doi: 10.3934/dcdss.2020394

Lipschitz stability in determination of coefficients in a two-dimensional Boussinesq system by arbitrary boundary observation

University of Tunis El Manar, National Engineering School of Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis, Tunisia

* Corresponding author: Mourad Bellassoued

Received  November 2019 Revised  April 2020 Published  June 2020

Fund Project: The authors are supported by the Tunisian Research Program PHC-UTIQUE 19G-1507

In this paper, we consider the inverse problem of determining two spatially varying coefficients appearing in the two-dimensional Boussinesq system from observed data of velocity vector and the temperature in a given arbitrarily subboundary. Based on Carleman estimates, we prove a Lipschitz stability result.

Citation: Mourad Bellassoued, Chaima Moufid. Lipschitz stability in determination of coefficients in a two-dimensional Boussinesq system by arbitrary boundary observation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020394
References:
[1]

M. Bellassoued, M. Cristofol and E. Soccorsi, Inverse boundary value problem for the dynamical heterogeneous Maxwell's system, Inverse Problems, 28 (2012), 095009, 18 pp. doi: 10.1088/0266-5611/28/9/095009.  Google Scholar

[2]

M. BellassouedO. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM Journal on Mathematical Analysis, 40 (2008), 238-265.  doi: 10.1137/070679971.  Google Scholar

[3]

M. Bellassoued and B. Riahi, Carleman estimate for Biot consolidation system in poro-elasticity and application to inverse problems, Mathematical Methods in the Applied Sciences, 39 (2016), 5281-5301.  doi: 10.1002/mma.3914.  Google Scholar

[4]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1080/00036811.2014.986847.  Google Scholar

[5]

M. Bellassoued and M. Yamamoto, Carleman estimates and an inverse heat source problem for the thermoelasticity system, Inverse Problems, 27 (2011), 015006, 18 pp. doi: 10.1088/0266-5611/27/1/015006.  Google Scholar

[6]

M. Bellassoued and M. Yamamoto, Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases, Applicable Analysis, 91 (2012), 35-67.  doi: 10.1080/00036811.2010.534731.  Google Scholar

[7]

M. Bellassoued and M. Yamamoto, Carleman estimate and inverse source problem for Biot's equations describing wave propagation in porous media, Inverse Problems, 29 (2013), 115002, 20 pp. doi: 10.1088/0266-5611/29/11/115002.  Google Scholar

[8]

M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation, Journal des Mathéematiques Pures et Appliquées, 85 (2006), 193-224.  doi: 10.1016/j.matpur.2005.02.004.  Google Scholar

[9]

A. L. BukhgeimJ. ChengV. Isakov and M. Yamamoto, Uniqueness in determining damping coefficients in hyperbolic equations, Analytic Extension Formulas and their Applications, Int. Soc. Anal. Appl. Comput., Kluwer Acad. Publ., Dordrecht, 9 (2001), 27-46.  doi: 10.1007/978-1-4757-3298-6_3.  Google Scholar

[10]

A. Bukhghe$\mathbf{v}$ım and M. Klibanov, Global uniqueness of class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247.   Google Scholar

[11]

T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 26 (1939), 9 pp.  Google Scholar

[12]

M. ChoulliO. Y. ImanuvilovJ.-P. Puel and M. Yamamoto, Inverse source problem for linearized Navier-Stokes equations with data in arbitrary sub-domain, Applicable Analysis, 92 (2013), 2127-2143.  doi: 10.1080/00036811.2012.718334.  Google Scholar

[13]

J. S. FanM. Di CristoY. Jiang and G. Nakamura, Inverse viscosity problem for the Navier-Stokes equation, Journal of Mathematical Analysis and Applications, 365 (2010), 750-757.  doi: 10.1016/j.jmaa.2009.12.012.  Google Scholar

[14]

J. S. Fan, Y. Jiang and G. Nakamura, Inverse problems for the Boussinesq system, Inverse Problems, 25 (2009), 085007, 10 pp. doi: 10.1088/0266-5611/25/8/085007.  Google Scholar

[15]

J. S. FanF. C. Li and G. Nakamura, Regularity criteria for the boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain, Discrete Contin. Dyn. Syst., 36 (2016), 4915-4923.  doi: 10.3934/dcds.2016012.  Google Scholar

[16]

