# Viscous Conservation Laws in 1d With Measure Initial Data

## TL;DR

This paper studies one-dimensional viscous conservation laws with measure initial data, establishing existence and uniqueness of solutions using decay estimates for viscous Hamilton-Jacobi equations.

## Contribution

It introduces a framework for solving viscous conservation laws with measure initial data, extending classical results to more general initial conditions.

## Key findings

- Existence of solutions for measure initial data.
- Uniqueness of solutions under weak convexity conditions.
- Decay estimates for viscous Hamilton-Jacobi equations used in proofs.

## Abstract

The one-dimensional viscous conservation law is considered on the whole line $$ u_t + f(u)_x=\eps u_{xx},\quad (x,t)\in\RR\times\overline{\RP},\quad \eps>0, $$ subject to positive measure initial data. The flux $f\in C^1(\RR)$ is assumed to satisfy a $p-$condition, a weak form of convexity. Existence and uniqueness of solutions is established. The method of proof relies on sharp decay estimates for viscous Hamilton-Jacobi equations.