Geodesic
Géométrie des systèmes hyperboliques de lois de conservation
Mémoires de la Société Mathématique de France, no. 56 (1994) , 132 p.

Voir la notice du livre provenant de la source Numdam

On étudie les systèmes hyperboliques de lois de conservation en dimension un d'espace. L'espace des états apparaît naturellement muni d'une structure affine. Les systèmes physiques possèdent des lois de conservation excédentaires, ou “entropies”, et on montre que les propriétés d'intégrabilité des champs de directions propres sont liées à l'existence de ces entropies. La dégénérescence linéaire et la présence d'une entropie non-dégénérée sont deux caractéristiques des systèmes d'origine physique. On montre qu'elles entraînent une propriété de “rigidité” du feuilletage de contact associé au champ linéairement dégénéré, qui est explicitée sur un certain nombre d'exemples. L'étude asymptotique de la stabilité des oscillations permises par la dégénérescence linéaire conduit à la notion d'hyperbolicité globale, que l'on étudie dans le cadre de la géométrie transverse du feuilletage de contact, et pour laquelle des critères généraux sont dégagés, tel le “non-enlacement” de ce feuilletage.

We study hyperbolic systems of conservation laws in one space dimension. State space appears naturally endowed with an affine structure. Physical systems posess extraneous conservation laws, the so-called “entropies”, and we show that the integrability properties of eigendirection fieids are strongly related to the existence of these entropies. Linear degeneracy and existence of a non-degenerate (e.g. strictly convex) entropy are characteristic features of physical systems. We show that they entail a “rigidity” property of the contact foliation associated with the linearly degenerated field, which is evidenced on some examples. In studying stability of oscillations permitted by linear degeneracy, one is led to the notion of global hyperbolicity, which is studied in the framework of contact foliation's transverse geometry. It results in several criteria of global hyperbolicity, such as the “unlinking” of contact foliation, for a certain class of systems.

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