# American Institute of Mathematical Sciences

March  2013, 3(1): 21-49. doi: 10.3934/mcrf.2013.3.21

## Stability estimates for a Robin coefficient in the two-dimensional Stokes system

 1 Université Pierre et Marie Curie-Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, 75005 Paris, France 2 INRIA, Projet REO, Rocquencourt, BP 105, 78153 Le Chesnay cedex, France, France

Received  February 2012 Revised  October 2012 Published  February 2013

In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim [11] and under the assumption that the velocity of a given reference solution stays far from $0$ on a part of the boundary where Robin conditions are prescribed.
Citation: Muriel Boulakia, Anne-Claire Egloffe, Céline Grandmont. Stability estimates for a Robin coefficient in the two-dimensional Stokes system. Mathematical Control and Related Fields, 2013, 3 (1) : 21-49. doi: 10.3934/mcrf.2013.3.21
##### References:
 [1] A.-C. Egloffe, "Étude de Quelques Problèmes Inverses pour le Système de Stokes. Application aux Poumons," Ph.D thesis, Paris 6 University, 2012. [2] F. Boyer and P. Fabrie, "Éléments D'analyse pour L'étude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52, Springer-Verlag, Berlin, 2006. [3] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. [4] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Springer-Verlag, Berlin, 1976. [5] A. Lunardi, "Interpolation Theory," $2^{nd}$ edition, Appunti, Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes, Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009. [6] H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [7] P.-A Raviart and J.-M. Thomas, "Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [8] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [9] R. Dautray and J.-L. Lions, "Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques," Vol. 5, Spectre des Opérateurs, With the collaboration of Michel Artola, Michel Cessenat, Jean Michel Combes and Bruno Scheurer, Reprinted from the 1984 edition, INSTN: Collection Enseignement, Masson, Paris, 1988. [10] M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude, preprint, arXiv:1201.2552, 2012. [11] A. L. Bukhgeĭm, Extension of solutions of elliptic equations from discrete sets, Journal of Inverse and Ill-Posed Problems, 1 (1993), 17-32. doi: 10.1515/jiip.1993.1.1.17. [12] G. Alessandrini, L. Del Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Problems, 19 (2003), 973-984. doi: 10.1088/0266-5611/19/4/312. [13] A. Quarteroni and A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations, Multiscale Modeling & Simulation, 1 (2003), 173-195. doi: 10.1137/S1540345902408482. [14] I. E. Vignon-Clementel, C. A. Figueroa, K. E. Jansen and C. A. Taylor, Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 3776-3796. doi: 10.1016/j.cma.2005.04.014. [15] L. Baffico, C. Grandmont and B. Maury, Multiscale modeling of the respiratory tract, Mathematical Models & Methods in Applied Sciences, 20 (2010), 59-93. doi: 10.1142/S0218202510004155. [16] M. Bellassoued, J. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, Journal of Mathematical Analysis and Applications, 343 (2008), 328-336. doi: 10.1016/j.jmaa.2008.01.066. [17] S. Chaabane, I. Fellah, M. Jaoua and J. Leblond, Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems, 20 (2004), 47-59. doi: 10.1088/0266-5611/20/1/003. [18] J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Mathematical Models & Methods in Applied Sciences, 18 (2008), 107-123. doi: 10.1142/S0218202508002620. [19] S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements, Inverse Problems, 15 (1999), 1425-1438. doi: 10.1088/0266-5611/15/6/303. [20] E. Sincich, Lipschitz stability for the inverse Robin problem, Inverse Problems, 23 (2007), 1311-1326. doi: 10.1088/0266-5611/23/3/027. [21] K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace, ESAIM: Control, Optimization and Calculus of Variations, 9 (2003), 621-635. doi: 10.1051/cocv:2003030. [22] L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains, Applicable Analysis, 89 (2010), 1745-1768. doi: 10.1080/00036810903393809. [23] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: The case of $C^{1,1}$ domains, Mathematical Modelling and Numerical Analysis, 44 (2010), 715-735. doi: 10.1051/m2an/2010016. [24] G. Alessandrini and E. Sincich, Detecting nonlinear corrosion by electrostatic measurements, Applicable Analysis, 85 (2006), 107-128. doi: 10.1080/00036810500277702. [25] C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Communications in Partial Differential Equations, 21 (1996), 573-596. doi: 10.1080/03605309608821198.

