Sharp Decay Estimates and Vanishing Viscosity for Diffusive Hamilton-Jacobi Equations

TL;DR

This paper derives precise decay estimates for solutions to viscous and non-viscous Hamilton-Jacobi equations, focusing on the influence of viscosity and initial conditions with minimal regularity.

Contribution

It provides sharp decay bounds for the gradient and time derivative, considering various growth conditions on the Hamiltonian and minimal initial regularity.

Findings

Decay estimates depend explicitly on viscosity.

Results cover Hamiltonians with superlinear and sublinear growth.

Estimates hold for initial data that are only continuous and bounded or non-negative.

Abstract

Sharp temporal decay estimates are established for the gradient and time derivative of solutions to a viscous Hamilton-Jacobi equation as well the associated Hamilton-Jacobi equation. Special care is given to the dependence of the estimates on the viscosity. The initial condition being only continuous and either bounded or non-negative. The main requirement on the Hamiltonians is that it grows superlinearly or sublinearly at infinity, including in particular H(r) = r^p for r non-negatif and p positif and different from 1.