Global solutions of nonlinear wave-Klein-Gordon system in two spatial dimensions: A prototype of strong coupling case

Abstract

In this article we will develop some techniques aimed at the strong couplings in two-dimensional wave-Klein-Gordon system. We distinguish the roles of different type of decay factors and develop a method which permits us to “exchange” one type of decay into the other. Then a global existence result of a model problem is established. We also give a sketch of the Klein-Gordon-Zakharov model system and establish the associate global existence result.

Introduction

This article belongs to a research project in which we attempt to understand the effects of different quadratic terms coupled in diagonalized wave-Klein-Gordon system in $2+1$ dimensional space-time. In the previous works for example [1] and [2], we mainly concentrate on the so-called weak coupling cases, i.e., in the wave equation there is no pure Klein-Gordon terms. In the present work we start an investigation on the strong coupling case. We will develop several technical tools and establish the global existence result for the following model system:

$$\{\begin{array}{cc}\hfill & \square u=A_1^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }u+A_3^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }v+A_4^{\alpha }v{\partial }_{\alpha }u\hfill \\ \hfill & +B_2^{\alpha \beta }{\partial }_{\alpha }v{\partial }_{\beta }v+B_3^{\alpha }v{\partial }_{\alpha }v+K_2{v}^2,\hfill \\ \hfill & \square v+c^{2}v=A_5^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }u.\hfill \end{array}$$

In (1.1), we remark the general strong coupling terms $B_2^{\alpha \beta }{\partial }_{\alpha }v{\partial }_{\beta }v+B_3^{\alpha }v{\partial }_{\alpha }v+K_2{v}^2$. The quadratic form $A_1,A_3$ and $A_5$ are supposed to be null. The rest are constant-coefficient (multi-)linear forms. In fact in the wave equation we have included all possible quadratic semi-linear terms on $(\partial u,\partial v,v)$.

In the mean time, we will show an other application of these techniques which is the following model system formulated form the Klein-Gordon-Zakharov system in ${\mathbb{R}}^{2+1}$ introduced and studied in [3]:

$$\{\begin{array}{cc}\hfill & \square E+E=E\mathrm{\Delta }u,\hfill \\ \hfill & \square u=|E{|}^2,\hfill \end{array}$$

where $u:{\mathbb{R}}^{2+1}\to \mathbb{R}$ a scalar and $E:{\mathbb{R}}^{2+1}\to {\mathbb{R}}^2$ a vector. We will give an alternative approach to the global existence of this system.

For clarity the initial data are supposed to be compactly supported on the initial hyperboloid ${\mathcal{H}}_2$ or equivalently, on $\{t=2\}$. This is not an essential restriction because with the so-called Euclidean-hyperboloidal foliation (see for example [4] for an one-dimensional case), the argument here can be easily generalized to non-compactly-supported initial data with sufficient spatial decreasing rate.

As explained in many existing works, the problem of global existence of wave-Klein-Gordon system is more delicate in $2+1$ dimension than in $3+1$ dimensional case because of the slow decay rate of both wave and Klein-Gordon equations in lower dimension. We recall [5], [6] for the methods based on Fourier analysis (which are in non-diagonalized quasi-linear case) and [2], [3] for the analysis in physical space-time. It also worth to mention the following results [7], [8], [9], [10], [11], [12] on wave equations in ${\mathbb{R}}^{2+1}$ and [13], [14] on Klein-Gordon equations in ${\mathbb{R}}^{2+1}$ dimension.

The main challenge of (1.1) comes from the insufficiency of the so-called principle decay, which is due to the strong coupling terms and the interaction terms ${\partial }_{\alpha }u{\partial }_{\beta }v,v{\partial }_{\alpha }u$ coupled in wave equation. The objective of the present work is mainly concentrated on this difficulty. We will give a detailed explanation in the coming two subsections.

The systems with strong coupling arise naturally in many physical or geometrical context. For example the Einstein-Klein-Gordon system

$${\tilde{\square }}_g{g}_{\alpha \beta }=F_{\alpha \beta }(g;\partial g,\partial g)+T_1(\partial \varphi ,\partial \varphi )+T_2(\varphi ,\varphi ),{\tilde{\square }}_g\varphi +c^2\varphi =0$$

where $T_1,T_2$ are quadratic forms. The wave map system formulated in [15]

$$\square u=-2\sum _{k=2}^n{\varphi }^k{\partial }_1{\varphi }^k+\text{cubic terms}\square {\varphi }^k+{\varphi }^k=2{\varphi }^k{\partial }_{1}u+\text{cubic terms},k=2,\cdots n,$$

as well as the already mentioned Klein-Gordon-Zakharov model system (1.2).

