Viscous Hamilton–Jacobi equations in exponential Orlicz hearts

Blessing, Jonas; Kupper, Michael

doi:10.1016/j.matpur.2022.05.018

Journal de Mathématiques Pures et Appliquées

2022

DOI: 10.1016/j.matpur.2022.05.018

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Viscous Hamilton–Jacobi equations in exponential Orlicz hearts

Jonas Blessing

Michael Kupper

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Nonlinear Semigroups Built on Generating Families and their Lipschitz Sets

Blessing

Kupper

2022

Potential Anal

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Under suitable conditions on a family (I(t))t≥ 0 of Lipschitz mappings on a complete metric space, we show that, up to a subsequence, the strong limit $S(t):=\lim _{n\to \infty }(I(t 2^{-n}))^{2^{n}}$ S ( t ) : = lim n → ∞ ( I ( t 2 − n ) ) 2 n exists for all dyadic time points t, and extends to a strongly continuous semigroup (S(t))t≥ 0. The common idea in the present approach is to find conditions on the generating family (I(t))t≥ 0, which can be transferred to the semigroup. The construction relies on the Lipschitz set, which is invariant under iterations and allows to preserve Lipschitz continuity to the limit. Moreover, we provide a verifiable condition which ensures that the infinitesimal generator of the semigroup is given by $\lim _{h\downarrow 0}\tfrac {I(h)x-x}{h}$ lim h ↓ 0 I ( h ) x − x h , whenever this limit exists. The results are illustrated with several examples of nonlinear semigroups such as robustifications and perturbations of linear semigroups.

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Nonlinear Semigroups Built on Generating Families and their Lipschitz Sets

Blessing

Kupper

2022

Potential Anal

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Viscous Hamilton–Jacobi equations in exponential Orlicz hearts

Cited by 1 publication

References 32 publications

Nonlinear Semigroups Built on Generating Families and their Lipschitz Sets

Nonlinear Semigroups Built on Generating Families and their Lipschitz Sets

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