Journal of Evolution Equations

, Volume 7, Issue 1, pp 145–175

Entropy formulation for fractal conservation laws


DOI: 10.1007/s00028-006-0253-z

Cite this article as:
Alibaud, N. J. evol. equ. (2007) 7: 145. doi:10.1007/s00028-006-0253-z


Using an integral formula of Droniou and Imbert (2005) for the fractional Laplacian, we define an entropy formulation for fractal conservation laws with pure fractional diffusion of order λ ∈]0, 1]. This allows to show the existence and the uniqueness of a solution in the L framework. We also establish a result of controled speed of propagation that generalizes the finite propagation speed result of scalar conservation laws. We finally let the non-local term vanish to approximate solutions of scalar conservation laws, with optimal error estimates for BV initial conditions as Kuznecov (1976) for λ = 2 and Droniou (2003) for λ ∈]1, 2].

Mathematics Subject Classification (2000):

35B30 35L65 35L82 35S10 35S30 


Fractional Laplacian fractal conservation laws entropy formulation vanishing viscosity method error estimates 

Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité Montpellier IIMontpellier cedex 5France

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