### 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 language | English |
---|---|

Pages (from-to) | 299-331 |

Number of pages | 33 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 182 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Oct 2006 |

Externally published | Yes |

### Cite this

*Archive for Rational Mechanics and Analysis*,

*182*(2), 299-331. https://doi.org/10.1007/s00205-006-0429-2

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*Archive for Rational Mechanics and Analysis*, vol. 182, no. 2, pp. 299-331. https://doi.org/10.1007/s00205-006-0429-2

**Fractal first-order partial differential equations.** / Droniou, Jérôme; Imbert, Cyril.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Fractal first-order partial differential equations

AU - Droniou, Jérôme

AU - Imbert, Cyril

PY - 2006/10/1

Y1 - 2006/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33747469213&partnerID=8YFLogxK

U2 - 10.1007/s00205-006-0429-2

DO - 10.1007/s00205-006-0429-2

M3 - Article

VL - 182

SP - 299

EP - 331

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -