中科院数学与系统科学研究院

华罗庚数学科学中心

学术报告

报告人：Prof. Nicolas Burq(Université Paris-Sud , France)

题  目：Almost sure global existence and scattering for the one dimensional Schr？dinger equation (I)(II) (III)

时  间：2019.03.05（星期二）, 14:40-16:40

2019.03.08（星期五）, 15:00-17:00

2019.03.10（星期日）, 15:00-17:00

地  点：数学院南楼N913室

摘  要：

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 Schr？dinger 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 Schr？dinger 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.