Pathwise mild solutions for quasilinear stochastic partial differential equations

TL;DR

This paper establishes the existence of local-in-time mild solutions for a broad class of quasilinear stochastic PDEs, combining deterministic PDE theory with evolution semigroup methods, and applies to models like SKT.

Contribution

It introduces a novel fixed point approach to prove local solutions for quasilinear SPDEs, including cross-diffusion systems, expanding the theoretical understanding of their regularity.

Findings

Existence of local-in-time mild solutions for broad quasilinear SPDEs

Application of theory to the SKT model

Examples of blow-up and ill-posed operators

Abstract

Stochastic partial differential equations (SPDEs) have become a key modelling tool in applications. Yet, there are many classes of SPDEs, where the existence and regularity theory for solutions is not completely developed. Here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear Cauchy problems, including - among others - cross-diffusion systems as a key application. Our solutions are local-in-time and are derived via a fixed point argument in suitable function spaces. The key idea is to combine the classical theory of deterministic quasilinear parabolic partial differential equations (PDEs) with recent theory of evolution semigroups. We also show, how to apply our theory to the Shigesada-Kawasaki-Teramoto (SKT) model. Furthermore, we provide examples of blow-up and ill-posed operators, which can occur after finite-time showing that solutions can only be local-in-time for general quasilinear SPDEs, while they might be global-in-time for special subclasses of problems.