# On the well-posedness of SPDEs with singular drift in divergence form

## TL;DR

This paper establishes the existence and uniqueness of strong solutions for a class of stochastic PDEs with singular divergence-form drift and multiplicative noise, broadening the understanding of well-posedness in such equations.

## Contribution

It proves well-posedness for SPDEs with maximal monotone divergence drift, extending previous results to more general multivalued cases with minimal assumptions.

## Key findings

- Existence of strong solutions under broad conditions
- Uniqueness of solutions for the class of SPDEs considered
- Self-contained proof of well-posedness

## Abstract

We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form $\operatorname{div} \gamma(\nabla \cdot)$, where $\gamma$ is a maximal monotone graph in $\mathbb{R}^n \times \mathbb{R}^n$ obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that $\gamma$ is single-valued, a solution with better integrability and regularity properties is constructed. The proof given here, however, is self-contained.