Nonlocal dissipation measure and L1 kinetic theory for fractional conservation laws

Abstract

We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L1 initial data, general self-adjoint pure jump Lévy operators, and locally Lipschitz nonlinearities of porous medium kind possibly strongly degenerate. The cornerstone of the formulation and the uniqueness proof is an adequate explicit representation of the dissipation measure associated to the diffusion. This measure is a function in our nonlocal framework. Our approach is inspired from the second order theory unlike the cutting technique previously introduced for bounded entropy solutions. The latter technique no longer seems to fit the kinetic setting. This is moreover the first time that the more standard and sharper tools of the second order theory are faithfully adapted to fractional conservation laws.