Viscous Hamilton-Jacobi equations in exponential Orlicz hearts

TL;DR

This paper introduces a semigroup approach to viscous Hamilton-Jacobi equations using exponential Orlicz hearts, enabling continuous dependence on initial data and establishing regularity and invariance properties.

Contribution

It develops a novel framework employing exponential Orlicz hearts for analyzing viscous Hamilton-Jacobi equations, extending nonlinear semigroup theory to this setting.

Findings

Solution depends continuously on initial data.

Invariant symmetric Lipschitz set identified.

A priori estimates and Sobolev regularity established.

Abstract

We provide a semigroup approach to the viscous Hamilton-Jacobi equation. It turns out that exponential Orlicz hearts are suitable spaces to handle the (quadratic) non-linearity of the Hamiltonian. Based on an abstract extension result for nonlinear semigroups on spaces of continuous functions, we represent the solution of the viscous Hamilton-Jacobi equation as a strongly continuous convex semigroup on an exponential Orlicz heart. As a result, the solution depends continuously on the initial data. We further determine the symmetric Lipschitz set which is invariant under the semigroup. This automatically yields a priori estimates and regularity in Sobolev spaces. In particular, on the domain restricted to the symmetric Lipschitz set, the generator can be explicitly determined and linked with the viscous Hamilton-Jacobi equation.