Abstracts

Jean-François Coulombel Continuous and discrete initial boundary value problems The aim of these two lectures is to review some basic aspects of hyperbolic systems of partial differential equations, both at the continuous level (existence and uniqueness of solutions to the Cauchy problem and to the initial boundary value problem) and at the discrete level (when one wishes to approximate the continuous solution by a finite difference scheme). For linear constant coefficients systems, the analysis of the Cauchy problem relies on the Fourier transform, while the analysis of the initial boundary value problem relies on a mixed Laplace-Fourier transform. One of the most striking results in the theory is a characterization, by means of a certain algebraic condition, of boundary conditions that give a well-posed problems. Some simple numerical tests will be presented to highlight stability and instability phenomena.

Benjamin Texier Approximations of pseudo-differential flows I will put forward a micro-local approach to stability of reference solutions in systems of partial differential equations.The idea is to look forunstable spectrum at the symbolic level, as opposed to unstable spectrum for the associated differential operators. I will give model results in this direction, based on an approximation lemma for pseudo-differential flows, and draw a comparison with Garding'sinequality.

Erik Wahlén Existence and stability theory for solitary water waves with weak surface tension I will discuss some recent progress in the theory of solitary water waves with weak surface tension. In the case of a two-dimensional fluid, it is well known that there are solitary waves which look like periodic wave trains modulated by localised envelopes described by the nonlinear Schrödinger equation. I will explain how such solutions can be constructed by minimising the energy subject to the constraint of fixed momentum. This has the advantage of giving information about the stability of the waves. I will also discuss some work in progress on a variational existence theory for solitary waves in three dimensions.

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