Optimal Control/Partial Differential Equations
Uniqueness and partial identification in a geometric inverse problem for the Boussinesq system
Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 665-670.

We analyze the inverse problem of the identification of a rigid body immersed in a fluid governed by the stationary Boussinesq system. First, we establish a uniqueness result. Then, we present a new method for the partial identification of the body. The proofs use local Carleman estimates, differentiation with respect to domains, data assimilation techniques and controllability results for PDEs.

On analyse le problème inverse de l'identification d'un corps rigide dans un fluide régi par le système stationnaire de Boussinesq. On établit d'abord un résultat d'unicité. Ensuite on présente une nouvelle méthode pour l'identification partielle du corps. Les preuves utilisent des estimations locales de Carleman, la différentiation par rapport au domaine, des techniques d'assimilation de données et des résultats de contrôlabilité des EDPs.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.03.006
Doubova, Anna 1; Fernández-Cara, Enrique 1; González-Burgos, Manuel 1; Ortega, Jaime 2, 3

1 Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080 Sevilla, Spain
2 Departamento de Ciencias Básicas, Universidad del Bío-Bío, Casilla 447, Campus Fernando May, Chillán, Chile
3 Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170/3 Correo 3, Santiago, Chile
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     title = {Uniqueness and partial identification in a geometric inverse problem for the {Boussinesq} system},
     journal = {Comptes Rendus. Math\'ematique},
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Doubova, Anna; Fernández-Cara, Enrique; González-Burgos, Manuel; Ortega, Jaime. Uniqueness and partial identification in a geometric inverse problem for the Boussinesq system. Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 665-670. doi : 10.1016/j.crma.2006.03.006. http://www.numdam.org/articles/10.1016/j.crma.2006.03.006/

[1] C. Alvarez, C. Conca, L. Friz, O. Kavian, J.H. Ortega, An inverse problem for the Stokes system, in press

[2] Bello, J.A.; Fernández-Cara, E.; Lemoine, J.; Simon, J. The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier–Stokes flow, SIAM J. Control Optim., Volume 35 (1997) no. 2, pp. 626-640

[3] A. Doubova, E. Fernández-Cara, J.H. Ortega, A geometric inverse problem for the Navier–Stokes equation, in press

[4] Fabre, C.; Lebeau, G. Prolongement unique des solutions de l'équation de Stokes, Comm. Partial Differential Equations, Volume 21 (1996), pp. 573-596

[5] O. Kavian, Four lectures on parameter identification in elliptic partial differential operators, Lectures at the University of Sevilla, Spain, 2002

[6] F. Murat, J. Simon, Quelques résultats sur le contrôle par un domaine géométrique, Rapport du L.A. 189, no. 74003, Université Paris VI, 1974

[7] Puel, J.P. A nonstandard approach to a data assimilation problem, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 335 (2002) no. 2, pp. 161-166

[8] Simon, J. Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optim., Volume 2 (1980), pp. 649-687

[9] Zuazua, E. Finite-dimensional null controllability for the semilinear heat equation, J. Math. Pures Appl. (9), Volume 76 (1997) no. 3, pp. 237-264

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