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(2019.3.5)Prof. Nicolas Burq:Almost sure global existence and scattering for the one dimensional Schrodinger equation (I)(II) (III)
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Update time: 2019-03-04
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Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:

Prof. Nicolas Burq,Université Paris-Sud , France

Inviter:  
Title:
Almost sure global existence and scattering for the one dimensional Schrodinger equation (I)(II) (III)
Time & Venue:
2019.3.5 14:40-16:40;2019.3.8,10 15:00-17:00 N913
Abstract:
In this mini course, I will give an introduction to the theory of random data nonlinear PDE’s, on one of the most simple example of dispersive PDE’s: the one dimensional nonlinear Schrodinger equation on the line $\mathbb{R}$. 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 Schrodinger 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 locally 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 wil show how this description gives

1) (Almost sure) Global well posedness for p>1 and asymptotic behaviour of solutions (nonscattering type)

2) (Almost sure) scattering for p>3.”
This is based on joint works with L. Thomann and N. Tzvetkov, and more recently with L. Thomann. The prerequisite in probability for the course are essentially elementary probability theory.

 

 

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