Well-posedness for cubic Nonlinear Schrodinger equation with randomized initial conditions

When

November 3, 2023 | 2:30 p.m. – 3:30 p.m.
Refreshments provided

Where

University Hall 4002

Speaker

Juraj Foldes

Abstract

During the talk, we will discuss the local solutions of the super-critical cubic Schrödinger equation (NLS) on the whole space with general differential operator. Although such a problem is known to be ill-posed, we show that the random initial data yield almost sure local well-posedness. Using estimates in directional spaces, we improve and extend known results for the standard Schrödinger equation in various directions: higher dimensions, more general operators, weaker regularity assumptions on the initial conditions. In particular, we show that in 3D, the classical cubic NLS is stochastically, locally well-posed for any initial data with regularity in H^a for any a>0 compared to the known results a>1/6 . The proofs are based on precise estimates in frequency space using various tools from Harmonic analysis. This is a joint project with Jean-Baptise Casteras (Lisbon University) and Gennady Uraltsev (University of Virginia, University of Arkansas).