Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type

TL;DR

This paper establishes the well-posedness of a broad class of doubly nonlinear stochastic PDEs of divergence type, using variational methods and compactness arguments without restrictive assumptions on nonlinearities.

Contribution

It proves existence and uniqueness for doubly nonlinear stochastic PDEs with multivalued maximal monotone operators, extending previous results to more general nonlinearities and less restrictive conditions.

Findings

Proved well-posedness under classical Leray-Lions conditions.

No restrictive smoothness or growth assumptions on the nonlinear operator β.

Established uniform estimates and compactness results for solutions.

Abstract

We prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form $dX_t-\text{div}\,\gamma(\nabla X_t)\,dt+\beta(X_t)\,dt\ni B(t,X_t)\,dW_t$, where $\gamma$ and $\beta$ are the two nonlinearities, assumed to be multivalued maximal monotone operators everywhere defined on $\mathbb{R}^d$ and $\mathbb{R}$ respectively, and $W$ is a cylindrical Wiener process. Using variational techniques, suitable uniform estimates (both pathwise and in expectation) and some compactness results, well-posedness is proved under the classical Leray-Lions conditions on $\gamma$ and with no restrictive smoothness or growth assumptions on $\beta$. The operator $B$ is assumed to be Hilbert-Schmidt and to satisfy some classical Lipschitz conditions in the second variable.