Doubly nonlinear stochastic evolution equations II

TL;DR

This paper studies the existence and uniqueness of solutions for doubly nonlinear stochastic partial differential equations, extending classical deterministic results to include stochastic effects in models of dissipative media.

Contribution

It establishes well-posedness results for doubly nonlinear SPDEs, including martingale and strong solutions, using regularization and Itô calculus in minimal regularity settings.

Findings

Proves existence of martingale solutions for doubly nonlinear SPDEs.

Shows well-posedness of strong solutions under additional assumptions.

Extends deterministic theory to stochastic cases in dissipative media models.

Abstract

Complementing the analysis in [41], we investigate the well-posedness of SPDEs problems of doubly nonlinear type. These arise ubiquitously in the modelization of dissipative media and correspond to generalized balance laws between conservative and nonconservative dynamics. We extend the reach of the classical deterministic case by allowing for stochasticity. The existence of martingale solutions is proved via a regularization technique, hinging on the validity of a It\^o formula in a minimal regularity setting. Under additional assumptions, the well-posedness of stochastically strong solutions is also shown.