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Nicolas Burq: Almost sure global existence and scattering for the one dimensional Schrödinger equation

Time: Tue 2019-04-02 14.00 - 15.00

Location: Seminar Hall Kuskvillan, Institut Mittag-Leffler

Lecturer: Nicolas Burq, Université Paris-Sud

Abstract: In this talk, I will present results on one of the most simple example of dispersive PDE’s: the one dimensional nonlinear Schrödinger equation on the line \(\mathbb{R}\),
\((i \partial_t + \partial_x^2) u + |u|^{p-1} u =0 \)

More precisely, I will define essentially on \(L^2 (\mathbb {R})\), the space of initial data, probability measures for which I can describe the (nontrivial) evolution by the linear flow of the Schrödinger equation

\((i\partial_t+\partial _x2)u=0, (t,x) \in\mathbb{R} \times \mathbb{R}.\) 

These mesures are essentially supported on \(L^2( \mathbb{R})\).

Then I will show that the nonlinear equation

\((i\partial_t + \partial_x^2 ) u - |u|^{p-1} u =0, (t, x) \in \mathbb{R}\times \mathbb{R},\)
is globally well posed on the support of the measure.
Finally I will describe precisely the evolution by the nonlinear flow of the measure defined previously in terms of the linear evolution (quasi-invariance). Lastly I will show how this description gives
-- (Almost sure) Global well posedness for p>1 and asymptotic behaviour of solutions (nonscattering type),
 -- (Almost sure) scattering for p>3.

Belongs to: Department of Mathematics
Last changed: Mar 29, 2019