We will explain the challenge comes from the strong coupling. For the convenience of discussion we recall some notation. We are working in ${\mathbb{R}}^{2+1}=\{(t,x)|t\in \mathbb{R},x\in {\mathbb{R}}^2\}$ and recall that

$$\mathcal{K}=\{t>r+1\},s=\sqrt{t^2-r^2}=\sqrt{t^2-|x{|}^2},{\mathcal{H}}_s=\{(t,x)|t=\sqrt{s^2+r^2}\}.$$

For a wave-Klein-Gordon system with unit propagation speed, if the initial data are posed on $\{t=2\}$ with support contained in $\{|x|<1\}$, then the solution is supported in $\mathcal{K}$. So we restrict ourselves in $\mathcal{K}$. More details on hyperboloidal foliation can be found in [16].

Return to the discussion on strong coupling. In dimension $3+1$, the strong coupling terms are not critical because of the decay enjoyed by the Klein-Gordon component:

$$|\partial v|+|v|\simeq {(s/t)}^{3/2}{s}^{-3/2+\delta }$$

where δ measure the increasing rate of its standard energy on hyperboloids. For clarity we call the factor $s^{-3/2+\delta }$ the principle decay and the factor ${(s/t)}^{3/2}$ the conical decay. The above decay is integrable with respect to s (in $\mathcal{K},t^{-1/2}<s/t\le 1$), so the strong coupling terms always enjoy integrable $L^2$ bounds.

However in $2+1$ dimensional case the role of these terms change dramatically. Even supposing that v enjoys uniform standard energy bounds, one can only obtain the following decay (via Klainerman-Sobolev inequality):

$$|\partial v|+|v|\simeq (s/t)s^{-1}$$

which is not integrable.

What is even worse is that, the strong couplings destroy “completely” the conformal invariance of the wave equation in (1.1). More precisely, consider the term ${\partial }_{\alpha }v{\partial }_{\beta }v$ and regard the conformal energy estimate (2.8), we need to integrate ${\Vert s{\partial }_{\alpha }v{\partial }_{\beta }v\Vert }_{L^2({\mathcal{H}}_s)}$ with respect to s. This term does not decrease even if we suppose that v enjoys uniform standard energy bound:

$${\Vert s{\partial }_{\alpha }v{\partial }_{\beta }v\Vert }_{L^2({\mathcal{H}}_s)}\simeq 1.$$

This leads to at least a $s^{+1}$ increasing rate of conformal energy.

In contrast, in the weak coupling case the quadratic semi-linear terms containing at least one factor of wave component (or its derivatives). These terms are more friendly because the wave component can be expected to enjoy better decay and $L^2$ bounds due to its conformal energy bounds. However, one can not expect such bounds on Klein-Gordon component. To be more precise, recall the following bounds due to Klainerman-Sobolev inequality:

$$(s/t)|\partial u|+{(s/t)}^{-1}|{\underset{\_}{\partial }}_{a}u|\simeq s^{-2}{\mathcal{E}}_2^N{(s,u)}^{1/2},$$

$${\Vert {(s/t)}^{2}s{\partial }_{\alpha }u\Vert }_{L^2({\mathcal{H}}_s)}\simeq {\mathcal{E}}_2^N{(s,u)}^{1/2},$$

where ${\mathcal{E}}_2^N(s,u)$ represents the N-order conformal energy defined on hyperboloid ${\mathcal{H}}_s$ (see in detail in Subsection 2.4). For example, let us consider the mixed term ${\partial }_{\alpha }u{\partial }_{\beta }v$ coupled in wave equation and suppose that v enjoys uniform standard energy bound. Then recall (1.6) one obtains

$$s{\partial }_{\alpha }u{\partial }_{\beta }v\simeq {(s/t)}^{-2}{s}^{-1}{(s/t)}^{2}s|{\partial }_{\alpha }u|$$

where ${(s/t)}^{2}s|{\partial }_{\alpha }u|$ can be controlled by conformal energy. If we demand null condition on the coefficients of this term, there will be an additional conical decay ${(s/t)}^2$ which will offset the ${(s/t)}^{-2}$. Then this bound will be sufficient to recover a slowly increasing conformal energy bound (see for example [17]). In an other word, in weak coupling case the wave component can be expected to enjoy slowly increasing conformal energy bound, which seems to be impossible in strong coupling case. This is the fundamental difference between the strong and weak coupling cases.

