Entropy solutions to the Dirichlet problem for nonlinear diffusion equations with conservative noise

This paper is concerned with the Dirichlet problem for nonlinear diffusion equations

where $D$ is a bounded domain in $\mathbb{R}^{d}$ with smooth boundary, and $\Phi:\mathbb{R\rightarrow\mathbb{R}}$ is a monotone function. The noise $\{W^{k}\}_{k\in\mathbb{N}}$ is a sequence of independent standard Brownian motions defined on a complete filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$, and the stochastic integral is in the Stratonovich sense. The Einstein summation convention is used throughout this paper.

A typical example of 1.1 and also our primary motivation of this paper is the porous medium equation in random environment, which describes the flow of an ideal gas in a homogeneous porous medium: let $u$ be the gas density that satisfies

where $\varepsilon\in(0,1)$ is the porosity of the medium, and $\bm{V}$ is the randomly perturbed velocity field of the form

where $\bm{V}_{0}$ is derived from Darcy’s law

Then, we can informally derive a stochastic porous medium equation

For more details of the model, we refer to [V0́7] and references therein. Moreover, equations of type 1.1 also arise as limits of interacting particle systems driven by common noise from mean field models [LL06a, LL06b, LL07], and as simplified models of fluctuations in non-equilibrium statistical physics [DSZ16]; for more applications we refer to [FG19, DG20] and the references therein.

The well-posedness of the Cauchy problem for stochastic nonlinear diffusion equations with general noise has been investigated in various frameworks, for example, the variational approach in the space $H^{-1}$ (cf. [BDPR08, BDPR16, Cio20] etc.), the kinetic formulation (cf. [DHV16, GH18, FG21a] etc.), and the entropy formulation (cf. [BVW15, DGG19, DGT20] etc.). The case of linear gradient noise, say $\sigma(x,u)=h(x)u$ in 1.1, has been studied in [DG19, T2̈0, Cio20], and the general case was discussed in [FG19] by a kinetic approach with rough path techniques (cf. [LPS13, LPS14, GS15, GS17a, GS17b]), requiring $\sigma\in C^{\gamma}_{b}(\mathbb{T}^{d}\times\mathbb{R})$ for some $\gamma>5$. The paper [DG20] relaxed the regularity assumption to $\sigma\in C^{3}_{b}(\mathbb{T}^{d}\times\mathbb{R})$ and treated general diffusion nonlinearity $\Phi$ under the entropy formulation. The recent work [FG21b] introduced a concept of stochastic kinetic solution based on the concept of renormalized solutions, and proved the existence and uniqueness results under a very general setting that can even cover the case $\sigma(x,u)=f(x)\sqrt{u}$ with $f\in C^{2}_{b}(\mathbb{T}^{d})$.

In this paper, we adopt the entropy approach to study 1.1. The notion of entropy solution was proposed for nonlinear PDEs in 1970s to tackle the difficulty that the weak solution may not be unique (see, for example, [Kru70, Ser99, ABK00, Daf05] for scalar conservation laws and [Car99] for degenerate parabolic equations). The entropy solution discriminates the physical admissible solution and maintains the uniqueness theoretically.

The entropy formulation has been naturally extended to nonlinear SPDEs. Many works have been done for first order equations (see, for example, [Kim03] for additive noise and [FN08, VW09, CDK12, BVW12, BVW14] for multiplicative noise). For second order equations, most works considered the Cauchy problem in the whole space or a torus; for instance, [BVW15] studied the Cauchy problem of parabolic-hyperbolic SPDEs with the noise term $\sigma(x,u)\mathrm{d}W$ and Lipschitz functions $\Phi$. Stochastic nonlinear diffusion equations in a torus are considered in [DGG19] for the noise term $\sigma(x,u)\mathrm{d}W$ and in [DG20] for the conservative noise $\nabla\cdot\sigma(x,u)\circ\mathrm{d}W$. The obstacle problem with the noise term $\sigma(x,u)\mathrm{d}W$ in a torus is introduced in [LT21].

