# Asymptotic Lipschitz regularity of viscosity solutions of Hamilton-Jacobi equations

## TL;DR

This paper establishes the long-term Lipschitz regularity of viscosity solutions to evolutionary Hamilton-Jacobi equations with convex, coercive Hamiltonians, given continuous initial data.

## Contribution

It proves the asymptotic Lipschitz regularity of viscosity solutions for a broad class of Hamilton-Jacobi equations, extending understanding of solution regularity over time.

## Key findings

- Viscosity solutions become Lipschitz continuous asymptotically.
- Results apply to equations with convex, coercive Hamiltonians.
- Provides a new regularity result for evolutionary Hamilton-Jacobi equations.

## Abstract

For each continuous initial data $\varphi(x)\in C(M,\mathbb{R})$, we obtain the asymptotic Lipschitz regularity of the viscosity solution of the following evolutionary Hamilton-Jacobi equation with convex and coercive Hamiltonians: \partial_tu(x,t)+H(x,\partial_xu(x,t))=0, u(x,0)=\varphi(x).