In the above discussion we have applied the term principle decay and conical decay. Due to their importance, let us give more detailed explanation on their different roles. In general we write the decay of a term in following form,

$${(s/t)}^{\beta }{s}^{-\alpha }.$$

The principle decay determines how fast the function decreases far from the light cone $\partial \mathcal{K}=\{r=t-1\}$. It also measures the homogeneity of the solution. The conical decay describes how much additional decay the solution enjoys near the light-cone. However when $\beta <0$, in order to offset this increasing rate near light-cone, one needs to pay principle decay. Remark that ${(s/t)}^{-1}\le s$ in $\mathcal{K}$, then one has

$${(s/t)}^{-\beta }{s}^{-\alpha }\le s^{-\alpha +\beta },\beta \ge 0.$$

We denote by $-\alpha +{[\beta ]}^{-}$ the total decay, where ${[\beta ]}^{-}$ denotes the negative part of β, which is −β when $\beta \le 0$ and 0 when $\beta >0$. Remark that the $L^2$ norm of the gradient of wave component can not be bounded directly by standard energy on hyperboloid (see in detail (2.4)). We need to pay a conical decay $(s/t)$. For example, to bound the $L^2$ norm of ${\partial }_{\alpha }u{\partial }_{\beta }v$, one needs to make the following calculation:

$${\Vert {\partial }_{\alpha }u{\partial }_{\beta }v\Vert }_{L^2({\mathcal{H}}_s)}\le C{\Vert (s/t){\partial }_{\alpha }u\Vert }_{L^2({\mathcal{H}}_s)}{\Vert {(s/t)}^{-1}{\partial }_{\beta }v\Vert }_{L^{\infty }({\mathcal{H}}_s)}$$

and demand how much total decay the term ${\partial }_{\beta }v$ enjoys. When this decay is integrable with respect to s, one concludes that ${\Vert {\partial }_{\alpha }u{\partial }_{\beta }v\Vert }_{L^2({\mathcal{H}}_s)}$ does not blow up the standard energy. The main difficulty comes when this decay is not integrable and it can be classified into two types. One is the lack of conical decay and the other is the lack of principle decay. Both may lead to insufficiency of total decay and even blow-up in finite time. We show some typical examples.

A first example is the following semi-linear wave equation:

$$\square u={({\partial }_{t}u)}^2$$

in ${\mathbb{R}}^{3+1}$. It is known that all non-zero initial data leads to finite time blow-up (see [18]). This can be observed through (1.10). In fact even if u enjoys uniformly standard energy bound (one can not demand more because for free linear wave equation it is conserved), ${(s/t)}^{-1}|{\partial }_{t}u|\sim {(s/t)}^{-1}{(s/t)}^{1/2}{s}^{-3/2}\sim {(s/t)}^{-1/2}{s}^{-3/2}$. The principle decay is integrable but the lack of conical decay will offset the principle decay by $s^{1/2}$ and make the total decay non-integrable.

The second example is the situation in [19] in ${\mathbb{R}}^{3+1}$ where one regarded the null quadratic term of wave component $N^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }u$. With the above observation, we also arrive at (1.10). However this time the null condition supplies a supplementary conical decay ${(s/t)}^2$ (see in detail (2.27)) which makes ${\Vert N^{\alpha \beta }{\partial }_{\alpha }u{\partial }_{\beta }u\Vert }_{L^2({\mathcal{H}}_s)}$ integrable. The role of classical null conditions is that they supply additional conical decay.