The results on the Dirichlet problem for stochastic nonlinear diffusion equations are relatively few. The recent papers [BGV20, Hen21] adopted the variational approach from [RRW07] with the monotone condition and affine noise, thus not covering our setting; [DGT20] considered entropy solutions of stochastic porous medium equations with the noise term $\sigma(x,u)\mathrm{d}W$. A recent paper [Cli23] is quite relevant to our paper, studying kinetic solutions to the Dirichlet problem for porous medium equations with nonlinear gradient noise driven by rough path. Thanks to the entropy approach, our result is built on the same regularity conditions with the existing work on the Cauchy problem in torus (cf. [DG20]), and do not require extra technical assumptions like [Cli23, Condition (2.4)] that prevents the space characteristics from escaping the domain. According to the recent work [FG21b], it is a very interesting question how to relax the regularity condition on $\sigma$ for the Dirichlet problem.

The strategy of the proof of our main result (2.6) basically follows from [DGG19, DGT20, DG20], combining the method of strong entropy condition (called the $(\star)$-property in this paper) and Kruzhkov’s doubling variables technique (cf. [Kru70]). The notion of strong entropy condition was introduced by [FN08] to tackle the uniqueness issue of stochastic scalar conservation laws. The strategy can be summarized to two steps:

1. (1)

   $L_{1}$-estimates: to derive an estimate for $\mathbb{E}\|u(t,\cdot)-\tilde{u}(t,\cdot)\|_{L_{1}(D)}$, providing one of the entropy solutions $u$ and $\tilde{u}$ (with different initial data) satisfies the $(\star)$-property.

2. (2)

   Approximation: to construct a sequence of non-degenerate equations whose solutions have the $(\star)$-property and converge to the entropy solution of 1.1.

Kruzhkov’s doubling variables technique plays a key role in the proof of the $L_{1}$-estimates.

The new difficulty arising in the Dirichlet problem, comparing to the Cauchy problem (cf. [DG20]), is how to deal with the boundary integral terms that may emerge when applying the divergence theorem in the proof of the $L_{1}$-estimates. In the paper [DGT20] where the noise term is $\sigma(x,u)\mathrm{d}W$, a weighted space with a weight function $w\in H_{0}^{1}$ satisfying $\Delta w=-1$ is introduced to eliminate the boundary terms. However, in the case of gradient noise, there will be a new “trouble” term $|(u-\tilde{u})\nabla w|$ appearing in the estimate, which cannot be dominated by any “good” terms like $|u-\tilde{u}|w$.

There are three key points in our approach to overcome the above difficulty: i) to expand the set of test functions in the definition of entropy solution (see 2.2), ii) to refine the strong entropy condition, and iii) to construct subtly a pair of the convex function $\eta$ and the test function $\phi$ to avoid the appearance of boundary terms. Specifically, when applying Kruzhkov’s doubling variables technique to our problem, we have to estimate both $(\tilde{u}(s,x)-u(t,y))^{+}$ and $(u(t,x)-\tilde{u}(s,y))^{+}$ rather than $|u(t,x)-\tilde{u}(s,y)|$ as in [DGG19, DGT20, DG20]. Inspired by [Car99, BVW14], we make use of a partition of unity and shifted mollifiers to keep the test function in $C_{c}^{\infty}(D)$ with respect to the variable $y$; and for the variable $x$, we choose a sequence of smooth convex functions to approach $\eta(r)=r^{+}$. Those carefully chosen functions along our modified definition of entropy solution and the refined strong entropy condition let the boundary terms vanish. More specific details are given at the beginning of Section 3 and Remarks 3.5, 3.12 and 3.15. It is worth noting that our approach avoids the involvement of weight functions and considers the $L_{1}$-estimates in standard Sobolev spaces, and our method may also apply to nonlinear diffusion equations with the noise term $\sigma(x,u)\mathrm{d}W$ to obtain the $L_{1}$-estimates without weight.

