Journal of Evolution Equations

, Volume 7, Issue 1, pp 145–175

Entropy formulation for fractal conservation laws

Article

Abstract.

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 

Keywords:

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

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Copyright information

© Birkhäuser Verlag, Basel 2007

Authors and Affiliations

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

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