Programme

Title: Well posedness of ODE's for nonsmooth velocities and in non Euclidean ambient spaces: a survey. Abstract: In this lecture I will make a survey of the theory of regular lagrangian flows, that allows to prove existence, uniqueness and stability for ODE's associated to vector fields with little regularity, beyond the classical Cauchy-Lipschitz theory. Initiated in a minal work by Di Perna and Lions who dealt with Sobolev vector fields, the RLF axiomatization provides natural links with Probability. More recent developments, including the regularity of the flow map, counterexamples and the case when even the ambient space is not smooth, will be discussed.

Title: Global solutions for 3D gravity water waves. Abstract: The aim of this talk is to present work in progress on the problem of local and global well-posedness for gravity water waves in the small-data, low-regularity regime, for fluids of infinite depth and infinite extent in spatial dimensions n≥3. More specifically, our ongoing work seeks to simultaneously achieve substantial improvements in both the local and global theory for these equations, reaching nearly optimal Sobolev regularity thresholds. On the local side, we obtain sharp improvements in the well-posedness theory for the gravity wave problem: up to scaling regularity in dimensions n≥4, and within 1/12 of a derivative above scaling in dimension n=3.On the global side, we establish global well-posedness for small initial data in a critical Besov space, without requiring spatial localization assumptions. These results fit within the framework of the “Nonlocalized Data Global Well-posedness Conjecture” , jointly propose

Title: Soliton and breather resolution for the cubic Szegö flow on the line. Abstract: We investigate the long--time behaviour of the solutions of the cubic Szegö equation on the line in the Sobolev space $H^{1/2}(\R)$. We prove that, for every datum of which the Lax operator has simple positive spectrum, the solution asymptotically decouples as an infinite sum of traveling quasi-periodic breather solutions. Under an additional generic condition on the data, we prove that these traveling breather solutions are in fact soliton solutions, leading to a soliton resolution theorem. This talk is based on a recent joint work with Sandrine Grellier.