Sharp estimates for pseudodifferential operators with symbols of limited smoothness and commutators

We consider here pseudo-differential operators whose symbol $\sigma(x,\xi)$ is not infinitely smooth with respect to $x$. Decomposing such symbols into four -sometimes five- components and using tools of paradifferential calculus, we derive sharp estimates on the action of such pseudo-differential operators on Sobolev spaces and give explicit expressions for their operator norm in terms of the symbol $\sigma(x,\xi)$. We also study commutator estimates involving such operators, and generalize or improve the so-called Kato-Ponce and Calderon-Coifman-Meyer estimates in various ways.Comment: Accepted for publication in Journal of Functional Analysi

Derivation of asymptotic two-dimensional time-dependent equations for ocean wave propagation

A general method for the derivation of asymptotic nonlinear shallow water and deep water models is presented. Starting from a general dimensionless version of the water-wave equations, we reduce the problem to a system of two equations on the surface elevation and the velocity potential at the free surface. These equations involve a Dirichlet-Neumann operator and we show that all the asymptotic models can be recovered by a simple asymptotic expansion of this operator, in function of the shallowness parameter (shallow water limit) or the steepness parameter (deep water limit). Based on this method, a new two-dimensional fully dispersive model for small wave steepness is also derived, which extends to uneven bottom the approach developed by Matsuno \cite{matsuno3} and Choi \cite{choi}. This model is still valid in shallow water but with less precision than what can be achieved with Green-Naghdi model, when fully nonlinear waves are considered. The combination, or the coupling, of the new fully dispersive equations with the fully nonlinear shallow water Green-Naghdi equations represents a relevant model for describing ocean wave propagation from deep to shallow waters

Mathematics for 2d Interfaces

We present here a survey of recent results concerning the mathematical analysis of instabilities of the interface between two incompressible, non viscous, fluids of constant density and vorticity concentrated on the interface. This configuration includes the so-called Kelvin-Helmholtz (the two densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and the water waves (one of the densities is zero) problems. After a brief review of results concerning strong and weak solutions of the Euler equation, we derive interface equations (such as the Birkhoff-Rott equation) that describe the motion of the interface. A linear analysis allows us to exhibit the main features of these equations (such as ellipticity properties); the consequences for the full, non linear, equations are then described. In particular, the solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily analytic if they are above a certain threshold of regularity (a consequence is the illposedness of the initial value problem in a non analytic framework). We also say a few words on the phenomena that may occur below this regularity threshold. Finally, special attention is given to the water waves problem, which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor configurations. Most of the results presented here are in 2d (the interface has dimension one), but we give a brief description of similarities and differences in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese

A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations

We study the well-posedness of the initial value problem for a wide class of singular evolution equations. We prove a general well-posedness theorem under three assumptions easy to check: the first controls the singular part of the equation, the second the behavior of the nonlinearities, and the third one assumes that an energy estimate can be found for the linearized system. We allow losses of derivatives in this energy estimate and therefore construct a solution by a Nash-Moser iterative scheme. As an application to this general theorem, we prove the well-posedness of the Serre and Green-Naghdi equation and discuss the problem of their validity as asymptotic models for the water-waves equations