Porous media equations with nonlinear gradient noise and Dirichlet boundary conditions

TL;DR

This paper proves the existence and uniqueness of solutions for porous media equations with nonlinear gradient noise across various regimes, establishing their continuous dependence on noise and generating a random dynamical system.

Contribution

It extends the theory of porous media equations with nonlinear gradient noise by establishing pathwise solutions, uniqueness, and continuous dependence for a broad range of parameters and initial data.

Findings

Existence of pathwise solutions for all m>0

Uniqueness of solutions when initial data is positive

Solutions depend continuously on the driving noise

Abstract

We establish pathwise existence of solutions for porous media and fast diffusion equations with nonlinear gradient noise, in the full regime $m\in(0,\infty)$ and for any initial data in $L^2$. Moreover, if the initial data is positive, solutions are pathwise unique. In turn, the solution map of these equations is a continuous function of the driving noise and it generates an associated random dynamical system. Finally, in the regime $m\in\{1\}\cup(2,\infty)$, all the aforementioned results also hold for signed initial data.