Entropy solutions for stochastic porous media equations

TL;DR

This paper develops an entropy framework for stochastic porous media equations with nonlinear, spatially inhomogeneous forcing, establishing well-posedness and contraction properties for a broad class of operators and nonlinearities.

Contribution

It introduces an entropy formulation for stochastic porous media equations with general nonlinearities and proves well-posedness and contraction in this setting.

Findings

Established entropy solutions for stochastic porous media equations.

Proved well-posedness and $L_1$-contraction properties.

Extended results to a wide range of nonlinear operators and diffusion exponents.

Abstract

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well - posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\Delta (|u|^{m-1}u)$ for all $m\in(1,\infty)$, and H\"older continuous diffusion nonlinearity with exponent $1/2$.