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

[17]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Second edition, Grundlehren der Mathematischen Wissenschaften, 256. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[18]

O. Y. Émanuilov, Controllability of parabolic equations, Sbornik Mathematics, 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.  Google Scholar

[19]

O. Y. Imanuvilov, Local exact controllability for the 2-D Boussinesq equations with the Navier slip boundary conditions, Control and Partial Differential Equations, ESAIM Proc., Soc. Math. Appl. Indust., Paris, 4 (1990), 153-170.  doi: 10.1051/proc:1998026.  Google Scholar

[20]

O. ImanuvilovV. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data, Communications on Pure and Applied Mathematics, 56 (2003), 1366-1382.  doi: 10.1002/cpa.10097.  Google Scholar

[21]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.  Google Scholar

[22]

V. Isakov and N. Kim, Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress, Discrete and Continuous Dynamical Systems, 27 (2010), 799-825.  doi: 10.3934/dcds.2010.27.799.  Google Scholar

[23]

L. B. Jin and J. S. Fan, Uniform regularity for the 2D Boussinesq system with a slip boundary condition, Journal of Mathematical Analysis and Applications, 400 (2013), 96-99.  doi: 10.1016/j.jmaa.2012.10.051.  Google Scholar

[24]

L. B. Jin, J. S. Fan, G. Nakamura and Y. Zhou, Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition, Boundary Value Problems, 2012 (2012), 5 pp. doi: 10.1186/1687-2770-2012-20.  Google Scholar

[25]

M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation, Applicable Analysis, 85 (2006), 515-538.  doi: 10.1080/00036810500474788.  Google Scholar

[26]

H. P. LiR. H. Pan and W. Z. Zhang, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion, Journal of Hyperbolic Differential Equations, 12 (2015), 469-488.  doi: 10.1142/S0219891615500137.  Google Scholar

[27]

S. A. Lorca and J. L. Boldrini, The initial value problem for a generalized Boussinesq model, Nonlinear Analysis, Ser. A: Methods and Applications, 36 (1999), 457-480.  doi: 10.1016/S0362-546X(97)00635-4.  Google Scholar

[28]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[29]

Y. M. QinX. SuY. Wang and J. L. Zhang, Global regularity for a two-dimensional nonlinear Boussinesq system, Mathematical Methods in the Applied Sciences, 40 (2017), 2042-2056.  doi: 10.1002/mma.4118.  Google Scholar

[30]

Y. Z. Sun and Z. F. Zhang, Global regularity for the initial boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085.  doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[31]

C. Wang and Z. F. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Advances in Mathematics, 228 (2011), 43-62.  doi: 10.1016/j.aim.2011.05.008.  Google Scholar

[32]

B. Wu and J. J. Liu, Conditional stability and uniqueness for determining two coefficients in a hyperbolic-parabolic system, Inverse Problems, 27 (2011), 075013, 18 pp. doi: 10.1088/0266-5611/27/7/075013.  Google Scholar

[33]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75 pp. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

[34]

G. H. Yuan and M. Yamamoto, Lipschitz stability in the determination of the principal part of a parabolic equation, ESAIM Control Optim. Calc. Var., 15 (2009), 525-554.  doi: 10.1051/cocv:2008043.  Google Scholar

[35]

K. Zhao, 2D inviscid heat conductive Boussinesq equations on a bounded domain, Michigan Mathematical Journal, 59 (2010), 329-352.  doi: 10.1307/mmj/1281531460.  Google Scholar

[36]

D. G. Zhou, Global regularity of the two-dimensional Boussinesq equations without diffusivity in bounded domains, Nonlinear Analysis: Real World Applications, 43 (2018), 144-154.  doi: 10.1016/j.nonrwa.2018.02.009.  Google Scholar

show all references

References:
[1]

M. Bellassoued, M. Cristofol and E. Soccorsi, Inverse boundary value problem for the dynamical heterogeneous Maxwell's system, Inverse Problems, 28 (2012), 095009, 18 pp. doi: 10.1088/0266-5611/28/9/095009.  Google Scholar

[2]

M. BellassouedO. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM Journal on Mathematical Analysis, 40 (2008), 238-265.  doi: 10.1137/070679971.  Google Scholar

[3]

