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Local feedback stabilisation to a non-stationary solution for a damped non-linear wave equation
Quantification of the unique continuation property for the nonstationary Stokes problem
1. | Sorbonne Universités, UPMC Paris 06, UMR 7598, LJLL, F-75005, Paris, France |
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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