Viscous Hamilton–Jacobi equations in exponential Orlicz hearts

Abstract

We provide a semigroup approach to viscous Hamilton–Jacobi equations. 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. Furthermore, we determine the so-called 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.