American Institute of Mathematical Sciences

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March  2016, 6(1): 27-52. doi: 10.3934/mcrf.2016.6.27

Quantification of the unique continuation property for the nonstationary Stokes problem

Muriel Boulakia 1,

1. 

Sorbonne Universités, UPMC Paris 06, UMR 7598, LJLL, F-75005, Paris, France

Received  December 2014 Revised  September 2015 Published  January 2016

The purpose of this work is to establish stability estimates for the unique continuation property of the nonstationary Stokes problem. These estimates hold without prescribing boundary conditions and are of logarithmic type. They are obtained thanks to Carleman estimates for parabolic and elliptic equations. Then, these estimates are applied to an inverse problem where we want to identify a Robin coefficient defined on some part of the boundary from measurements available on another part of the boundary.
Keywords: unsteady Stokes system., Inverse problems, identification of parameters, unique continuation property.
Mathematics Subject Classification: 35B35, 35R30, 76D0.
Citation: Muriel Boulakia. Quantification of the unique continuation property for the nonstationary Stokes problem. Mathematical Control and Related Fields, 2016, 6 (1) : 27-52. doi: 10.3934/mcrf.2016.6.27
References:
[1]

G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), 755-806.

[2]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004.

[3]

G. Alessandrini and E. Sincich, Detecting nonlinear corrosion by electrostatic measurements, Appl. Anal., 85 (2006), 107-128. doi: 10.1080/00036810500277702.

[4]

L. Baffico, C. Grandmont and B. Maury, Multiscale modeling of the respiratory tract, Math. Models Methods Appl. Sci., 20 (2010), 59-93. doi: 10.1142/S0218202510004155.

[5]

A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015, 25pp. doi: 10.1088/0266-5611/26/12/125015.

[6]

M. Bellassoued, J. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math. Anal. Appl., 343 (2008), 328-336. doi: 10.1016/j.jmaa.2008.01.066.

[7]

M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude, Math. Methods Appl. Sci., 36 (2013), 2429-2448. doi: 10.1002/mma.2762.

[8]

M. Bellassoued, O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the navier-stokes equations and an application to a lateral cauchy problem,, preprint., (). 

[9]

M. Boulakia, A. Egloffe and C. Grandmont, Stability estimates for a Robin coefficient in the two-dimensional Stokes system, Math. Control Relat. Fields, 3 (2013), 21-49. doi: 10.3934/mcrf.2013.3.21.

[10]

M. Boulakia, A. Egloffe and C. Grandmont, Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem, Inverse Problems, 29 (2013), 115001, 21pp. doi: 10.1088/0266-5611/29/11/115001.

[11]

L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains, Appl. Anal., 89 (2010), 1745-1768. doi: 10.1080/00036810903393809.

[12]

F. Boyer and P. Fabrie, Éléments D'analyse Pour L'étude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles, volume 52 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-29819-3.

[13]

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.

[14]

A.-C. Egloffe, Étude de Quelques Problèmes Inverses Pour le Système de Stokes. Application Aux Poumons, PhD thesis, Université Paris VI, 2012.

[15]

C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Comm. Partial Differential Equations, 21 (1996), 573-596. doi: 10.1080/03605309608821198.

[16]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), 1501-1542. doi: 10.1016/j.matpur.2004.02.010.

[17]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, volume 34 of Lecture Notes Series, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.

[18]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[19]

L. Hörmander, The Analysis of Linear Partial Differential Operators, III, volume 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1985. Pseudodifferential operators.

[20]

M. Ignatova and I. Kukavica, Strong unique continuation for the Navier-Stokes equation with non-analytic forcing, J. Dynam. Differential Equations, 25 (2013), 1-15. doi: 10.1007/s10884-012-9282-1.

[21]

O. Yu. Imanuvilov, On exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 3 (1998), 97-131. doi: 10.1051/cocv:1998104.

[22]

O. Y. Imanuvilov and M. Yamamoto, Equivalence of two inverse boundary value problems for the navier-stokes equations, http://arxiv.org/pdf/1501.02550v1.pdf, 2015.

[23]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585. doi: 10.1002/cpa.3160130402.

[24]

E. M. Landis, Some questions in the qualitative theory of elliptic and parabolic equations, Amer. Math. Soc. Transl. (2), 20 (1962), 173-238.

[25]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747. doi: 10.1051/cocv/2011168.

[26]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097.

[27]

C.-L. Lin, G. Uhlmann and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst., 28 (2010), 1273-1290. doi: 10.3934/dcds.2010.28.1273.

[28]

K.-D. Phung, Remarques sur l'observabilité pour l'équation de laplace, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 621-635. doi: 10.1051/cocv:2003030.

[29]

A. Quarteroni and A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations, Multiscale Model. Simul., 1 (2003), 173-195 (electronic). doi: 10.1137/S1540345902408482.

[30]

L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques, Comm. Partial Differential Equations, 16 (1991), 789-800. doi: 10.1080/03605309108820778.

[31]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.

[32]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182. doi: 10.1007/BF02387373.

[33]

S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates, Inverse Problems, 24 (2008), 023001, 81pp. doi: 10.1088/0266-5611/24/2/023001.

[34]

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, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3776-3796. doi: 10.1016/j.cma.2005.04.014.

[35]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013. doi: 10.1088/0266-5611/25/12/123013.

show all references

References:
[1]

G. Alessandrini, E. Beretta, E. Rosset and S. Vessella, Optimal stability for inverse elliptic boundary value problems with unknown boundaries, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29 (2000), 755-806.

