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

TL;DR

This paper establishes the existence, uniqueness, and stability of entropy solutions for nonlinear degenerate diffusion equations with conservative noise, extending the theory to stochastic porous medium-like equations with boundary conditions.

Contribution

It introduces a novel approach combining Kruzhkov's technique with a revised entropy condition for equations with conservative noise, ensuring well-posedness in bounded domains.

Findings

Proved existence of entropy solutions under Dirichlet boundary conditions.

Established uniqueness and $L_1$-stability of solutions.

Extended entropy solution theory to stochastic nonlinear diffusion equations.

Abstract

Motivated by porous medium equations with randomly perturbed velocity field, this paper considers a class of nonlinear degenerate diffusion equations with nonlinear conservative noise in bounded domains. The existence, uniqueness and $L_{1}$-stability of non-negative entropy solutions under the homogeneous Dirichlet boundary condition are proved. The approach combines Kruzhkov's doubling variables technique with a revised strong entropy condition that is automatically satisfied by the solutions of approximate equations.