Entropy formulation for fractal conservation laws

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 lambda is an element of]0, 1]. This allows to show the existence and the uniqueness of a solution in the L-infinity 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 lambda = 2 and Droniou (2003) for lambda is an element of]1, 2].