In some more recent works more delicate techniques are developed. For example in the situation of [20], the standard energies only enjoy slowly increasing energy bounds. By Klainerman-Sobolev inequality the wave component satisfies

$$|{\partial }_{\alpha }u|\simeq {(s/t)}^{-1/2}{s}^{-3/2+\delta }.$$

However in this case the null condition is no longer valid. This is the case of lack of conical decay. The sharp decay estimate in [20] (integration along characteristics) and many other techniques are in fact an exchange of principle decay into conical decay. Remark that the principle decay is sufficient and has a margin (between $-3/2+\delta$ and −1), or in another word, the insufficiency of total decay only occurs in the region near light-cone. The sharp decay estimate in [20] sacrifices some principle decay to recover the insufficiency of conical decay, and arrive at $|\partial u|\simeq (s/t)s^{-1}$, i.e., we lose some principle decay of order $(-1/2+\delta )$ and recover a conical decay of order $(3/2)$). Of course, some delicate structures of Einstein equation are also applied for this improvement. Many systems in dimension $3+1$ enjoy the above property because of (1.12). We emphasize that this type of techniques in fact do not demand very much principle decay (in fact <−1 is sufficient) and there is still some margin left.

Then we take a look at the Klein-Gordon-Zakharov model system (1.2) in ${\mathbb{R}}^{2+1}$. Clearly this is a strong coupling system. However it is not critical if we regard the principle decay. The main observation is that the term EΔw coupled in Klein-Gordon equation only contains the Hessian form of the wave component. The gradient ∂w does not appear in right-hand-side of the system. Remark that the Hessian form enjoys a faster principle decay (see for example in Proposition 2.4). In fact

$$|\partial \partial w|\simeq {(s/t)}^{-1}{s}^{-2+\delta }+{(s/t)}^{-2}|\square u|$$

and the last term is quadratic, i.e., it can be expected to enjoy better decay than linear ones. When regarding (1.2), $\partial \partial w\simeq s^{-1}$ is sufficient. So there is a margin between $(-2+\delta )$ and $(-1)$. Of course, to make the argument in [3] work, there are several non-trivial works in order to overcome the insufficiency of conical decay. Our techniques are also applicable on (1.2) (though it falls to be a special case of (1.1)). In Section 9 we give a sketch in order to show the difference between critical principle decay and non-critical principle decay.

In the same manner, the system treated in [5] also enjoys the above structure of Hessian form. When restricted to compactly supported initial data, our method is also applicable (see also a generalization in [3]).

In the case of (1.1), the difficulty comes from the other side: we can show that sufficient conical decay is available but the principle decay is at the critical level, i.e., the uniform standard energy bounds only leads to $s^{-1}$ principle decay for both wave and Klein-Gordon components and this is not integrable. Regarding the interaction ${\partial }_{\alpha }u{\partial }_{\beta }v$ coupled in wave equation, this will lead to non-uniformly-bounded standard energy on wave component. The existing techniques such as in [20] will not be applicable because there is no margin of principle decay, i.e., even if we are far from light-cone, the decay of ∂u is still insufficient. The null conditions or the fast decay of Klein-Gordon component (Proposition 2.5) will not aid because they only affect the conical decay. So we need to develop a series of techniques in a somehow inverse sens, i.e., they permit us to exchange surplus conical decay into principle decay.

The main idea is to make the critical principle decay $s^{-1}$ sufficient, and when necessary, we can accept a loss on conical decay. Let us consider the energy bounds

$${\mathcal{E}}_2^N{(s,u)}^{1/2}\simeq s$$

where ${\mathcal{E}}_2^N(s,u)$ represents the N order conformal energy (see in detail in Subsection 2.4). The importance is that to recover this bound, one only needs

$${\Vert \square u\Vert }_{L^2({\mathcal{H}}_s)}\le C{s}^{-1}$$

and there will not be logarithmic loss. By Klainerman-Sobolev inequality, (1.13) leads to the following decay

$$|\partial u|\simeq {(s/t)}^{-1}{s}^{-1}.$$

If we only regard the principle decay, this is sufficient. In an other word, now we have sufficient principle decay, but we have payed ${(s/t)}^{-1}$. The main use of the techniques to be established, is to show that this prise is acceptable.

In the present article we concentrate only on the techniques of “paying conical for principle”. The main system (1.1) is a model in order to show our mechanism. The choice of this system is made under the following two considerations. First, the system should not be too trivial in order to show the necessity and potential of these techniques. Second, the system should not be too general or complicated such that the main ideas are covered under too much technical details. In our opinion the system (1.1) balances well the above two points. We omit all terms that can be treated through existing techniques and preserve all semi-linear terms on $(\partial u,\partial v,v)$ in wave equation.

The Klein-Gordon-Zakharov model system is not in the form of (1.1), however our techniques are also applicable and the proof is somehow shorter. So we take it as a secondary example.

