Well-posedness and Lewy-Stampaccia Inequalities for Nonlinear Stochastic Evolution Equations

Explore a 37-minute conference talk on well-posedness and Lewy-Stampaccia inequalities for nonlinear stochastic evolution equations, presented by Aleksandra Zimmermann at the Workshop on "Stochastic Partial Differential Equations" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI). Delve into the analysis of equations such as the stochastic p-Laplace equation with nonlinear first-order perturbations, featuring leading operators that are nonlinear, second-order pseudomonotone of Leray-Lions type. Examine the inclusion of Lipschitz continuous zero-order perturbations and multiplicative noise terms given by stochastic integrals with respect to Q-Wiener processes. Learn about the well-posedness of initial value problems with random initial data on bounded domains under homogeneous Dirichlet boundary conditions. Understand the approach of using singular perturbations with higher-order operators to obtain weak convergence, and how the theorems of Prokhorov and Skorokhod are applied to establish the existence of martingale solutions. Discover how pathwise uniqueness is derived from an L1-contraction principle, and how the Gyöngy-Krylov method is utilized to obtain stochastically strong solutions. Gain insights into how these well-posedness results form the foundation for studying variational inequalities and Lewy-Stampaccia inequalities in the context of nonlinear stochastic evolution equations.