This paper is organized as follows. Section 2 describes the entropy formulation and presents the main theorem. Section 3, which is the main part of this paper, introduces a refined strong entropy condition and derives the $L_{1}$-estimates for the difference of two entropy solutions. Section 4 constructs the approximate equations and proves the strong entropy condition of their solutions as well as their solvability. Section 5 completes the proof of the main theorem. Two auxiliary lemmas are proved in the final section.

We conclude the introduction with some notation. Fix $T>0$. Define $\Omega_{T}\coloneqq\Omega\times[0,T]$ and $D_{T}\coloneqq[0,T]\times D$. Define $|D|$ and $\overline{D}$ as the volume and closure of $D$, respectively. $L_{p}$ and $H_{p}^{k}$ are the usual Lebesgue spaces and Sobolev spaces. Denote by $H_{p,0}^{k}$ the closure of $C_{c}^{\infty}$ in $H_{p}^{k}$. When $p=2$, we simplify the notation by $H^{k}\coloneqq H_{2}^{k}$ and $H_{0}^{k}\coloneqq H_{2,0}^{k}$. Moreover, if a function space is given on $\Omega$ or $\Omega_{T}$, we understand it to be defined with respect to $\mathcal{F}_{T}$ and the predictable $\sigma$-field, respectively. Let $E$ be a Banach space and $U=D$ or $\mathbb{R}$. For a function $f:U\rightarrow E$, we define

We define a non-negative smooth mollifier $\rho:\mathbb{R}\rightarrow\mathbb{R}$, such that $\text{supp}\,\rho\subset(0,1)$, $\rho\leq 2$ and $\int_{\mathbb{R}}\rho(r)\mathrm{d}r=1$. For $\delta>0$, we set $\rho_{\delta}(r)\coloneqq\delta^{-1}\rho(\delta^{-1}r)$ as a sequence of mollifiers.

First of all, we rewrite 1.1 into an Itô form (the notation follows from [DG20]):

where $i,j=1,\dots,d$ and

We denote by $\Pi(\Phi,\xi)$ the Dirichlet problem 2.1 with given $\Phi$ and $\xi$.

Throughout this paper, we denote

for a function $g:D\times\mathbb{R}\rightarrow\mathbb{R}$, we use the notation

and drop $x$ in the above if $g$ does not depend on $x\in D$. Our condition on the nonlinearity $\Phi$ is the same with [DGG19, DGT20, DG20].

The following definition of entropy solution is based on the formulation in [DGT20, DG20] with a slight but significant modification inspired by [BVW14, Definition 1]. Define two sets

The regularity assumption on coefficients coincides with [DG20, Assumption 2.3] for the Cauchy problem in a torus.

Due to the physical background, we focus on non-negative solutions to the concerned problem, for which we need the following natural condition.

The main result of this paper is stated as follows.

We conclude this section by proving the non-negativity of entropy solutions under our assumptions.

The uniqueness for entropy solutions of stochastic partial differential equations is usually a challenging problem. The seminal work [FN08] introduced a notion of strong entropy condition, called the $(\star)$-property in what follows, to deal with the stochastic integral in the $L_{1}$-estimates and proved the uniqueness of the strong entropy solution (namely, an entropy solution that satisfies the $(\star)$-property) for the Cauchy problem of stochastic scalar conservation laws. Recently, a series of papers [DGG19, DGT20, DG20] improved this technique and proved the uniqueness of entropy solutions to stochastic porous medium equations. The basic idea is to estimate the $L_{1}$-difference between an entropy solution and a strong entropy solution, which leads to the uniqueness of entropy solutions by proving that the entropy solutions constructed from approximation always satisfy the $(\star)$-property. The key technique in the proof of the $L_{1}$-estimates is Kruzhkov’s doubling variables method [Kru70], which is a classical method to study the uniqueness problem for deterministic conservation laws.

