Fractal Conservation Laws: Global Smooth Solutions and Vanishing Regularization

We consider the parabolic regularization of a scalar conservation law in which the Laplacian operator has been replaced by a fractional power of itself. Using a splitting method, we prove the existence of a solution to the problem and, thanks to the Banach fixed point theorem, its uniqueness and regularity. We also show that, as the regularization vanishes, the solution converge to the entropy solution of the scalar conservation law. We only present here the outlines of the proofs; we refer the reader to [4] and [5] for the details. Mathematics Subject Classification (2000). 35L65, 35S30, 35A35, 35B65.