Existence and uniqueness of solution for semi linear conservation laws with velocity field in $L^\infty$

TL;DR

This paper proves existence and uniqueness of solutions for semi-linear conservation laws with velocity fields in $L^ Infty$, extending previous results using fixed point methods, including in penalized cases.

Contribution

It introduces new fixed point proofs for existence and uniqueness of semi-linear conservation laws with bounded velocity fields, extending prior work.

Findings

Proves existence of solutions for semi-linear conservation laws with bounded velocity.

Establishes uniqueness of solutions in the same setting.

Handles penalized cases for the conservation laws.

Abstract

In this paper we extend results obtained in [3] and [5]. By considering a semi linear conservation law with velocity in $L^\infty$, we prove by fixed point arguments existence and uniqueness result and even in a penalized situation.