Nous nous intéressons à lʼidentification dʼun coefficient de Robin défini sur une partie non accessible du bord, à partir de mesures disponibles sur une autre partie de celui-ci, dans le système de Stokes non stationnaire. Nous prouvons une estimation de stabilité lipschitzienne sous lʼhypothèse a priori que le coefficient de Robin est défini dans un sous-ensemble compact et convexe dʼun sous-espace vectoriel de dimension finie de lʼespace des fonctions continues. Pour ce faire, nous utilisons un théorème prouvé par L. Bourgeois permettant dʼétablir des inégalités de stabilité lipschitzienne pour une classe de problèmes inverses dans un cadre abstrait.
We are interested in the inverse problem of recovering a Robin coefficient defined on some non-accessible part of the boundary from available data on another part of the boundary in the non-stationary Stokes system. We prove a Lipschitz stability estimate under the a priori assumption that the Robin coefficient lives in some compact and convex subset of a finite dimensional vectorial subspace of the set of continuous functions. To do so, we use a theorem proved by L. Bourgeois and which establishes Lipschitz stability estimates for a class of inverse problems in an abstract framework.
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Anne-Claire Egloffe 1
@article{CRMATH_2013__351_13-14_527_0, author = {Anne-Claire Egloffe}, title = {Lipschitz stability estimate in the inverse {Robin} problem for the {Stokes} system}, journal = {Comptes Rendus. Math\'ematique}, pages = {527--531}, publisher = {Elsevier}, volume = {351}, number = {13-14}, year = {2013}, doi = {10.1016/j.crma.2013.06.010}, language = {en}, }
Anne-Claire Egloffe. Lipschitz stability estimate in the inverse Robin problem for the Stokes system. Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 527-531. doi : 10.1016/j.crma.2013.06.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.06.010/
[1] Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4), Volume 29 (2000), pp. 755-806
[2] Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math. Anal. Appl., Volume 343 (2008), pp. 328-336
[3] Stability estimates for a Robin coefficient in the two-dimensional Stokes system, Math. Control Relat. Fields, Volume 1 (2013), pp. 21-49
[4] Unique continuation estimates for the Stokes system. Application to an inverse problem, Inverse Probl. (2013) (in press)
[5] A remark on Lipschitz stability for inverse problems, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013), pp. 187-190
[6] On the detection of a moving obstacle in an ideal fluid by a boundary measurement, Inverse Probl., Volume 24 (2008), pp. 2842-2866
[7] Lipschitz stability estimate in the inverse Robin problem for the Stokes system, 2013 (Inria research report, RR-8222)
[8] Prolongement unique des solutions de lʼequation de Stokes, Commun. Partial Differ. Equ., Volume 21 (1996), pp. 573-596
[9] Lipschitz stability for the inverse Robin problem, Inverse Probl., Volume 23 (2007), pp. 1311-1326
[10] Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., Volume 256 (2009), pp. 2842-2866
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