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 & 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, (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 (2006).

[3]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", $2^{nd}$ edition, 140 (2003).

[4]

J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Springer-Verlag, (1976).

[5]

A. Lunardi, "Interpolation Theory,", $2^{nd}$ edition, (2009).

[6]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Mathématiques Appliquées pour la Maîtrise, (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, (1983).

[8]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (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, (1984).

[10]

M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude,, preprint, (2012).

[11]

A. L. Bukhgeĭm, Extension of solutions of elliptic equations from discrete sets,, Journal of Inverse and Ill-Posed Problems, 1 (1993), 17. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1088/0266-5611/15/6/303.

[20]

E. Sincich, Lipschitz stability for the inverse Robin problem,, Inverse Problems, 23 (2007), 1311. doi: 10.1088/0266-5611/23/3/027.

[21]

K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace,, ESAIM: Control, 9 (2003), 621. 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. 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. doi: 10.1051/m2an/2010016.

[24]

G. Alessandrini and E. Sincich, Detecting nonlinear corrosion by electrostatic measurements,, Applicable Analysis, 85 (2006), 107. 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. doi: 10.1080/03605309608821198.

show all references

References:
[1]

A.-C. Egloffe, "Étude de Quelques Problèmes Inverses pour le Système de Stokes. Application aux Poumons,", Ph.D thesis, (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 (2006).

[3]

R. A. Adams and J. J. F. Fournier, "Sobolev Spaces,", $2^{nd}$ edition, 140 (2003).

[4]

J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Springer-Verlag, (1976).

[5]

A. Lunardi, "Interpolation Theory,", $2^{nd}$ edition, (2009).

[6]

H. Brezis, "Analyse Fonctionnelle. Théorie et Applications,", Collection Mathématiques Appliquées pour la Maîtrise, (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, (1983).

[8]

A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (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, (1984).

[10]

M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude,, preprint, (2012).

[11]

A. L. Bukhgeĭm, Extension of solutions of elliptic equations from discrete sets,, Journal of Inverse and Ill-Posed Problems, 1 (1993), 17. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1088/0266-5611/15/6/303.

[20]

E. Sincich, Lipschitz stability for the inverse Robin problem,, Inverse Problems, 23 (2007), 1311. doi: 10.1088/0266-5611/23/3/027.

[21]

K.-D. Phung, Remarques sur l'observabilité pour l'équation de Laplace,, ESAIM: Control, 9 (2003), 621. 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. 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. doi: 10.1051/m2an/2010016.

[24]

G. Alessandrini and E. Sincich, Detecting nonlinear corrosion by electrostatic measurements,, Applicable Analysis, 85 (2006), 107. 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. doi: 10.1080/03605309608821198.

[1]

Xiaofei Cao, Guowei Dai. Stability analysis of a model on varying domain with the Robin boundary condition. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 935-942. doi: 10.3934/dcdss.2017048

[2]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367

[3]

Lucie Baudouin, Emmanuelle Crépeau, Julie Valein. Global Carleman estimate on a network for the wave equation and application to an inverse problem. Mathematical Control & Related Fields, 2011, 1 (3) : 307-330. doi: 10.3934/mcrf.2011.1.307

[4]

Raffaela Capitanelli. Robin boundary condition on scale irregular fractals. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1221-1234. doi: 10.3934/cpaa.2010.9.1221

[5]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[6]

Jie Liao, Xiao-Ping Wang. Stability of an efficient Navier-Stokes solver with Navier boundary condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 153-171. doi: 10.3934/dcdsb.2012.17.153

[7]

Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609

[8]

J. F. Padial. Existence and estimate of the location of the free-boundary for a non local inverse elliptic-parabolic problem arising in nuclear fusion. Conference Publications, 2011, 2011 (Special) : 1176-1185. doi: 10.3934/proc.2011.2011.1176

[9]

Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231

[10]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393

[11]

R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497

[12]

Samia Challal, Abdeslem Lyaghfouri. The heterogeneous dam problem with leaky boundary condition. Communications on Pure & Applied Analysis, 2011, 10 (1) : 93-125. doi: 10.3934/cpaa.2011.10.93

[13]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615

[14]

Eemeli Blåsten, Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Problems & Imaging, 2015, 9 (3) : 709-723. doi: 10.3934/ipi.2015.9.709

[15]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343

[16]

Gen Nakamura, Michiyuki Watanabe. An inverse boundary value problem for a nonlinear wave equation. Inverse Problems & Imaging, 2008, 2 (1) : 121-131. doi: 10.3934/ipi.2008.2.121

[17]

Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985

[18]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems & Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014

[19]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks & Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[20]

Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic & Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865

2017 Impact Factor: 0.631

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (8)

[Back to Top]