A variational approach to dissipative SPDEs with singular drift

TL;DR

This paper establishes the global well-posedness of a broad class of dissipative stochastic partial differential equations with singular drift and general multiplicative noise, using a novel variational approach.

Contribution

It introduces a variational method to prove well-posedness for SPDEs with singular, maximal monotone drift and very general noise conditions, extending existing theory.

Findings

Proved existence and uniqueness of solutions under minimal assumptions.

Developed a new combination of variational techniques and a priori estimates.

Handled highly singular nonlinearities without growth restrictions.

Abstract

We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation, on solutions to regularized equations.