Regularization by Nonlinear Noise for PDEs: Well-posedness and Finite Time Extinction

This work focuses on the global existence, uniqueness and the Feller property for a class of stochastic partial differential equations by adding a suitable nonlinear noise, while the corresponding deterministic equations may only have local solutions. In particular, we discover a new phenomenon that for a potentially explosive deterministic system, an appropriate intervention of nonlinear noise can not only prevent blow-up but also lead to the finite time extinction of the associated stochastic system. Furthermore, our main results have broad applications, including not only all locally monotone stochastic equations in the variational framework (cf. LR2, LR1, RSZ1), but also several new models such as stochastic p-Laplace equations with heat sources, stochastic 3D Navier-Stokes equations, stochastic quasi-geostrophic equations and stochastic surface growth models, which provide positive answers to some longstanding open problems in this field.