Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators

Abstract

UDC 519.21 We study an important class of stochastic nonlinear evolution problems with pseudomonotone elliptic parts and establish the existence of probabilistic weak (or martingale) solutions. No solvability theory has been developed so far for these equations despite numerous works involving various generalizations of the monotonicity condition. Key to our work is a sign result for the Ito differential of an approximate solution that we establish, as well as several compactness results of the analytic and probabilistic nature, and a characterization of pseudomonotone operators due to F. E. Browder.