Fractal first-order partial differential equations

Jérôme Droniou, Cyril Imbert

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119 Citations (Scopus)

Abstract

The present paper is concerned with semi-linear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton-Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one hand, the key a priori estimates for the scalar conservation law and the Hamilton-Jacobi equation and, on the other hand, the smoothing effect of the operator. As far as Hamilton-Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows us to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.

Original languageEnglish
Pages (from-to)299-331
Number of pages33
JournalArchive for Rational Mechanics and Analysis
Volume182
Issue number2
DOIs
Publication statusPublished - 1 Oct 2006
Externally publishedYes

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