LECTURES ON NONLINEAR DISPERSIVE EQUATIONS I

CONTENTS J. Bona Derivation and some fundamental properties of nonlinear dispersive waves equations F. Planchon Schr\"odinger equations with variable coecients P. Rapha\"el On the blow up phenomenon for the L^2 critical non linear Schrodinger Equatio

On decaying properties of nonlinear Schr\"odinger equations

In this paper we discuss quantitative (pointwise) decay estimates for solutions to the 3D cubic defocusing Nonlinear Schr\"odinger equation with various initial data, deterministic and random. We show that nonlinear solutions enjoy the same decay rate as the linear ones. The regularity assumption on the initial data is much lower than in previous results (see \cite{fan2021decay} and the references therein) and moreover we quantify the decay, which is another novelty of this work. Furthermore, we show that the (physical) randomization of the initial data can be used to replace the $L^1$-data assumption (see \cite{fan2022note} for the necessity of the $L^1$-data assumption). At last, we note that this method can be also applied to derive decay estimates for other nonlinear dispersive equations.Comment: 24 pages. Comments are welcome

On a Bubble algorithm for the cubic Nonlinear Schr{\"o}dinger equation

Based on very recent and promising ideas, stemming from the use of bubbles, we discuss an algorithm for the numerical simulation of the cubic nonlinear Schr{\"o}dinger equation with harmonic potential (cNLS) in any dimension, that could easily be extended to other polynomial nonlinearities. This algorithm consists in discretizing the initial function as a sum of modulated complex gaussian functions (the bubbles), each one having its own set of parameters, and then updating the parameters according to cNLS. Numerically, we solve exactly the linear part of the equation and use the Dirac-Frenkel-MacLachlan principle to approximate the nonlinear part. We then obtain a grid free algorithm in any dimension whose efficiency compared with spectral methods is illustrated by numerical examples

Infinite energy solutions to the Navier-Stokes equations in the half-space and applications

This short note serves as an introduction to the papers arXiv:1711.01651 and arXiv:1711.04486 by Maekawa, Miura and Prange. These two works deal with the existence of mild solutions on the one hand and local energy weak solutions on the other hand to the Navier-Stokes equations in the half-space $\R^3_+$. We emphasize a concentration result for (sub)critical norms near a potential singularity. The contents of these notes were presented during the X-EDP seminar at IH\'ES in October 2017.Comment: 18 pages. This is a review article. This note will be published in the proceedings of the "S\'eminaire Laurent Schwartz-EDP et applications

On heat equations associated with fractional harmonic oscillators

We establish some fixed-time decay estimates in Lebesgue spaces for the fractional heat propagator e−t Hβ , t,β > 0, associated with the harmonic oscillator H = − + |x| 2. We then prove some local and global wellposedness results for nonlinear fractional heat equation