Article

Journal of Evolution Equations

, Volume 7, Issue 1, pp 145-175

First online:

Entropy formulation for fractal conservation laws

  • Nathaël AlibaudAffiliated withDépartement de mathématiques, Université Montpellier II Email author 

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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