Wong–Zakai Approximation for a Class of SPDEs with Fully Local Monotone Coefficients and Its Application

Abstract

In this article, we establish the Wong–Zakai approximation result for a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients perturbed by a multiplicative Wiener noise. This class of SPDEs encompasses various fluid dynamic models and also includes quasi-linear SPDEs, the convection–diffusion equation, the Cahn–Hilliard equation, and the two-dimensional liquid crystal model. It has been established that the class of SPDEs in question is well-posed, however, the existence of a unique solution to the associated approximating system cannot be inferred from the solvability of the original system. We employ a Faedo–Galerkin approximation method, compactness arguments, and Prokhorov’s and Skorokhod’s representation theorems to ensure the existence of a probabilistically weak solution for the approximating system. Furthermore, we also demonstrate that the solution is pathwise unique. Moreover, the classical Yamada–Watanabe theorem allows us to conclude the existence of a probabilistically strong solution (analytically weak solution) for the approximating system. Subsequently, we establish the Wong–Zakai approximation result for a class of SPDEs with fully local monotone coefficients. We utilize the Wong–Zakai approximation to establish the topological support of the distribution of solutions to the SPDEs with fully local monotone coefficients. Finally, we explore the physically relevant stochastic fluid dynamics models that are covered by this work’s functional framework.