Séminaire d'analyse (archives)

In this talk, we study the influence of a Coriolis forcing on water waves. First, we present a local wellposedness result on the Castro-Lannes equations which generalize the so called Zakharov/Craig-Sulem formulation in the rotational framework. Then, we study different asymptotic models in shallow waters. First, we fully justify on large times the Boussinesq equations, asymptotic model in a weakly nonlinear regime, and then we fully justify the Poincaré waves and the Ostrovsky equation.

We consider a class of quasi linear Schrödinger equations and we prove the existence and the stability of a Cantor families of quasi-periodic, small amplitude solutions. We deal with reversible autonomous nonlinearities and we apply an abstract Nash-Moser/KAM algorithm for the construction of invariant tori which allows us to find analytic solutions.

In this talk we analyse the spectrum of the dissipative Schrödinger operator on binary tree-shaped networks. As applications, we study the stability of the Schrödinger system using a Riesz basis as well as the transfer function associated to the system. Moreover, we study the dispersive effects associated to the Schrödinger operator with potential on star-shaped network and to the free Schrödinger operator on a tadpole graph.

On abordera le problème de la formation des Mach stems dans un fluide compressible, notamment via la formation de singularités sur le front d'une onde de choc. La justification rigoureuse d'une telle formation d'un Mach stem passe par un problème d'optique géométrique fortement non-linéaire dont on montrera qu'il admet des solutions approchées à tout ordre. Il s'agit d'un travail en collaboration avec Mark Williams (University of North Carolina).

In this talk, the dissipative wave equations in an outside of compact regions with smooth boundary are treated. Assume that the dissipation coefficient depends only on the space variable. First, in this setting, a sufficient condition showing uniformly decaying estimates of the local energy is given. As an application of this condition, a decaying estimate of the local energy near the boundary is obtained if the dissipation coefficient is positive near concave part of the boundary in some sense. Second, the cases that the dissipation coefficient is not small in far field are considered. In this case, an optimal decay of the total energy corresponding to the decay rate for high frequency waves is discussed.