keywords
- Kinetic theory
- Control theory
- Linear Boltzmann equation
- Large time behaviour
- Hypocoercivity
- Geometric control conditions