Fractal first order partial differential equations

Abstract : The present paper is concerned with semilinear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton-Jacobi equations. The starting point is a new formula for the operator. It permits to prove the key a priori estimate that stands for the scalar conservation law and the Hamilton-Jacobi equation. The smoothing effect of the operator is also put in light and used to solve both equations. 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 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.
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Article dans une revue
Archive for Rational Mechanics and Analysis, Springer Verlag, 2006, 182 (2), pp.299--331
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https://hal.archives-ouvertes.fr/hal-00004462
Contributeur : Cyril Imbert <>
Soumis le : mercredi 8 février 2006 - 17:15:19
Dernière modification le : jeudi 11 janvier 2018 - 06:15:40
Document(s) archivé(s) le : lundi 20 septembre 2010 - 14:14:25

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  • HAL Id : hal-00004462, version 2

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Jérôme Droniou, Cyril Imbert. Fractal first order partial differential equations. Archive for Rational Mechanics and Analysis, Springer Verlag, 2006, 182 (2), pp.299--331. 〈hal-00004462v2〉

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