Viscous Hamilton–Jacobi equations in exponential Orlicz hearts
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Zusammenfassung
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.
Zusammenfassung in einer weiteren Sprache
Nous proposons une approche par semigroupe de l'équation visqueuse de Hamilton–Jacobi. Il s'avère que les cœurs d'Orlicz exponentiels sont des espaces appropriés pour traiter la non-linéarité (quadratique) du Hamiltonien. Sur la base d'un résultat d'extension abstrait pour les semigroupes non linéaires sur des espaces de fonctions continues, nous représentons la solution de l'équation de Hamilton–Jacobi visqueuse comme un semigroupe convexe fortement continu sur un cœur d'Orlicz exponentiel. Par conséquent, la solution dépend continuellement des données initiales. Nous déterminons en outre ce que l'on appelle l'ensemble symétrique de Lipschitz qui est invariant sous le semigroupe. Cela donne automatiquement des estimations a priori et une régularité dans les espaces de Sobolev. En particulier, sur le domaine restreint à l'ensemble symétrique de Lipschitz, le générateur peut être déterminé explicitement et lié à l'équation visqueuse de Hamilton–Jacobi.
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BLESSING, Jonas, Michael KUPPER, 2022. Viscous Hamilton–Jacobi equations in exponential Orlicz hearts. In: Journal de Mathématiques Pures et Appliquées. Elsevier. 2022, 163, pp. 654-672. ISSN 0021-7824. eISSN 1776-3371. Available under: doi: 10.1016/j.matpur.2022.05.018BibTex
@article{Blessing2022Visco-58053,
year={2022},
doi={10.1016/j.matpur.2022.05.018},
title={Viscous Hamilton–Jacobi equations in exponential Orlicz hearts},
volume={163},
issn={0021-7824},
journal={Journal de Mathématiques Pures et Appliquées},
pages={654--672},
author={Blessing, Jonas and Kupper, Michael}
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