Strong solutions to SPDEs with monotone drift in divergence form

TL;DR

This paper establishes the existence and uniqueness of strong solutions for a broad class of nonlinear stochastic PDEs with divergence form drift, accommodating highly nonlinear growth conditions without extra assumptions.

Contribution

It introduces a novel approach combining a priori estimates and weak compactness to handle fully nonlinear SPDEs with super-polynomial growth in the drift.

Findings

Proved well-posedness for a class of nonlinear SPDEs

Allowed nonlinear drift growth faster than polynomial

Established continuous dependence on initial data

Abstract

We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. A well-posedness result is obtained through a combination of a priori estimates on regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, and weak compactness arguments. The result is essentially sharp, in the sense that no extra hypotheses are needed, bar continuity of the nonlinear function in the drift, with respect to the deterministic theory.