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Nonlinear SPDEs and Maximal Regularity: An Extended Survey.

Antonio Agresti^1,2, Mark Veraar³

¹Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box , 5031 2600 GA Delft, The Netherlands.

Nonlinear Differential Equations and Applications : Nodea

|September 4, 2025

Summary

This survey explores well-posedness for stochastic evolution equations using maximal regularity. It introduces critical spaces for sharp blow-up criteria and regularization in nonlinear stochastic partial differential equations (SPDEs).

Keywords:

Allen–Cahn equations Blow-up criteria Cahn–Hilliard equation Critical spaces Fluid dynamic models Local and global well-posedness Navier-Stokes equations Parabolic equations Quasi-geostrophic equations Reaction-diffusion equations Regularization Serrin criteria Stochastic evolution equations Stochastic maximal regularity Stochastic partial differential equations Variational setting

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Area of Science:

Stochastic Analysis
Partial Differential Equations
Mathematical Physics

Background:

The well-posedness of stochastic evolution equations is crucial for modeling complex systems.
Maximal regularity techniques offer powerful tools for analyzing these equations.
Existing theories often lack sharp criteria for blow-up and instantaneous regularization.

Purpose of the Study:

To present recent advancements in the well-posedness theory of stochastic evolution equations.
To introduce and apply a novel framework based on critical spaces.
To refine, unify, and extend previous results on nonlinear stochastic partial differential equations (SPDEs).

Main Methods:

Employing maximal regularity techniques for stochastic evolution equations.
Developing an abstract notion of critical spaces, coinciding with scaling-invariant spaces for nonlinear SPDEs.
Applying the abstract framework to specific SPDEs, including Navier-Stokes and reaction-diffusion systems.

Main Results:

Established sharp blow-up criteria and instantaneous regularization results for nonlinear SPDEs.
Provided unified and refined analyses of existing theories.
Derived new Serrin-type blow-up criteria for the Navier-Stokes equations.

Conclusions:

The critical space framework provides a unified approach to well-posedness for a wide range of SPDEs.
The results advance the understanding of blow-up phenomena and regularization properties.
Identified open problems in both abstract stochastic evolution equations and concrete SPDEs.