Local Hölder continuity of weak solutions to a diffusive shallow medium equation
Introduction
In this work we prove the local Hölder continuity of bounded weak solutions to the doubly nonlinear equation where is an open bounded set, and are given sufficiently regular functions, and the parameters and are restricted to the range Note that , 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 , although the equation is typically used in models with only two spatial dimensions.
In the case , (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 , we recover the parabolic -Laplace equation, whereas taking 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 , 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 . This corresponds to the value for the exponent appearing in the flow law for polycrystalline ice, see [14] for details. In the range , 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 .
Regardless of the range of , the function 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 is the height of the medium measured with respect to the same level. Thus, one always looks for solutions satisfying . Although and depend on the ground level, their difference is invariant. Therefore, it is natural to reformulate the equation in terms of , see (2.1) of Section 2. It turns out that this formulation is also mathematically convenient. The right-hand side 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 and . While the article focused on the case due to the applications in shallow water dynamics, it was also pointed out that the value of 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 -Laplace equation, see also [8]. Systems of parabolic -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 , the equation satisfied by a certain power of 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 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 and 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 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 . Formally applying the chain rule as in [24], we can write the equation in the form where 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 For the right-hand side 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 in the present article. Thus, , and we have the mollified weak formulation:
Lemma 4.1 Let 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 we define the function We note that for and for we have the estimates Moreover, is a solution to the differential equation for and we have
Lemma 5.1 Let be a weak solution. Suppose that and set
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 we define
In what follows, we always distinguish between two cases. We are either in the degenerate regime if or we are in the non-degenerate case where 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 to be chosen later, we define . We will work with the space-time cylinder where and is chosen so small that the cylinder is contained in . 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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