## Applied Math Colloquium: Gennady Uraltsev (Virginia)

#### Schrodinger, Solitons, and Harmonic Analysis!

Friday, April 14, 2023 · 2 - 3 PM

Title: Almost sure local well-posedness of the nonlinear Schrödinger equation

using directional estimatesAbstract: The nonlinear Schrödinger equation (NLS) on R^d is a

prototypical dispersive equation, i.e. it is characterized bydifferent frequencies travelling 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 setting.

Deterministic local well-posedness for the NLS is well-understood: it

holds only for initial data with regularity above a certain

energy-critical threshold.

We show how directional behavior combined with multilinear tree

expansions for the solutions provide the framework to deal with

randomized initial data in any positive regularity for the cubic power

nonlinearity in dimension three. This approach improves our

understanding of the structure of the solutions and sheds light on NLS

in dimensions greater than or equal to three and potentially with other power

nonlinearities.

**Almost sure local well-posedness for cubic nonlinear Schrodinger equation with higher order operators**, with: J-B. Casteras, J. Földes.