Thomas Alazard

On the water-wave equations with surface tension

The purpose of this article is to clarify the Cauchy theory of the water waves equations as well in terms of regularity indexes for the initial conditions as for the smoothness of the bottom of the domain (namely no regularity assumption is assumed on the bottom). Our main result is that, following the approach developped by T. Alazard and G. Metivier in [1], after suitable paralinearizations, the system can be arranged into an explicit symmetric system of Schrodinger type. We then show that the smoothing effect for the (one dimensional) surface tension water waves proved by H. Christianson, V. M. Hur, and G. Staffilani in [9], is in fact a rather direct consequence of this reduction, which allows also to lower the regularity indexes of the initial data, and to obtain the natural weights in the estimates.

Low mach number flows and combustion

We prove uniform existence results for the full Navier–Stokes equations for time intervals which are independent of the Mach number, the Reynolds number, and the Peclet number. We consider general equations of state and give an application for the low Mach number limit combustion problem introduced by Majda in [Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer‐Verlag, New York, 1984].

Loss of regularity for supercritical nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse, namely that there is a loss of regularity, in the spirit of the result due to G. Lebeau in the case of the wave equation. As a consequence, the Cauchy problem for energy-supercritical equations is not well-posed in the sense of Hadamard. We reduce the problem to a supercritical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functional à la Brenier.

Supercritical geometric optics for nonlinear Schrodinger equations

We consider the small time semi-classical limit for nonlinear Schrödinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y. Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows.

Gravity Capillary Standing Water Waves

The paper deals with the 2D gravity-capillary water waves equations in their Hamiltonian formulation, addressing the question of the nonlinear interaction of a plane wave with its reflection off a vertical wall. The main result is the construction of small amplitude, standing (namely periodic in time and space, and not travelling) solutions of Sobolev regularity, for almost all values of the surface tension coefficient, and for a large set of time-frequencies. This is an existence result for a quasi-linear, Hamiltonian, reversible system of two autonomous pseudo-PDEs with small divisors. The proof is a combination of different techniques, such as a Nash–Moser scheme, microlocal analysis and bifurcation analysis.

WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity

We consider the semi-classical limit for the Gross-Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a point-wise description of the wave function as the Planck constant goes to zero, so long as no singularity appears in the limit system. For a cubic-quintic nonlinearity, we show that working with analytic data may be necessary and sufficient to obtain a similar result.

Semi-classical limit of Schrödinger–Poisson equations in space dimension n⩾3

Abstract We prove the existence of solutions to the Schrodinger–Poisson system on a time interval independent of the Planck constant, when the doping profile does not necessarily decrease at infinity, in the presence of a subquadratic external potential. The lack of integrability of the doping profile is resolved by working in Zhidkov spaces, in space dimension at least three. We infer that the main quadratic quantities (position density and modified momentum density) converge strongly as the Planck constant goes to zero. When the doping profile is integrable, we prove pointwise convergence.

The Water-Wave Equations: From Zakharov to Euler

Starting from the Zakharov/Craig–Sulem formulation of the water-wave equations, we prove that one can define a pressure term and hence obtain a solution of the classical Euler equations. It is proved that these results hold in rough domains, under minimal assumptions on the regularity to ensure, in terms of Sobolev spaces, that the solutions are C 1.

Boundary observability of gravity water waves

Consider a three-dimensional fluid in a rectangular tank, bounded by a flat bottom, vertical walls and a free surface evolving under the influence of gravity. We prove that one can estimate its energy by looking only at the motion of the points of contact between the free surface and the vertical walls. The proof relies on the multiplier technique, the Craig-Sulem-Zakharov formulation of the water-wave problem, a Pohozaev identity for the Dirichlet to Neumann operator, previous results about the Cauchy problem and computations inspired by the analysis done by Benjamin and Olver of the conservation laws for water waves.

Some Recent Asymptotic Results in Fluid Mechanics

The general equations of fluid mechanics are the law of mass conservation, the Navier-Stokes equation, the law of energy conservation and the laws of thermodynamics. These equations are merely written in this generality. Instead, one often prefers simplified forms. To obtain reduced systems, the easiest route is to introduce dimensionless numbers which quantify the importance of various physical processes. Many recent works are devoted to the study of the classical solutions when such a dimensionless number goes to zero. A few results in this field are here reviewed.