中国科学院数学与系统科学研究院

数学研究所

数学科学全国重点实验室

学术报告

偏微分方程研讨班

Speaker: Professor Nicolas Burq

(Université Paris-Saclay)

Inviter: 张平 院士

Language: English

Title:Nonlinear interpolation and the flow map of quasilinear equations

Time&Venue: 2025年6月18日（星期三）15:00-17:00

& 南楼N913

Abstract: Solving a quasilinear evolution Partial Differential Equation requires usually two quite distinct steps: first implementing an iteration scheme where some high regularity norms are bounded and second implementing a contraction scheme at some lower regularity level. This double scheme implies existence of the solution at the high regularity level and continuity of the flow (i.e. the data-to-solution map) at the low regularity level. Starting from this point many works have been dealing with recovering the continuity of the flow at the high regularity level. In this talk, I will present an abstract interpolation result which shows that actually this continuity property of the flow follows automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame {\em a priori} estimates for higher regularities and contraction for weaker norms.

I will illustrate on some simple examples (burger’s equations and nonlinear Schrödinger equations)  the robustness of the approach. This is a joint work with T.Alazard, M. Ifrim, D. Tataru and C. Zuily.

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Title:Probabilistic well-posedness and Gibbs measure evolution for the non linear 

Schr ̈odinger equation on the two dimensional sphere 

Time&Venue: 2025年6月19日（星期四）9:30-11:30

& 南楼N913

Abstract: In this talk I will present some recent work about the cubic nonlinear Schrödinger equation (NLS) with random initial data. Namely I will show how taking such random initial data can allow to beat the deterministic regularity threshold we established with P. Gérard and N. Tzvetkov in the 2000’ . I will present first the state of the art and then will explain the ideas involved to go  all the way up to the Gibb’s measure regularity threeshold (essentially $L^2$) and take benefit from this to exhibit global solutions at this level of regularity.This is a joint work with N. Camps, C. Sun and N. Tzvetkov.