# Lipschitz regularity for viscous Hamilton-Jacobi equations with $L^p$ terms

## TL;DR

This paper establishes Lipschitz regularity for solutions to viscous Hamilton-Jacobi equations with right-hand sides in Lebesgue spaces, using a duality approach and gradient analysis of a dual Fokker-Planck equation.

## Contribution

It introduces a novel duality method to prove Lipschitz regularity for viscous Hamilton-Jacobi equations with L^p right-hand sides, highlighting the regularizing effect of diffusion.

## Key findings

- Lipschitz regularity achieved for solutions with L^p data
- Duality approach links Hamilton-Jacobi and Fokker-Planck equations
- Regularizing effect due to diffusion and Hamiltonian coercivity

## Abstract

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable.