Nonlinear SPDEs and Maximal Regularity: An Extended Survey

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mn>2</mml:mn></mml:msup> </mml:math> -theory, many require a more general <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msup><mml:mi>L</mml:mi> <mml:mi>p</mml:mi></mml:msup> <mml:mrow><mml:mo>(</mml:mo> <mml:msup><mml:mi>L</mml:mi> <mml:mi>q</mml:mi></mml:msup> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> -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.