Joint BCAM-UPV/EHU Analysis and PDE seminar: Almost sure local well-posedness of the nonlinear Schrödinger equation using directional estimates

The nonlinear Schrödinger equation (NLS) on ℝd​​ is a prototypical dispersive equation, i.e. it is characterized by different frequencies traveling at different velocities and by the lack of a  smoothing effect over time.

Furthermore, NLS is a prototypical infinite-dimensional Hamiltonian system. Constructing an invariant measure for the NLS flow is a natural, albeit very difficult problem. It requires showing local well-posedness in low regularity spaces, in an appropriate probabilistic sense.

Deterministic local well-posedness for the NLS is well-understood: it holds only for initial data with regularity above a certain scaling-critical threshold.

We show how directional behavior of solutions can be used to obtain better interaction estimates to control the non-linearity. Combined with multilinear tree expansions for the solutions, this provides the framework to deal with randomized initial data in any positive regularity for the cubic power nonlinearity in dimension 3​​. This approach improves our understanding of the structure of the solutions and sheds light on NLS in dimensions d ≥ 3​​ and potentially with other power nonlinearities.