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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

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