From the Cauchy problem to the Dirichlet problem, the new technical difficulties mainly lie in how to handle the boundary terms. Considering the noise form $\sigma(x,u)\mathrm{d}W$, the paper [DGT20] introduced a weighted space to overcome the difficulties, where the weight function $w\in H_{0}^{1}$ satisfying $\Delta w=-1$ is used to remove the boundary terms in the estimate. However, in the case of 1.1, there will be a new “trouble” term $|(u-\tilde{u})\nabla w|$ appearing in the $L_{1}$-estimates, which cannot be dominated by any “good” terms like $|u-\tilde{u}|w$.

In order to deal with the boundary terms in our case, we do not introduce the weight function but modify the definition of the $(\star)$-property by subtly choosing the test functions. Since the modified definition is highly intricate, we give a heuristic explanation before the exact formulation.

We begin by addressing the problem of obtaining the $L_{1}$-difference between entropy solutions $u$ and $\tilde{u}$ of 1.1. Using Kruzhkov’s doubling variables technique (see, for example, [Kru70, LW12, DGG19, DGT20, DG20]), we wish to estimate the term

where $\rho_{\theta}(\cdot)\coloneqq\theta^{-1}\rho(\theta^{-1}\cdot)$ defined before Section 2 is a time mollifier, and $\varphi_{\varepsilon}\in C^{\infty}(\overline{D}\times\overline{D})$ is a spatial mollifier which satisfies $\lim_{\varepsilon\rightarrow 0^{+}}\varphi_{\varepsilon}(x,y)=\delta_{0}(x-y)$ for all $(x,y)\in\overline{D}\times\overline{D}$. When $\theta,\varepsilon\rightarrow 0^{+}$, we have the estimate for $\mathbb{E}\int_{D_{T}}|u(t,x)-\tilde{u}(t,x)|\mathrm{d}x\mathrm{d}t$.

How to select a suitable spatial mollifier is important but quite subtle: this is standard when $x$ is an interior point (see, e.g., [DGT20, proof of Proposition 4.2]), but much more complicated when $x$ is on the boundary. Following the idea from [Car99, BVW14], we introduce a partition of unity $\sum_{i=1}^{N}\psi_{i}\equiv 1$ on $\overline{D}$ as the set of localization functions, and then choose corresponding mollifiers $\varrho_{\varepsilon,i}\in C^{\infty}(\mathbb{R}^{d})$ such that $\text{supp}\,\varrho_{\varepsilon,i}(x-\cdot)\subset D$ for all $x\in\text{supp}\,\psi_{i}$ and sufficiently small $\varepsilon$. Then, the spatial mollifier $\varphi_{\varepsilon}(x,y)\coloneqq\sum^{N}_{i=1}\psi_{i}(x)\varrho_{\varepsilon,i}(x-y)$ satisfies

It is worth noting that this mollifier is asymmetric in the spatial variables, specifically, for all sufficiently small $\varepsilon$,

Consequently, this asymmetry makes it difficult to estimate 3.1. Instead, we respectively estimate both

Actually, when directly applying the methods in [DGG19, DGT20, DG20], the estimate of 3.1 contains the following terms

which may not be equal to

The reason for this discrepancy is that when formally applying the divergence theorem in $x$ to the first term in 3.3, the boundary term does not vanish due to the asymmetry of the spatial mollifier, and estimating it when $\varepsilon\rightarrow 0^{+}$ is challenging due to the potential absence of a trace for $u$ caused by its low regularity. While for the case of 3.2, this equivalence is guaranteed by the support of $\eta(\cdot)=(\cdot-\tilde{u}(s,y))^{+}$.

Note that the time mollifier is also asymmetric and always requires $s>t$. For this, we have an important observation: we only need the $(\star)$-property of the function at the larger time variable $s$. However, when applying this observation to 3.2, it necessitates the $(\star)$-property of both $u$ and $\tilde{u}$, which is not what we desire. Therefore, we turn to estimate