Global Dynamics of the 2d NLS with White Noise Potential and Generic Polynomial Nonlinearity

Using an approach introduced by Hairer–Labbé we construct a unique global dynamics for the NLS on T2 with a white noise potential and an arbitrary polynomial nonlinearity. We build the solutions as a limit of classical solutions (up to a phase shift) of the same equation with smoothed potentials. This is an improvement on previous contributions of us and Debussche–Weber dealing with quartic nonlinearities and cubic nonlinearities respectively. The main new ingredient are space–time estimates for the approximate nonlinear solutions exploiting the time averaging effect for dispersive equations (the previous works were based only on fixed time spatial estimates).