Nonlinear SPDEs and Maximal Regularity: An Extended Survey

TL;DR

This survey reviews recent advances in the well-posedness and regularity of nonlinear stochastic partial differential equations, emphasizing maximal regularity, critical spaces, and applications to fluid dynamics and reaction-diffusion systems.

Contribution

It introduces an abstract framework for critical spaces in nonlinear SPDEs, unifies previous results, and presents new criteria for blow-up and regularization, with applications to key models like Navier-Stokes.

Findings

Established new blow-up criteria for stochastic Navier-Stokes equations.

Unified and refined maximal regularity results for nonlinear SPDEs.

Applied abstract theory to concrete models including quasi-geostrophic and reaction-diffusion systems.

Abstract

In this survey, we provide an in-depth exposition of our recent results on the well-posedness theory for stochastic evolution equations, employing maximal regularity techniques. The core of our approach is an abstract notion of critical spaces, which, when applied to nonlinear SPDEs, coincides with the concept of scaling-invariant spaces. This framework leads to several sharp blow-up criteria and enables one to obtain instantaneous regularization results. Additionally, we refine and unify our previous results, while also presenting several new contributions. In the second part of the survey, we apply the abstract results to several concrete SPDEs. In particular, we give applications to stochastic perturbations of quasi-geostrophic equations, Navier-Stokes equations, and reaction-diffusion systems (including Allen--Cahn, Cahn--Hilliard and Lotka--Volterra models). Moreover, for the Navier--Stokes equations, we establish new Serrin-type blow-up criteria. While some applications are addressed using $L^2$-theory, many require a more general $L^p(L^q)$-framework. In the final section, we outline several open problems, covering both abstract aspects of stochastic evolution equations, and concrete questions in the study of linear and nonlinear SPDEs.