Journées Équations aux dérivées partielles

We review some recent results in which we develop a new method for proving global unique continuation for some conservative PDEs. The main tool is to prove some global propagation of analyticity. We first present some known results on the subject. Then, we sketch the abstract method we use, which relies on some finite determining mode property. We give applications to semilinear wave, plates and …

We address the question of the large-time behavior of solutions to reaction-diffusion equations in periodic media. We start with the description of the asymptotic shape of the invasion set, which is characterized by the Freidlin–Gärtner formula. We outline a proof of the formula that holds true for general types of reaction terms. We then present some recent results, obtained in collaboration wit…

This paper introduces the a-contraction theory for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> stability of discontinuous solutions to conservation laws and their inviscid limits. It can be viewed as an extension of the Weak–Strong stability principle, first introduced by Dafermos and DiPerna, to the stabil…

We present recent results on probabilistic well-posedness of the two dimensional NLS, posed on the sphere. These results deal with low regularity solutions. The construction of such solutions is beyond the scope of applicability of the deterministic methods of Burq-Gérard-Tzvetkov developed between 2000 and 2004.

This note is based on a talk given by the first author at the conference Journées Équations aux dérivées partielles 2025. We consider the intermediate long wave equation (ILW), modeling the internal wave propagation of the interface in a stratified fluid of finite depth, connecting the deep-water regime (= the BO regime) and the shallow-water regime (= the KdV regime). Exploiting the complete int…

Here, we are interested in proving observability results for the Stokes system and for various boundary conditions. In particular, we address the Dirichlet, Navier, and Neumann conditions. After reviewing approaches to the same type of problem for parabolic equations and a brief review of the literature for the Stokes problem, we present the main results and some ideas from the proof.

We briefly review the main results from Germain–La–Menegaki (2024) and Escobedo–Germain–La–Menegaki (2025) which (i) proves nonlinear stability of the non-singular Rayleigh–Jeans equilibria for the kinetic wave equation associated with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -FPUT chain and (ii) classifies the entropy maximizers for a more gene…

We present two results of enhanced long-time stability for the nonlinear Schrödinger equation posed on rescaled tori with Diophantine properties. This proceeding for the conference “Journées Équations aux dérivées partielles” is the opportunity to give a glimpse on modern methods at the interface of PDEs and classical mechanics that are currently being developed to study the long-time dynamics of…

We review asymptotic stability of solitary waves for nonlinear dispersive equations set on the line. Our focus is threefold: first, the nonlinear Schrödinger equation; second, the notion of full asymptotic stability (which states that perturbations of a solitary wave decompose globally into a solitary wave and a decaying solution); and third, spectral methods. Besides this focus, we summarize the…