Quantitative and qualitative properties for Hamilton-Jacobi PDEs via the nonlinear adjoint method

TL;DR

This paper introduces new integral estimates for Hamilton-Jacobi equations, leading to improved convergence rates, regularizing effects, and Liouville-type results, using duality and advection-diffusion analysis.

Contribution

It provides novel integral estimates and applies duality techniques to derive new properties and convergence results for Hamilton-Jacobi PDEs.

Findings

L^p convergence rates for vanishing viscosity approximations

Regularizing effects in the Euclidean space

Liouville-type theorems for solutions

Abstract

We provide some new integral estimates for solutions to Hamilton-Jacobi equations and we discuss several consequences, ranging from $L^p$-rates of convergence for the vanishing viscosity approximation to regularizing effects for the Cauchy problem in the whole Euclidean space and Liouville-type theorems. Our approach is based on duality techniques \`a la Evans and a careful study of advection-diffusion equations. The optimality of the results is discussed by several examples.