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October  2016, 21(8): 2379-2407. doi: 10.3934/dcdsb.2016052

Stability estimates for Navier-Stokes equations and application to inverse problems

Mehdi Badra 1, , Fabien Caubet 2, and  Jérémi Dardé 2,

1. 

Laboratoire LMA, UMR CNRS 5142, Université de Pau et des Pays de l’Adour, 64013 Pau Cedex
2. 

Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France, France

Received  November 2015 Revised  June 2016 Published  September 2016

In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.
Keywords: Cauchy problem., Carleman inequalities, Stability estimate, inverse problems, Navier-Stokes equations.
Mathematics Subject Classification: Primary: 35R30, 35Q30; Secondary: 76D07, 76D0.
Citation: Mehdi Badra, Fabien Caubet, Jérémi Dardé. Stability estimates for Navier-Stokes equations and application to inverse problems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2379-2407. doi: 10.3934/dcdsb.2016052
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003.

[2]

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.

[3]

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.

[4]

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

[5]

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.

[6]

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.

[7]

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.

[8]

F. Ben Belgacem, Why is the cauchy problem severely ill-posed?, Inverse Problems, 23 (2007), 823-836. doi: 10.1088/0266-5611/23/2/020.

[9]

M. Boulakia, A.-C. 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.-C. 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, A duality-based method of quasi-reversibility to solve the cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016, 21pp. doi: 10.1088/0266-5611/26/9/095016.

[12]

H. Cao, M. V. Klibanov and S. V. Pereverzev, A carleman estimate and the balancing principle in the quasi-reversibility method for solving the cauchy problem for the laplace equation, Inverse Problems, 25 (2009), 035005, 21pp. doi: 10.1088/0266-5611/25/3/035005.

[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]

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.

[15]

J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123. doi: 10.1142/S0218202508002620.

[16]

J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Probl. Imaging, 10 (2016), 379-407. doi: 10.3934/ipi.2016005.

[17]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Translated from the French. Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976.

[18]

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.

[19]

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

[20]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[21]

O. Y. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., 16 (2003), 883-913. doi: 10.1155/S107379280321117X.

[22]

M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Appl. Numer. Math., 94 (2015), 46-74. doi: 10.1016/j.apnum.2015.02.003.

[23]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[24]

R. Lattès and J.-L. Lions, The Method of Quasi-reversibility. Applications to Partial Differential Equations, Modern Analytic and Computational Methods in Science and Mathematics. American Elsevier Publishing Co., New-York, 1969.

[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]

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.

[27]

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.

[28]

G. Savaré, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations, 22 (1997), 869-899. doi: 10.1080/03605309708821287.

[29]

E. Sincich, Lipschitz stability for the inverse Robin problem, Inverse Problems, 23 (2007), 1311-1326. doi: 10.1088/0266-5611/23/3/027.

[30]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[31]

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.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003.

[2]

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.

[3]

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.

[4]

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

[5]

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.

[6]

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.

[7]

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.

[8]

F. Ben Belgacem, Why is the cauchy problem severely ill-posed?, Inverse Problems, 23 (2007), 823-836. doi: 10.1088/0266-5611/23/2/020.

[9]

M. Boulakia, A.-C. 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.-C. 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, A duality-based method of quasi-reversibility to solve the cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016, 21pp. doi: 10.1088/0266-5611/26/9/095016.

[12]

H. Cao, M. V. Klibanov and S. V. Pereverzev, A carleman estimate and the balancing principle in the quasi-reversibility method for solving the cauchy problem for the laplace equation, Inverse Problems, 25 (2009), 035005, 21pp. doi: 10.1088/0266-5611/25/3/035005.

[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]

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.

[15]

J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123. doi: 10.1142/S0218202508002620.

[16]

J. Dardé, Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems, Inverse Probl. Imaging, 10 (2016), 379-407. doi: 10.3934/ipi.2016005.

[17]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Translated from the French. Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976.

[18]

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.

[19]

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

[20]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, volume 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[21]

O. Y. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, Int. Math. Res. Not., 16 (2003), 883-913. doi: 10.1155/S107379280321117X.

[22]

M. V. Klibanov, Carleman estimates for the regularization of ill-posed Cauchy problems, Appl. Numer. Math., 94 (2015), 46-74. doi: 10.1016/j.apnum.2015.02.003.

[23]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, Inverse and Ill-posed Problems Series. VSP, Utrecht, 2004. doi: 10.1515/9783110915549.

[24]

R. Lattès and J.-L. Lions, The Method of Quasi-reversibility. Applications to Partial Differential Equations, Modern Analytic and Computational Methods in Science and Mathematics. American Elsevier Publishing Co., New-York, 1969.

[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]

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.

[27]

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.

[28]

G. Savaré, Regularity and perturbation results for mixed second order elliptic problems, Comm. Partial Differential Equations, 22 (1997), 869-899. doi: 10.1080/03605309708821287.

[29]

E. Sincich, Lipschitz stability for the inverse Robin problem, Inverse Problems, 23 (2007), 1311-1326. doi: 10.1088/0266-5611/23/3/027.

[30]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[31]

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.

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