Nonlinear Stochastic parabolic partial differential equations with a monotone operator of the Ladyzenskaya-Smagorinsky type, driven by a Levy noise

TL;DR

This paper proves the global existence of solutions for nonlinear stochastic PDEs with monotone operators of Ladyzenskaya-Smagorinsky type driven by Levy noise, using approximation methods beyond classical Galerkin techniques.

Contribution

It introduces a novel approach combining operator approximations and martingale solution theory for complex stochastic PDEs with Levy noise.

Findings

Established global existence of martingale solutions.

Extended solution theory to Levy-driven nonlinear PDEs.

Developed approximation techniques for monotone operators.

Abstract

The aim of this article is to show the global existence of both martingale and pathwise solutions of stochastic equations with a monotone operator, of the Ladyzenskaya-Smagorinsky type, driven by a general Levy noise. The classical approach based on using directly the Galerkin approximation is not valid. Instead, our approach is based on using appropriate approximations for the monotone operator, Galerkin approximations and on the theory of martingale solutions.