Nonlinear Hyperbolic Systems: Nondegenerate Flux, Inner Speed Variation, and Graph Solutions
Glass, Olivier; LeFloch, Philippe F. (2007), Nonlinear Hyperbolic Systems: Nondegenerate Flux, Inner Speed Variation, and Graph Solutions, Archive for Rational Mechanics and Analysis, 185, 3, p. 409-480. http://dx.doi.org/10.1007/s00205-006-0046-0
TypeArticle accepté pour publication ou publié
Lien vers un document non conservé dans cette basehttp://arxiv.org/abs/math/0701041v1
Nom de la revueArchive for Rational Mechanics and Analysis
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Résumé (EN)We study the Cauchy problem for general nonlinear strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time regularity of the graph solutions (X,U) introduced by LeFloch, and propose a geometric version of our scheme; in turn, the spatial component X of a graph solution can be chosen to be continuous in both time and space, while its component U is continuous in space and has bounded variation in time.
Mots-clésCauchy problem; PDEs
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