This paper studies the unique continuation problem for the heat equation. We prove a so-called conditional stability estimate for the solution. We are interested in local estimates that are Hölder stable with the weakest possible norms of data on the right-hand side. Such an estimate is useful for the convergence analysis of computational methods dealing with data assimilation. We focus on the case of a known solution at initial time and in some subdomain but that is unknown on the boundary. To the best of our knowledge, this situation has not yet been studied in the literature.
Cette contribution traite du problème de continuation unique pour l’équation de la chaleur. Nous prouvons une estimée conditionnelle de stabilité pour la solution de ce problème. Nous sommes intéressés par une estimée locale qui est Hölder-stable avec les normes les plus faibles possibles pour le terme de droite. Une telle estimée est utile pour l’analyse de convergence des méthodes de calcul traitant du problème d’assimilation de données. Nous nous intéressons en particulier au cas où la solution est connue à l’instant initial et dans un certain sous-domaine mais est inconnue sur la frontière du domaine. Cette situation ne semble pas avoir été déjà traitée dans la littérature.
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Erik Burman 1; Guillaume Delay 2; Alexandre Ern 3; Lauri Oksanen 4
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@article{CRMATH_2023__361_G9_1521_0,
author = {Erik Burman and Guillaume Delay and Alexandre Ern and Lauri Oksanen},
title = {A stability estimate for data assimilation subject to the heat equation with initial datum},
journal = {Comptes Rendus. Math\'ematique},
pages = {1521--1530},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
year = {2023},
doi = {10.5802/crmath.506},
language = {en},
}
TY - JOUR AU - Erik Burman AU - Guillaume Delay AU - Alexandre Ern AU - Lauri Oksanen TI - A stability estimate for data assimilation subject to the heat equation with initial datum JO - Comptes Rendus. Mathématique PY - 2023 SP - 1521 EP - 1530 VL - 361 PB - Académie des sciences, Paris DO - 10.5802/crmath.506 LA - en ID - CRMATH_2023__361_G9_1521_0 ER -
%0 Journal Article %A Erik Burman %A Guillaume Delay %A Alexandre Ern %A Lauri Oksanen %T A stability estimate for data assimilation subject to the heat equation with initial datum %J Comptes Rendus. Mathématique %D 2023 %P 1521-1530 %V 361 %I Académie des sciences, Paris %R 10.5802/crmath.506 %G en %F CRMATH_2023__361_G9_1521_0
Erik Burman; Guillaume Delay; Alexandre Ern; Lauri Oksanen. A stability estimate for data assimilation subject to the heat equation with initial datum. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1521-1530. doi : 10.5802/crmath.506. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.506/
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