Applied Math seminar

Speaker: Aynur Bulut
University of Michigan

Title: Random data Cauchy problems for nonlinear Schr\”odinger and wave equations

Abstract: We will discuss recent progress on probabilistic local and global well-posedness results for the Nonlinear Schrödinger and Nonlinear Wave equations. In these problems one considers randomly chosen initial data, distributed as a Gaussian process, and with low regularity properties (supercritical with respect to the scaling of the nonlinearity). In particular, our data belong almost surely to the ill-posed regime for these problems, and probabilistic considerations are therefore essential. Tools used in the approach include sharp a priori bounds for the nonlinear evolutions and associated linearizations, algebraic structure arising from the Hamiltonian nature of the problems, and careful analysis of frequency interactions.