Stochastic PDEs and Lack of Regularity

We review results on the existence and uniqueness for a surface growth model with or without space--time white noise. If the surface is a graph, then this model has striking similarities to the three dimensional Navier-Stokes equations in terms of energy estimates and scaling properties, and in both models the question of uniqueness of global weak solutions remains open. In the physically relevant dimension d=2 and with the physically relevant space--time white noise driving the equation, the direct fixed-point argument for mild solutions fails, as there is not sufficient regularity for the stochastic forcing. The situation is the simplest case where the method of regularity structures introduced by Martin Hairer can be applied, although we follow here a significantly simpler approach to highlight the key problems. Using spectral Galerkin method or any other type of regularization of the noise, one can give a rigorous meaning to the stochastic PDE and show existence and uniqueness of local solutions in that setting. Moreover, several types of regularization seem to yield all the same solution. We finally comment briefly on possible blow up phenomena and show with a simple argument that many complex-valued solutions actually do blow up in finite time. This shows that energy estimates alone are not enough to verify global uniqueness of solutions. Results in this direction are known already for the 3D-Navier Stokes by Li and Sinai, treating complex valued solutions, and more recently by Tao by constructing an equation of Navier-Stokes type with blow up.