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

Spectral Methods in Mathematical Physics

02 April 14:00 - 15:00

Nicolas Burq - Université Paris-Sud

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.
Søren Fournais
Aarhus University
Rupert Frank
LMU Munich
Benjamin Schlein
University of Zurich, UZH
Simone Warzel
TU Munich


Rupert Frank


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