Parabolic SPDEs in critical spaces
Critical spaces for nonlinear PDEs are important due to scaling invariance, and in particular this plays a central role in uid dynamics. In this talk we introduce and discuss local/global well-posedness and blow-up criteria for stochastic partial dierential equations (SPDEs) in critical spaces. Our results extend the celebrated theory of Pruss, 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. During the talk we give applications to stochastic reaction-diusion equations and stochastic Navier-Stokes equations with gradient noise.
This is a joint work with Mark Veraar (TU Delft).