28 avril 2021

Mercredi 28 avril 10:30-11:30 Zihui He (Karlsruhe)
Solvability of the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient

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Lieu : https://bbb.imo.universite-paris-saclay.fr/b/nic-jog-f24-z6j

Résumé : We will show the existence and some regularity properties of (a class of) weak solutions to the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient. To establish these results, we analyze a fourth-order nonlinear elliptic equation for the stream functions. The density function and the viscosity coefficient can have large variations.
We will also give some explicit solutions for the parallel, concentric, and radial flows with piecewise constant viscosity coefficients, and their regularity properties will be discussed. This talk is based on joint work with X. Liao.

Solvability of the two-dimensional stationary incompressible inhomogeneous Navier–Stokes equations with variable viscosity coefficient  Version PDF

Mercredi 28 avril 16:00-17:00 Nicolas Camps (LMO)
Generic dynamics for Nonlinear Schrödinger equation

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Lieu : en ligne

Résumé : I will be interested in some « generic » large-time behavior of solutions to the nonlinear Schrödinger equation. Solutions to this partial differential equation display some rich dynamics resulting from the interplay between dispersion and nonlinear interactions.
The aim of this talk is to show how tools coming from analysis and probability theory can be combined to understand these dynamics from a mathematical perspective.
First, I will present the notion of dispersion and its features. I will also give some insights on the emergence of structures resulting from nonlinear interactions and that can easily be observed in Nature.
Then, after giving you the harmonic analysis toolbox used to quantify dispersion and nonlinear interactions, I will express some expected long term dynamics such as scattering and the emergence of soliton solutions in a mathematical formalism.
In the second part of the talk, we will see how probability theory is used to go further in the analysis of the Schrödinger equation by constructing measures on the phase space. On the support of these measures, the dynamic of the flow is well-defined and possibly well-understood depending on the context. This active domain of research, called « probabilistic Cauchy theory », was initiated by Jean Bourgain 30 years ago.

Notes de dernières minutes : TBA

Generic dynamics for Nonlinear Schrödinger equation  Version PDF