SPDE in Hilbert Space with Locally Monotone Coefficients

TL;DR

This paper establishes the existence and uniqueness of strong solutions for a broad class of stochastic partial differential equations in Hilbert spaces with locally monotone coefficients, extending classical results to more general settings.

Contribution

It generalizes the classical Krylov-Rozovskii result to locally monotone coefficients, enabling analysis of various complex SPDEs with non-monotone perturbations.

Findings

Proves existence and uniqueness of solutions for SPDEs with locally monotone coefficients.

Applicable to multiple types of SPDEs including reaction-diffusion, Burgers, Navier-Stokes, p-Laplace, and porous media equations.

Extends classical theory to more general, non-monotone coefficient cases.

Abstract

In this paper we prove the existence and uniqueness of strong solutions for SPDE in Hilbert space with locally monotone coefficients, which is a generalization of the classical result of Krylov and Rozovskii for monotone coefficients. Our main result can be applied to different types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equation, stochastic 2-D Navier-Stokes equation, stochastic $p$-Laplace equation and stochastic porous media equation with some non-monotone perturbations.