In fact the techniques to be developed can be applied on more general systems. For example when there are pure Klein-Gordon terms coupled in Klein-Gordon equation, the normal form method developed by [21], [13] and [14] are applicable and compatible with these techniques. When considering quasi-linear systems, these techniques can be easily adapted to curved metric.

In a coming work, these techniques will be applied on a special type of strong coupling system deduced form (1.4). This will give a preliminary answer to the problem posed in [15] on the stability of wave map in ${\mathbb{R}}^{2+1}$ case.

Now we state the main results on (1.1) and (1.2) and then give a brief description on the structure of the present article.

Theorem 1.1

Consider the Cauchy problem associate to (1.1) with the following initial data

$$u(2,x)=u_0,{\partial }_{t}u(2,x)=u_1,v(2,x)=v_0,{\partial }_{t}v(2,x)=v_1.$$

Suppose that $u_i$ and $v_i$ are supported in the unit disc $\{|x|<1\}$, sufficiently regular. Then there exists a positive constant ε such that if for $N\ge 14$ the following bounds hold:

$${\Vert u_0\Vert }_{H^{N+1}({\mathbb{R}}^2)}+{\Vert v_0\Vert }_{H^{N+1}({\mathbb{R}}^2)}+{\Vert u_1\Vert }_{H^N({\mathbb{R}}^2)}+{\Vert v_1\Vert }_{H^N({\mathbb{R}}^2)}\le \epsilon ,$$

then the associate local solution extends to time infinity. Furthermore, the following decay bounds hold:

$$|\partial u|\le C\epsilon t^{-1/2}{(1+|t-r|)}^{-1/2},|u|\le C\epsilon t^{-1/2}{(1+|t-r|)}^{1/2},|v|+|\partial v|\le C\epsilon t^{-1},$$

with C determined by the system and N.

Theorem 1.2

Consider the Cauchy problem associate to (1.2) with the following initial data

$$u(2,x)=u_0,{\partial }_{t}u(2,x)=u_1,E^a(2,x)=E_0^a,{\partial }_t{E}^a(2,x)=E_1^a.$$

Suppose that $u_i$ and $E_i^a$ are supported in the unit disc $\{|x|<1\}$, sufficiently regular. Then there exists a positive constant ε such that if for $N\ge 13$ the following bounds hold:

$${\Vert u_0\Vert }_{H^{N+2}({\mathbb{R}}^2)}+{\Vert E_0^a\Vert }_{H^{N+1}({\mathbb{R}}^2)}+{\Vert u_1\Vert }_{H^{N+1}({\mathbb{R}}^2)}+{\Vert E_1^a\Vert }_{H^N({\mathbb{R}}^2)}\le \epsilon ,$$

then the associate local solution extends to time infinity. Furthermore, the following decay bounds hold:

$$|\partial u|\le C\epsilon t^{-1/2}{(1+|t-r|)}^{-1/2},|u|\le C\epsilon t^{-1/2}{(1+|t-r|)}^{1/2},|E^a|+|\partial E^a|\le C\epsilon t^{-1},$$

with C determined by the system and N.

This article is composed by two parts. In the first part (from Section 2 to Section 4) we develop the necessary estimates and in the second part (from Section 5 to Section 9) we prove the global existence of the model systems (1.1) and (1.2).

For the convenience of the reader we recall some basic results of the hyperboloidal foliation in Section 2 and sketch some of their proofs in the Appendix. Then as explained in the previous subsections the main task is to surmount the loss of ${(s/t)}^{-1}$. This is done in two steps. The first step is an estimate on the fundamental solution of the wave equation made in Section 3. In our mechanism it is a necessary tool for recovering conical decay without paying principle decay on $|u|$. The second step, contained in Section 4, is an $L^{\infty }$ estimate on $|\partial u|$ via integration along hyperbolas.

The global existence of (1.1) is proved by the standard bootstrap argument. In Section 5 we state the bootstrap argument. In the set of bootstrap bounds there are the energy bounds on wave and Klein-Gordon components as well as the decay bound (5.3) on wave component. They will be improved in Section 5, Section 6 and Section 7 respectively.

In Section 9 we sketch the global existence of (1.2).

Acknowledgments

The present work belongs to a research project “Global stability of quasilinear wave-Klein-Gordon system in $2+1$ space-time dimension” (11601414), supported by NSFC.