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Fractal First-Order Partial Differential Equations

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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.

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Authors and Affiliations

  1. Département de Mathématiques, UMR CNRS 5149, CC 051, Université Montpellier II, Place Eugène Bataillon, 34095, Montpellier cedex 5, France

    Jérôme Droniou

  2. Département de Mathématiques, UMR CNRS 5149, CC 051,  ,  

    Cyril Imbert

  3. Polytech'Montpellier, Université Montpellier II, Place Eugène Bataillon, 34095, Montpellier cedex 5, France

    Cyril Imbert

Authors
  1. Jérôme Droniou
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  2. Cyril Imbert
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Correspondence to Jérôme Droniou.

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Communicated by Y. Brenier

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Droniou, J., Imbert, C. Fractal First-Order Partial Differential Equations. Arch Rational Mech Anal 182, 299–331 (2006). https://doi.org/10.1007/s00205-006-0429-2

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