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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https://hal.archives-ouvertes.fr/hal-00004462
Contributor : Cyril Imbert <>
Submitted on : Wednesday, February 8, 2006 - 5:15:19 PM
Last modification on : Thursday, January 11, 2018 - 6:15:40 AM
Long-term archiving on: Monday, September 20, 2010 - 2:14:25 PM

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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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