2019 Analysis of Partial Differential Equations

2019 Analysis of Partial Differential Equations


Title: Almost sure global existence and scattering for the one dimensional Schrdinger equation (I),(II) (III)

Speaker: Professor Nicolas Burq

From:Université Paris-Sud , France

Time: 2:40-4:40, march 5
3:00-5:00, March 8
3:00-5:00, March 10
In this mini course, I will give an introduction to the theory of random data non linear PDE’s, on one of the most simple example of dispersive PDE’s: the one dimensional non linear Schroedinger 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 (non trivial) evolution by the linear flow of the Schroedinger equation
$$ (i\partial_t + \partial_x^2 ) u =0, (t, x) \in \mathbb{R}\times \mathbb{R}$$
These mesures are essentially supported on $L^2( \mathbb{R})$.
Then I will show that the non linear 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 non linear 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 (non scattering 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