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arXiv · 2104.06433

Viscous Hamilton-Jacobi equations in exponential Orlicz hearts

Jonas Blessing·Michael Kupper
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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.

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Jonas Blessing, Michael Kupper. 2021-04-13. Viscous Hamilton-Jacobi equations in exponential Orlicz hearts. https://arxiv.org/abs/2104.06433

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