f v dp , ρ := R 3 f dp , dp :

PAR CHRISTOPHE CHEVERRY, OLIVIER GUÈS ET GUY MÉTIVIERRÉSUMÉ. -On s'intéresse à des systèmes quasilinéaires en multidimension d'espace. En présence de valeurs propres linéairement dégénérées, des transparences se produisent. Les calculs classiques d'optique géométrique conduisent à des équations qui sont linéaires. On montre dans cet article que les amplitudes des développements asymptotiques peuvent être augmentées pour atteindre des régimes non linéaires sans pour autant compromettre la construction de solutions approchées. On met en valeur les contraintes permettant de justifier l'existence de solutions exactes qui correspondent à de telles solutions approchées. En l'absence de ces contraintes, on classifie les différents types d'instabilité créés. 2003 Published by Elsevier SAS ABSTRACT. -Transparencies happen for quasilinear waves associated to linearly degenerate eigenvalues, leading to linear geometric optics. To reach nonlinear phenomena, we have to consider larger amplitudes. We show that approximate solutions can still be constructed. We find intrinsic conditions allowing to justify the approximation. In the absence of these conditions, instabilities are created. We analyse the different types of the underlying instabilities.

This article is devoted to the Relativistic Vlasov-Maxwell system in space dimension three. We prove the local existence and uniqueness of solutions for initial data (f 0 , E 0 , B 0 ) ∈ L ∞ × H 1 × H 1 , with f 0 compactly supported in momentum. This result is the consequence of the local smooth solvability for the "weak" topology associated with L ∞ × H 1 × H 1 . It is derived from a representation formula decoding how the momentum spreads and revealing that the domain of influence in momentum is controlled by mild information (involving only conserved quantities). We do so by developing a Radon Fourier analysis on the RVM system, leading to the study of a class of singular weighted integrals. In parallel, we implement our method to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas.

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In this work we prove that the shallow water flow, subject to strong wind forcing and linearized around an adequate stationary profile, develops for large times closed trajectories due to the propagation of Rossby waves, while Poincaré waves are shown to disperse. The methods used in this paper involve semi-classical analysis and dynamical systems for the study of Rossby waves, while some refined spectral analysis is required for the study of Poincaré waves, due to the large time scale involved which is of diffractive type.2010 Mathematics Subject Classification. 35Q86; 76M45; 35S30; 81Q20.