Elsevier

Nonlinear Analysis

Volume 185, August 2019, Pages 306-335
Nonlinear Analysis

Local Hölder continuity of weak solutions to a diffusive shallow medium equation

https://doi.org/10.1016/j.na.2019.03.013Get rights and content

Abstract

We show the local Hölder continuity of bounded weak solutions to a doubly nonlinear parabolic equation used in models for diffusion of shallow media.

Introduction

In this work we prove the local Hölder continuity of bounded weak solutions to the doubly nonlinear equation tu((uz)α|u|p2u)=f in ΩTΩ×(0,T),where ΩRn is an open bounded set, z:ΩR and f:ΩTR are given sufficiently regular functions, and the parameters α and p are restricted to the range p>2 and α>0.Note that p+α>2, which means that we are in the slow diffusion case. The term doubly nonlinear refers to the fact that the diffusion part depends nonlinearly both on the gradient and the solution itself. In this article we consider all n2, although the equation is typically used in models with only two spatial dimensions.

In the case z=0, (1.1) reduces to a standard doubly nonlinear equation which was studied already in [27] and [22], although their notion of weak solution differs from ours. If in addition we set α=0, we recover the parabolic p-Laplace equation, whereas taking p=2 we obtain the porous medium equation, after a formal application of the chain rule. Thus, (1.1) can be seen as a generalization of many previously studied equations.

In the range p>2, Eq. (1.1) has applications in models of the dynamics of glaciers in the so called shallow ice approximation, see for example Chapter 2 of [21] and the classic work [20]. Laboratory experiments, theoretical considerations and field measurements suggest that in most situations p4. This corresponds to the value 3 for the exponent appearing in the flow law for polycrystalline ice, see [14] for details. In the range p<2, which is outside the scope of this article, Eq. (1.1) is used to model shallow water dynamics in situations such as floods and dam breaks, see [1], [10], [15]. Due to the aforementioned applications, we propose the term diffusive shallow medium equation or more concisely, DSM equation, to describe Eq. (1.1) for all p>1.

Regardless of the range of p, the function z represents the elevation of the land on top of which the water or ice is moving, measured with respect to some arbitrarily chosen ground level. The value of u is the height of the medium measured with respect to the same level. Thus, one always looks for solutions satisfying uz. Although u and z depend on the ground level, their difference vuz is invariant. Therefore, it is natural to reformulate the equation in terms of v, see (2.1) of Section 2. It turns out that this formulation is also mathematically convenient. The right-hand side f is a source term which can account for snowfall in the case of glaciers, and rainfall, evaporation or infiltration in the shallow water setting.

Local boundedness of weak solutions to (1.1) was proven in [24] for sufficiently regular f and z. While the article focused on the case p<2 due to the applications in shallow water dynamics, it was also pointed out that the value of p does not really play a role in the arguments.

Hölder continuity for various nonlinear parabolic equations has been studied extensively in the past. The key technique is the method of intrinsic scaling introduced by DiBenedetto in [7] in order to study the degenerate parabolic p-Laplace equation, see also [8]. Systems of parabolic p-Laplace type were considered by DiBenedetto and Friedman in [9]. Hölder continuity for the doubly nonlinear so-called Trudinger’s equation was proven in the degenerate case by Kuusi, Siljander and Urbano in [19] whereas the singular case was treated by Kuusi, Laleoglu, Siljander and Urbano in [18]. Despite being doubly nonlinear, Trudinger’s equation has the advantage of being homogeneous, i.e. positive multiples of solutions are also solutions, which allowed the authors to use a Harnack inequality obtained previously in [16].

Doubly nonlinear parabolic equations lacking this type of homogeneity were studied in metric measure spaces by Henriques and Laleoglu in [13]. In the special case f=0=z, the equation satisfied by a certain power of v reduces to the model equation studied in [13]. In this sense, our result can be seen as a generalization of the result obtained in [13], although we restrict ourselves to the Euclidean setting and do not formulate general structure conditions.

To prove Hölder continuity, we use the method of intrinsic scaling to obtain the reduction of the oscillation. The iterative methods combining energy estimates with Sobolev inequalities date back to De Giorgi, see [6]. As in [5], [13], [18], [19], we need to distinguish between the degenerate regime, where the infimum of the solution is small compared to the supremum, and the non-degenerate regime in which the infimum and supremum are close. The two regimes require different intrinsic scalings.

