RG Analysis and Partial Differential Equations

Graduate seminar on Advanced topics in PDE

Organizers

Schedule

This seminar takes place regularly on Fridays, at 14:15. The seminar will take place in person in SR 0.011. Please join the pdg-l mailing list for further information.

March 28 - Tetsu Mizumachi (Hiroshima)

Title: On transverse stability of line solitary waves for 2D long wave models

Abstract:

In this talk, I will explain modulations of 1-solitons of the KP-II equation and how the idea works for elastic 2-line solitons or resonant solitons. I will also mention transverse stability of line solitary waves for the Benney-Luke equation and the 2D-Toda equation.

April 11 - Nikolay Tzvetkov (Lyon)

Title: Probabilistic well-posedness for the nonlinear Schrödinger equation on the two dimensional sphere

Abstract:

It is known that the minimal Sobolev regularity needed for the semi-linear, local well-posedness of the non linear Schrödinger equation, posed on a two dimensional domain depends heavily on the geometry of the domain. In this talk we will observe a similar phenomenon in the study of the probabilistic well-posedness. This is a joint work with Burq-Camps-Sun.

April 18 - easter friday

April 25 - Fred Lin (Bonn)

Title: On a smoothing inequality related to triangular Hilbert transform along general curve

Abstract:

A kind of smoothing inequality plays a central role in proving the boundedness of the triangular Hilbert transform along curve, its maximal variant and the associated Roth type problem. I will survey the recent developement in this field, then sketch the proof of such smoothing inequality. This is joint work with Martin Hsu.

May 02 - Tadahiro Oh (Edinburgh)

Title: Fourier restriction norm method adapted to controlled paths

Abstract:

Over the last decade, there has been a significant development in the study of stochastic dispersive PDEs, broadly interpreted with random initial data and/or additive stochastic forcing, where the difficulty comes from roughness in spatial regularity. In this talk, I consider pathwise well-posedness of stochastic dispersive PDEs with multiplicative noises, whose Ito solutions were constructed in 80's for the wave case and in 90's for the Schrödinger case, and present the first results on pathwise well-posedness for stochastic nonlinear wave equations (SNLW) and stochastic nonlinear Schrödinger equations (SNLS). The main challenge of this problem comes from the deficiency of temporal regularities. We overcome this issue by building a unified framework for controlled rough paths and the Fourier restriction norm method.