Analysis Seminar

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Probabilistic local well-posedness and scattering for the 4D cubic NLS

Speaker: Jonas LUEHRMANN, Johns Hopkins U.

Location: Warren Weaver Hall 1302

Date: Thursday, September 13, 2018, 11 a.m.

Synopsis:

We consider the Cauchy problem for the defocusing cubic nonlinear
Schrodinger equation (NLS) in four space dimensions. It is known that
for initial data at energy regularity, the solutions exist globally in
time and scatter. However, the problem is ill-posed for initial data at
super-critical regularity, i.e. for regularities below the energy
regularity.

In this talk we study the super-critical data regime for this Cauchy
problem from a probabilistic point of view, using a randomization
procedure that is based on a unit-scale decomposition of frequency space.
In the first part of the talk we will explain how the problem of
establishing almost sure local existence for the cubic NLS for such
random data has some features in common with proving local existence for
a derivative NLS equation. Our method is inspired by the local smoothing
estimates and functional frameworks from the Schrodinger maps
literature. In the second part of the talk we will turn to the long-time
dynamics of the solutions. We will present a conditional almost sure
scattering result and an almost sure scattering result for randomized
radial data.

This is joint work with Ben Dodson and Dana Mendelson.