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