dc.contributor.author | Alibaud, Nathaël | |
dc.contributor.author | Letnes, Jørgen Endal | |
dc.contributor.author | Jakobsen, Espen Robstad | |
dc.date.accessioned | 2024-09-17T12:26:23Z | |
dc.date.available | 2024-09-17T12:26:23Z | |
dc.date.created | 2024-06-13T13:20:53Z | |
dc.date.issued | 2024 | |
dc.identifier.citation | Journal des Mathématiques Pures et Appliquées. 2024, 188, 26-72. | en_US |
dc.identifier.issn | 0021-7824 | |
dc.identifier.uri | https://hdl.handle.net/11250/3152764 | |
dc.description.abstract | We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality: ˆ |Stu0−Stv0|ϕ0dx≤ ˆ |u0−v0|Gtϕ0dx, ∀ϕ0≥0,∀u0,∀v0, ( ) where St is the entropy solution semigroup of the anisotropic degenerate parabolic equation ∂tu+divF(u)=div(A(u)Du), and where we look for the smallest semigroup Gt satisfying ( ). This amounts to finding an optimal weighted L1contraction estimate for St. Our main result is that Gt is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation ∂tϕ=supξ{F(ξ)·Dϕ+tr(A(ξ)D2ϕ)}. Since weightedL1 contraction results are mainly used for possibly nonintegrable L∞solutions u, the natural spaces behind this duality are L∞for St and L1 for Gt. We therefore develop a corresponding L1 theory for viscosity solutions ϕ. But L1 itself is too large for well-posedness, and we rigorously identify the weakest L1 type Banach setting where we can have it – a subspace of L1 called L∞ int. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018]. | en_US |
dc.language.iso | eng | en_US |
dc.publisher | Elsevier | en_US |
dc.relation.uri | https://arxiv.org/abs/1812.02058 | |
dc.rights | Navngivelse 4.0 Internasjonal | * |
dc.rights.uri | http://creativecommons.org/licenses/by/4.0/deed.no | * |
dc.title | Optimal stability results and nonlinear duality for L∞ entropy and L1 viscosity solutions | en_US |
dc.title.alternative | Optimal stability results and nonlinear duality for L∞ entropy and L1 viscosity solutions | en_US |
dc.type | Journal article | en_US |
dc.type | Peer reviewed | en_US |
dc.description.version | publishedVersion | en_US |
dc.source.pagenumber | 26-72 | en_US |
dc.source.volume | 188 | en_US |
dc.source.journal | Journal des Mathématiques Pures et Appliquées | en_US |
dc.identifier.doi | 10.1016/j.matpur.2024.05.003 | |
dc.identifier.cristin | 2275992 | |
dc.relation.project | Norges forskningsråd: 325114 | en_US |
cristin.ispublished | true | |
cristin.fulltext | preprint | |
cristin.qualitycode | 2 | |