# Kinetic solutions for nonlocal scalar conservation laws

## TL;DR

This paper investigates the existence and uniqueness of kinetic solutions for nonlocal scalar conservation laws with super-critical diffusion, using a microscopic contraction approach and parabolic approximation, with applications to fractional Burgers-Fisher equations.

## Contribution

It introduces a novel approach for proving uniqueness and existence of kinetic solutions in nonlocal conservation laws involving super-critical diffusion operators.

## Key findings

- Proved uniqueness of kinetic solutions using microscopic contraction functional.
- Established existence of solutions via parabolic approximation.
- Showed Lipschitz continuity and continuous dependence of solutions.

## Abstract

This work is devoted to examine the uniqueness and existence of kinetic solutions for a class of scalar conservation laws involving a nonlocal super-critical diffusion operator. Our proof for uniqueness is based upon the analysis on 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 L\'{e}vy measures.