Renormalized solutions for stochastic $p$-Laplace equations with $L^1$-initial data: The multiplicative case

TL;DR

This paper develops a framework for renormalized solutions to stochastic p-Laplace equations with L^1 initial data under multiplicative noise, establishing existence and uniqueness results.

Contribution

It introduces the concept of renormalized solutions for stochastic p-Laplace equations with minimal initial data regularity and proves their well-posedness.

Findings

Existence of renormalized solutions under L^1 initial data.

Uniqueness of solutions in the renormalized framework.

Extension of solution concepts to stochastic p-Laplace equations with multiplicative noise.

Abstract

We consider a $p$-Laplace evolution problem with multiplicative noise on a bounded domain $D \subset \mathbb{R}^d$ with homogeneous Dirichlet boundary conditions for $1<p< \infty$. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic $p$-Laplace equations with $L^1$-initial data and study existence and uniqueness of solutions in this framework.