Sur le flot de l’équation d’Euler à surface libre

The Euler equation with free boundary, i.e the water waves system,describes the evolution of the interface between air and a perfect irrotational fluid.It is a system of two coupled equations : the Euler equation in the interior of thedomain and a kinematic equation describing the deformation of the domain. The four works that constitute the body of this thesis can be divided into three connected subjects on the Cauchy problem of the water waves system.In the continuation of the works in [2, 4, 5, 7] where the Cauchy problem forthe water waves system is shown to be well-posed and the flow map continuouson sufficiently regular Sobolev spaces we show :-In [60] that the water waves system with and without surface tension isquasi-linear in the strongest sense, i.e the flow map is not uniformly continuous.Moreover in the case with surface tension we show that in orderto have Lipschitz estimate on the flow map at least a loss of derivative. More generally for the Burgers equation augmented by a dispersiveterm of the form, we show that at least a loss of derivative is needed to ensure Lipschitz control on the flow.-In [61] we show that the results obtained in [60] are indeed optimal, that isfor the Burgers equation augmented with a dispersive term,alphain the flow map is indeed Lipschitz from for periodic data with 0 mean value. For the water waves system with surface tensionin two space dimension we show that after suitable re-normalisation that the flow map is Lipschitz under loss of derivative. In order to prove the results in [61] we developed a paradifferential generalisationof a complex Cole-Hopf type gauge transform first introduced by T. Taofor the Benjamin-Ono equation. In [62] we use this to improve upon knownresults on a numerical conjecture by Saut and Klein [45] on the dispersiveBurgers equation, which to the author’s knowledge is the first time the gaugetransform was implemented to that end for. In order to prove the different results in [60, 61, 62] we needed to study andrefine different known results in paradifferential calculus. More precisely in [59],we improve some estimates on the paracomposition operator introduced byAlinhac, give a proof of the change of variables in paradifferential operatorsand finally study the frequency cut-off after composition of paradifferentialoperators.