Almost sure local well-posedness of Nonlinear Schrödinger equation

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 $H^\varepsilon$ for any $\varepsilon > 0$, compared to the known results $\varepsilon > 1/6$ . 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), Itamar Oliviera (University of Birmingham), and Gennady Uraltsev (University of Virginia, University of Arkansas).

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsoptimization and control

Audience: researchers in the topic

Potomac region PDE seminar

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