The study of (1.1) is challenging for several reasons. First of all, the definition of weak solutions is nontrivial. As we concluded in [24], it is natural to require that a certain power of v=uz belongs to a Sobolev space. This can be compared with similar definitions for the porous medium equation cf. [3], [26] and doubly nonlinear equations in [2], [11], [25]. As a consequence, the powers of the terms in the energy estimates originating from the diffusion part of the equation differ from the powers of the terms corresponding to the parabolic part. Hence, special care needs to be taken in order to be able to use the energy estimates in the iterative arguments.

The presence of f and z produces extra terms in the energy estimates and logarithmic estimates which complicate the analysis. While source terms in nonlinear parabolic equations have been treated for example in [8], terms involving z naturally are a phenomenon particular to this equation. Nevertheless, we have treated these extra terms in a unified fashion whenever possible.

A common feature of many nonlinear parabolic equations is the use of a mollified weak formulation of the problem. In [24] we demonstrated how the exponential time mollification can be used to prove the energy estimates in a rigorous way. In this article we pay special attention to a detailed proof of the logarithmic estimates.

The paper is organized as follows. In Section 2 we formulate precise definitions and present the main result. In Section 3 we explain the notation and provide some technical tools and lemmas needed for the analysis. Section 4 is devoted to the energy estimates and Section 5 is concerned with the logarithmic estimates. In Section 6 we present the two regimes and prove the De Giorgi type lemma which is the starting point of the main argument. In Section 7 we consider the two alternatives, conclude that they imply the reduction of the oscillation, and show how Hölder continuity can be deduced from this result.

Section snippets

Setting and main result

In order to motivate the natural definition of weak solutions, we reformulate (1.1) in terms of vuz. Formally applying the chain rule as in [24], we can write the equation in the form tv(β1p|vβ+βvβ1z|p2(vβ+βvβ1z))=f,where β1+αp1>1.Throughout the article, we will focus on this form of the equation. In order to simplify the notation, we denote the vector field appearing in the diffusion part of (2.1) as A(v,vβ)=β1p|vβ+βvβ1z|p2(vβ+βvβ1z).For the right-hand side f:ΩTR and

Preliminaries

Here we introduce some notation and present auxiliary tools that will be useful in the course of the paper.

Energy estimates

In this section we prove the energy estimates which will be used in the iterative arguments. For this purpose we need a specific mollified weak formulation of the problem which only makes sense for solutions which possess some regularity in time. Fortunately, the results of [24] are still valid with virtually the same proofs as before, even though p>2 in the present article. Thus, vC0([0,T];Llocβ+1(Ω)), and we have the mollified weak formulation:

Lemma 4.1

Let v be a weak solution to the DSW equation

Logarithmic energy estimates

In this section we prove the logarithmic estimates which will be used to expand information about the solution forwards in time. For 0<δ<Γ we define the function ϕ(w)ϕΓ,δ(w)lnΓΓ+δw+ for w<Γ+δWe note that ϕ(w)=0 for wδ and for wΓ we have the estimates 0ϕ(w)lnΓδ and 0ϕ(w)1δ for wδ.Moreover, ϕ is a solution to the differential equation ϕ=(ϕ)2 for wδ and we have (ϕ2)=2ϕϕ on [0,Γ]and(ϕ2)=2(1+ϕ)(ϕ)2 on [0,Γ]{δ}.

Lemma 5.1

Let v be a weak solution. Suppose that Br(xo)×(t1,t2)ΩT and set Γ±=

De Giorgi type lemma

In this section we prove a De Giorgi type lemma, which serves as the initial step of the main argument. In order to formulate the lemma we need to define the two regimes that are used in the subsequent arguments. For numbers 0μμ+< we define ωβμ+βμβ.

In what follows, we always distinguish between two cases. We are either in the degenerate regime if μ12μ+ and θ=ω1(p1)β,or we are in the non-degenerate case where μ>12μ+ and (2μ+)1βωβ(2p)θ(12μ+)1βωβ(2p)holds true. We now prove some

Hölder continuity of weak solutions

In this chapter we will use the previously introduced tools and estimates to prove the Hölder continuity. We start by defining the geometry of the space–time cylinder in which we will prove the reduction of the oscillation. For a sufficiently large ΛN to be chosen later, we define θ+2Λ(p2)θ. We will work with the space-time cylinder Qϱ,θ+ϱp(xo,to) where (xo,to)ΩT and ϱ is chosen so small that the cylinder is contained in ΩT. We distinguish between two alternatives. The first alternative is

Acknowledgments

T. Singer has been supported by the DFG-Project, Germany SI 2464/1-1 “Highly nonlinear evolutionary problems” and both authors want to express their gratitude to the Academy of Finland.

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