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Applied Math Colloquium: Gennady Uraltsev (Virginia)

Schrodinger, Solitons, and Harmonic Analysis!


Mathematics/Psychology : 101

Date & Time

April 14, 2023, 2:00 pm3:00 pm


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

Abstract: The nonlinear Schrödinger equation (NLS) on R^d is a
prototypical dispersive equation, i.e. it is characterized by
different 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

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