[2]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004, 47pp. doi: 10.1088/0266-5611/25/12/123004.

[3]

G. Alessandrini and E. Sincich, Detecting nonlinear corrosion by electrostatic measurements, Appl. Anal., 85 (2006), 107-128. doi: 10.1080/00036810500277702.

[4]

L. Baffico, C. Grandmont and B. Maury, Multiscale modeling of the respiratory tract, Math. Models Methods Appl. Sci., 20 (2010), 59-93. doi: 10.1142/S0218202510004155.

[5]

A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015, 25pp. doi: 10.1088/0266-5611/26/12/125015.

[6]

M. Bellassoued, J. Cheng and M. Choulli, Stability estimate for an inverse boundary coefficient problem in thermal imaging, J. Math. Anal. Appl., 343 (2008), 328-336. doi: 10.1016/j.jmaa.2008.01.066.

[7]

M. Bellassoued, M. Choulli and A. Jbalia, Stability of the determination of the surface impedance of an obstacle from the scattering amplitude, Math. Methods Appl. Sci., 36 (2013), 2429-2448. doi: 10.1002/mma.2762.

[8]

M. Bellassoued, O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the navier-stokes equations and an application to a lateral cauchy problem,, preprint., (). 

[9]

M. Boulakia, A. Egloffe and C. Grandmont, Stability estimates for a Robin coefficient in the two-dimensional Stokes system, Math. Control Relat. Fields, 3 (2013), 21-49. doi: 10.3934/mcrf.2013.3.21.

[10]

M. Boulakia, A. Egloffe and C. Grandmont, Stability estimates for the unique continuation property of the Stokes system and for an inverse boundary coefficient problem, Inverse Problems, 29 (2013), 115001, 21pp. doi: 10.1088/0266-5611/29/11/115001.

[11]

L. Bourgeois and J. Dardé, About stability and regularization of ill-posed elliptic Cauchy problems: The case of Lipschitz domains, Appl. Anal., 89 (2010), 1745-1768. doi: 10.1080/00036810903393809.

[12]

F. Boyer and P. Fabrie, Éléments D'analyse Pour L'étude de Quelques Modèles D'écoulements de Fluides Visqueux Incompressibles, volume 52 of Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-29819-3.

[13]

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.

[14]

A.-C. Egloffe, Étude de Quelques Problèmes Inverses Pour le Système de Stokes. Application Aux Poumons, PhD thesis, Université Paris VI, 2012.

[15]

C. Fabre and G. Lebeau, Prolongement unique des solutions de l'equation de Stokes, Comm. Partial Differential Equations, 21 (1996), 573-596. doi: 10.1080/03605309608821198.

[16]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl. (9), 83 (2004), 1501-1542. doi: 10.1016/j.matpur.2004.02.010.

[17]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, volume 34 of Lecture Notes Series, Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul, 1996.

[18]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition.

[19]

L. Hörmander, The Analysis of Linear Partial Differential Operators, III, volume 274 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1985. Pseudodifferential operators.

[20]

M. Ignatova and I. Kukavica, Strong unique continuation for the Navier-Stokes equation with non-analytic forcing, J. Dynam. Differential Equations, 25 (2013), 1-15. doi: 10.1007/s10884-012-9282-1.

[21]

O. Yu. Imanuvilov, On exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 3 (1998), 97-131. doi: 10.1051/cocv:1998104.

[22]

O. Y. Imanuvilov and M. Yamamoto, Equivalence of two inverse boundary value problems for the navier-stokes equations, http://arxiv.org/pdf/1501.02550v1.pdf, 2015.

[23]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585. doi: 10.1002/cpa.3160130402.

[24]

E. M. Landis, Some questions in the qualitative theory of elliptic and parabolic equations, Amer. Math. Soc. Transl. (2), 20 (1962), 173-238.

[25]

J. Le Rousseau and G. Lebeau, On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747. doi: 10.1051/cocv/2011168.

[26]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356. doi: 10.1080/03605309508821097.

[27]

C.-L. Lin, G. Uhlmann and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst., 28 (2010), 1273-1290. doi: 10.3934/dcds.2010.28.1273.

[28]

K.-D. Phung, Remarques sur l'observabilité pour l'équation de laplace, ESAIM: Control, Optimisation and Calculus of Variations, 9 (2003), 621-635. doi: 10.1051/cocv:2003030.

[29]

A. Quarteroni and A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations, Multiscale Model. Simul., 1 (2003), 173-195 (electronic). doi: 10.1137/S1540345902408482.

[30]

L. Robbiano, Théorème d'unicité adapté au contrôle des solutions des problèmes hyperboliques, Comm. Partial Differential Equations, 16 (1991), 789-800. doi: 10.1080/03605309108820778.

[31]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.

[32]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182. doi: 10.1007/BF02387373.

[33]

S. Vessella, Quantitative estimates of unique continuation for parabolic equations, determination of unknown time-varying boundaries and optimal stability estimates, Inverse Problems, 24 (2008), 023001, 81pp. doi: 10.1088/0266-5611/24/2/023001.

[34]

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, Comput. Methods Appl. Mech. Engrg., 195 (2006), 3776-3796. doi: 10.1016/j.cma.2005.04.014.

[35]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013. doi: 10.1088/0266-5611/25/12/123013.

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