Abstract
This work is devoted to examining the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal supercritical diffusion operator. Our proof for uniqueness is based upon the analysis of a microscopic contraction functional, and the existence is enabled by a parabolic approximation. As an illustration, we obtain the existence and uniqueness of kinetic solutions for the generalized fractional Burgers--Fisher-type equations. Moreover, we demonstrate the kinetic solutions' Lipschitz continuity in time and continuous dependence on nonlinearities and Levy measures.
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