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Habilitation À Diriger Des Recherches Year : 2013

## Oscillating waves for nonlinear conservation laws

Stéphane Junca

#### Abstract

The manuscript presents my research on hyperbolic Partial Differential Equations (PDE), especially on conservation laws. My works began with this thought in my mind: ''Existence and uniqueness of solutions is not the end but merely the beginning of a theory of differential equations. The really interesting questions concern the behavior of solutions.'' (P.D. Lax, The formation and decay of shock waves 1974). To study or highlight some behaviors, I started by working on geometric optics expansions (WKB) for hyperbolic PDEs. For conservation laws, existence of solutions is still a problem (for large data, $L^\infty$ data), so I early learned method of characteristics, Riemann problem, $BV$ spaces, Glimm and Godunov schemes, \ldots In this report I emphasize my last works since 2006 when I became assistant professor. I use geometric optics method to investigate a conjecture of Lions-Perthame-Tadmor on the maximal smoothing effect for scalar multidimensional conservation laws. With Christian Bourdarias and Marguerite Gisclon from the LAMA (Laboratoire de \\ Mathématiques de l'Université de Savoie), we have obtained the first mathematical results on a $2\times2$ system of conservation laws arising in gas chromatography. Of course, I tried to put high oscillations in this system. We have obtained a propagation result exhibiting a stratified structure of the velocity, and we have shown that a blow up occurs when there are too high oscillations on the hyperbolic boundary. I finish this subject with some works on kinetic équations. In particular, a kinetic formulation of the gas chromatography system, some averaging lemmas for Vlasov equation, and a recent model of a continuous rating system with large interactions are discussed. Bernard Rousselet (Laboratoire JAD Université de Nice Sophia-Antipolis) introduced me to some periodic solutions related to crak problems and the so called nonlinear normal modes (NNM). Then I became a member of the European GDR: ''Wave Propagation in Complex Media for Quantitative and non Destructive Evaluation.'' In 2008, I started a collaboration with Bruno Lombard, LMA (Laboratoire de Mécanique et d'Acoustique, Marseille). We details mathematical results and challenges we have identified for a linear elasticity model with nonlinear interfaces. It leads to consider original neutral delay differential systems.

#### Domains

Mathematics [math] Analysis of PDEs [math.AP]

### Dates and versions

tel-00845827 , version 1 (17-07-2013)

### Identifiers

• HAL Id : tel-00845827 , version 1

### Cite

Stéphane Junca. Oscillating waves for nonlinear conservation laws. Analysis of PDEs [math.AP]. Université Nice Sophia Antipolis, 2013. ⟨tel-00845827⟩

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