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##### References:
 [1] A.-C. Egloffe, "Étude de Quelques Problèmes Inverses pour le Système de Stokes. Application aux Poumons," Ph.D thesis, Paris 6 University, 2012. [2] F. Boyer and P. Fabrie, "Éléments D'analyse pour L'étude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin) [Mathematics & Applications], 52, Springer-Verlag, Berlin, 2006. [3] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," $2^{nd}$ edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003. [4] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Springer-Verlag, Berlin, 1976. [5] A. Lunardi, "Interpolation Theory," $2^{nd}$ edition, Appunti, Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes, Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009. [6] H. Brezis, "Analyse Fonctionnelle. Théorie et Applications," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [7] P.-A Raviart and J.-M. Thomas, "Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles," Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. [8] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [9] R. Dautray and J.-L. Lions, "Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques," Vol. 5, Spectre des Opérateurs, With the collaboration of Michel Artola, Michel Cessenat, Jean Michel Combes and Bruno Scheurer, Reprinted from the 1984 edition, INSTN: Collection Enseignement, Masson, Paris, 1988. [10] M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude, preprint, arXiv:1201.2552, 2012. [11] A. L. Bukhgeĭm, Extension of solutions of elliptic equations from discrete sets, Journal of Inverse and Ill-Posed Problems, 1 (1993), 17-32. doi: 10.1515/jiip.1993.1.1.17. [12] G. Alessandrini, L. Del Piero and L. Rondi, Stable determination of corrosion by a single electrostatic boundary measurement, Inverse Problems, 19 (2003), 973-984. doi: 10.1088/0266-5611/19/4/312. [13] A. Quarteroni and A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations, Multiscale Modeling & Simulation, 1 (2003), 173-195. doi: 10.1137/S1540345902408482. [14] I. E. Vignon-Clementel, C. A. Figueroa, K. E. Jansen and C. A. Taylor, Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 3776-3796. doi: 10.1016/j.cma.2005.04.014. [15] L. Baffico, C. Grandmont and B. Maury, Multiscale modeling of the respiratory tract, Mathematical Models & Methods in Applied Sciences, 20 (2010), 59-93. doi: 10.1142/S0218202510004155. [16] M. Bellassoued, J. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, Journal of Mathematical Analysis and Applications, 343 (2008), 328-336. doi: 10.1016/j.jmaa.2008.01.066. [17] S. Chaabane, I. Fellah, M. Jaoua and J. Leblond, Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems, Inverse Problems, 20 (2004), 47-59. doi: 10.1088/0266-5611/20/1/003. [18] J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Mathematical Models & Methods in Applied Sciences, 18 (2008), 107-123. doi: 10.1142/S0218202508002620. [19] S. Chaabane and M. Jaoua, Identification of Robin coefficients by the means of boundary measurements, Inverse Problems, 15 (1999), 1425-1438. doi: 10.1088/0266-5611/15/6/303. [20] E. Sincich, Lipschitz stability for the inverse Robin problem, Inverse Problems, 23 (2007), 1311-1326. doi: 10.1088/0266-5611/23/3/027. [21] K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace, ESAIM: Control, Optimization and Calculus of Variations, 9 (2003), 621-635. doi: 10.1051/cocv:2003030. [22] L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains, Applicable Analysis, 89 (2010), 1745-1768. doi: 10.1080/00036810903393809. [23] L. Bourgeois, About stability and regularization of ill-posed elliptic Cauchy problems: The case of $C^{1,1}$ domains, Mathematical Modelling and Numerical Analysis, 44 (2010), 715-735. doi: 10.1051/m2an/2010016. [24] G. Alessandrini and E. Sincich, Detecting nonlinear corrosion by electrostatic measurements, Applicable Analysis, 85 (2006), 107-128. doi: 10.1080/00036810500277702. [25] C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Communications in Partial Differential Equations, 21 (1996), 573-596. doi: 10.1080/03605309608821198.
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