On uniform controllability of 1D transport equations in the vanishing viscosity limit
We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.We assume that the vector...
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Académie des sciences
2023-01-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.405/ |
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| author | Laurent, Camille Léautaud, Matthieu |
| author_facet | Laurent, Camille Léautaud, Matthieu |
| author_sort | Laurent, Camille |
| collection | DOAJ |
| description | We consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schrödinger operator. They improve, in this particular situation, the results obtained in the companion paper [38].The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [40], together with estimates on the spectral gap [33, 1]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest. |
| format | Article |
| id | doaj-art-2c896e5a83204f2d99bd9e597e965ec2 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-01-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-2c896e5a83204f2d99bd9e597e965ec22025-02-07T11:06:07ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-01-01361G126531210.5802/crmath.40510.5802/crmath.405On uniform controllability of 1D transport equations in the vanishing viscosity limitLaurent, Camille0Léautaud, Matthieu1CNRS UMR 7598 and Sorbonne Universités UPMC Univ Paris 06, Laboratoire Jacques-Louis Lions, F-75005, Paris, FranceLaboratoire de Mathématiques d’Orsay, UMR 8628, Université Paris-Saclay, CNRS, Bâtiment 307, 91405 Orsay Cedex FranceWe consider a one dimensional transport equation with varying vector field and a small viscosity coefficient, controlled by one endpoint of the interval. We give upper and lower bounds on the minimal time needed to control to zero, uniformly in the vanishing viscosity limit.We assume that the vector field varies on the whole interval except at one point. The upper/lower estimates we obtain depend on geometric quantities such as an Agmon distance and the spectral gap of an associated semiclassical Schrödinger operator. They improve, in this particular situation, the results obtained in the companion paper [38].The proofs rely on a reformulation of the problem as a uniform observability question for the semiclassical heat equation together with a fine analysis of localization of eigenfunctions both in the semiclassically allowed and forbidden regions [40], together with estimates on the spectral gap [33, 1]. Along the proofs, we provide with a construction of biorthogonal families with fine explicit bounds, which we believe is of independent interest.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.405/ |
| spellingShingle | Laurent, Camille Léautaud, Matthieu On uniform controllability of 1D transport equations in the vanishing viscosity limit Comptes Rendus. Mathématique |
| title | On uniform controllability of 1D transport equations in the vanishing viscosity limit |
| title_full | On uniform controllability of 1D transport equations in the vanishing viscosity limit |
| title_fullStr | On uniform controllability of 1D transport equations in the vanishing viscosity limit |
| title_full_unstemmed | On uniform controllability of 1D transport equations in the vanishing viscosity limit |
| title_short | On uniform controllability of 1D transport equations in the vanishing viscosity limit |
| title_sort | on uniform controllability of 1d transport equations in the vanishing viscosity limit |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.405/ |
| work_keys_str_mv | AT laurentcamille onuniformcontrollabilityof1dtransportequationsinthevanishingviscositylimit AT leautaudmatthieu onuniformcontrollabilityof1dtransportequationsinthevanishingviscositylimit |