Nonlocal Dissipation Measure and L 1 Kinetic Theory For Fractional Conservation Laws

TL;DR

This paper develops a kinetic formulation for scalar conservation laws with nonlocal, nonlinear diffusion, extending second order theory tools to fractional laws and handling L1 initial data and Levy operators.

Contribution

It introduces a novel kinetic formulation and explicit nonlocal dissipation measure for fractional conservation laws, bridging second order theory with nonlocal operators.

Findings

First kinetic formulation for fractional conservation laws

Explicit representation of nonlocal dissipation measure

Extension of second order tools to nonlocal, fractional setting

Abstract

We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L 1 initial data, general self-adjoint pure jump L{\'e}vy 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 nonlocal dissipation measure. This 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 our 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.