Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: an entropy formulation

Kai Du, Ruoyang Liu

TL;DR

This work resolves the well-posedness of the obstacle problem for a degenerate stochastic diffusion with Stratonovich gradient noise by introducing an entropy formulation that naturally incorporates the Skorohod constraint via inequality testing. The authors develop a vanishing-viscosity penalization scheme, proving existence, uniqueness, and $L_1$-stability of entropy solutions, even when the obstraction induces a low-regularity Radon measure in the equation. A key contribution is embedding the Skorohod condition into the entropy inequality, enabling handle of degenerate leading operators and gradient-noise SPDEs. The framework extends to deterministic porous medium equations with obstacles, yielding well-posedness and clarifying the link between entropy and variational solutions in the PME context, with potential implications for modeling diffusion with random barriers.

Abstract

In this paper, we establish the existence, uniqueness and stability results for the obstacle problem associated with a degenerate nonlinear diffusion equation perturbed by conservative gradient noise. Our approach revolves round introducing a new entropy formulation for stochastic variational inequalities. As a consequence, we obtain a novel well-posedness result for the obstacle problem of deterministic porous medium equations with nonlinear reaction terms.

Well-posedness of the obstacle problem for stochastic nonlinear diffusion equations: an entropy formulation

TL;DR

This work resolves the well-posedness of the obstacle problem for a degenerate stochastic diffusion with Stratonovich gradient noise by introducing an entropy formulation that naturally incorporates the Skorohod constraint via inequality testing. The authors develop a vanishing-viscosity penalization scheme, proving existence, uniqueness, and -stability of entropy solutions, even when the obstraction induces a low-regularity Radon measure in the equation. A key contribution is embedding the Skorohod condition into the entropy inequality, enabling handle of degenerate leading operators and gradient-noise SPDEs. The framework extends to deterministic porous medium equations with obstacles, yielding well-posedness and clarifying the link between entropy and variational solutions in the PME context, with potential implications for modeling diffusion with random barriers.

Abstract

In this paper, we establish the existence, uniqueness and stability results for the obstacle problem associated with a degenerate nonlinear diffusion equation perturbed by conservative gradient noise. Our approach revolves round introducing a new entropy formulation for stochastic variational inequalities. As a consequence, we obtain a novel well-posedness result for the obstacle problem of deterministic porous medium equations with nonlinear reaction terms.
Paper Structure (8 sections, 15 theorems, 159 equations)

This paper contains 8 sections, 15 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Entropy formulation and main results
  3. $L_{1}$-stability
  4. Approximation and the penalization scheme
  5. Comparison principle
  6. Existence
  7. The porous medium equation
  8. Auxiliary lemma

Key Result

Theorem 2.4

Let Assumptions assu:assumption for phi and assu:assumption for sigma be satisfied and $\psi\in C^{\kappa}_x(\bar{Q}_{T})$ with $\kappa\in (0,1]$. Then for any $\xi\in L_{m+1}(\Omega,\mathcal{F}_{0};L_{m+1}(\mathbb{T}^{d}))$ satisfying there exists an entropy solution $(u,\nu)$ to $\Pi_{\psi}(\Phi,f,\xi)$.

Theorems & Definitions (34)

  • Definition 2.1
  • Theorem 2.4: existence
  • Theorem 2.5: $L_1$-stability and uniqueness
  • Remark 2.6
  • proof : Proof of Theorem \ref{['thm:unque_main thm']}
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Definition 3.1: Definition 3.6 in dareiotis2020nonlinear
  • Proposition 3.2
  • ...and 24 more