Stochastic PDEs in critical spaces

MATHPHYS ANALYSIS SEMINAR

Critical spaces for non-linear equations are important due to scaling invariance, and in particular this plays a central role in fluid dynamics. In this talk we introduce and discuss local/global well-posedness, and blow-up criteria for stochastic parabolic evolution equations in critical spaces. Our results extend the celebrated theory of Prüss, Wilke and Simonett for deterministic PDEs. Due to the presence of noise it is unclear that a stochastic version of the theory is possible, but as we will show a suitable variation of the theory remains valid. We will also explain several features which are new in both the deterministic and stochastic framework. In particular, we discuss a new bootstrap method to prove regularization of solutions to (S)PDEs, which can also be applied in critical situations. Our theory is applicable to a large class of semilinear and quasilinear parabolic problems which includes many of the classical SPDEs. During the talk we give applications to stochastic reaction-diffusion equations and stochastic Navier-Stokes equations with gradient noise.

This is a joint work with Mark Veraar (TU Delft).