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
External document linkhttp://arxiv.org/abs/math/0701041v1
Journal nameArchive for Rational Mechanics and Analysis
MetadataShow full item record
Abstract (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.
Subjects / KeywordsCauchy problem; PDEs
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