M. Bellassoued and B. Riahi, Carleman estimate for Biot consolidation system in poro-elasticity and application to inverse problems, Mathematical Methods in the Applied Sciences, 39 (2016), 5281-5301.  doi: 10.1002/mma.3914.  Google Scholar

[4]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1080/00036811.2014.986847.  Google Scholar

[5]

M. Bellassoued and M. Yamamoto, Carleman estimates and an inverse heat source problem for the thermoelasticity system, Inverse Problems, 27 (2011), 015006, 18 pp. doi: 10.1088/0266-5611/27/1/015006.  Google Scholar

[6]

M. Bellassoued and M. Yamamoto, Carleman estimate with second large parameter for second order hyperbolic operators in a Riemannian manifold and applications in thermoelasticity cases, Applicable Analysis, 91 (2012), 35-67.  doi: 10.1080/00036811.2010.534731.  Google Scholar

[7]

M. Bellassoued and M. Yamamoto, Carleman estimate and inverse source problem for Biot's equations describing wave propagation in porous media, Inverse Problems, 29 (2013), 115002, 20 pp. doi: 10.1088/0266-5611/29/11/115002.  Google Scholar

[8]

M. Bellassoued and M. Yamamoto, Logarithmic stability in determination of a coefficient in an acoustic equation by arbitrary boundary observation, Journal des Mathéematiques Pures et Appliquées, 85 (2006), 193-224.  doi: 10.1016/j.matpur.2005.02.004.  Google Scholar

[9]

A. L. BukhgeimJ. ChengV. Isakov and M. Yamamoto, Uniqueness in determining damping coefficients in hyperbolic equations, Analytic Extension Formulas and their Applications, Int. Soc. Anal. Appl. Comput., Kluwer Acad. Publ., Dordrecht, 9 (2001), 27-46.  doi: 10.1007/978-1-4757-3298-6_3.  Google Scholar

[10]

A. Bukhghe$\mathbf{v}$ım and M. Klibanov, Global uniqueness of class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247.   Google Scholar

[11]

T. Carleman, Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 26 (1939), 9 pp.  Google Scholar

[12]

M. ChoulliO. Y. ImanuvilovJ.-P. Puel and M. Yamamoto, Inverse source problem for linearized Navier-Stokes equations with data in arbitrary sub-domain, Applicable Analysis, 92 (2013), 2127-2143.  doi: 10.1080/00036811.2012.718334.  Google Scholar

[13]

J. S. FanM. Di CristoY. Jiang and G. Nakamura, Inverse viscosity problem for the Navier-Stokes equation, Journal of Mathematical Analysis and Applications, 365 (2010), 750-757.  doi: 10.1016/j.jmaa.2009.12.012.  Google Scholar

[14]

J. S. Fan, Y. Jiang and G. Nakamura, Inverse problems for the Boussinesq system, Inverse Problems, 25 (2009), 085007, 10 pp. doi: 10.1088/0266-5611/25/8/085007.  Google Scholar

[15]

J. S. FanF. C. Li and G. Nakamura, Regularity criteria for the boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain, Discrete Contin. Dyn. Syst., 36 (2016), 4915-4923.  doi: 10.3934/dcds.2016012.  Google Scholar

[16]

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

[17]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Second edition, Grundlehren der Mathematischen Wissenschaften, 256. Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[18]

O. Y. Émanuilov, Controllability of parabolic equations, Sbornik Mathematics, 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.  Google Scholar

[19]

O. Y. Imanuvilov, Local exact controllability for the 2-D Boussinesq equations with the Navier slip boundary conditions, Control and Partial Differential Equations, ESAIM Proc., Soc. Math. Appl. Indust., Paris, 4 (1990), 153-170.  doi: 10.1051/proc:1998026.  Google Scholar

[20]

O. ImanuvilovV. Isakov and M. Yamamoto, An inverse problem for the dynamical Lamé system with two sets of boundary data, Communications on Pure and Applied Mathematics, 56 (2003), 1366-1382.  doi: 10.1002/cpa.10097.  Google Scholar

[21]

O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), 1229-1245.  doi: 10.1088/0266-5611/14/5/009.  Google Scholar

[22]

V. Isakov and N. Kim, Weak Carleman estimates with two large parameters for second order operators and applications to elasticity with residual stress, Discrete and Continuous Dynamical Systems, 27 (2010), 799-825.  doi: 10.3934/dcds.2010.27.799.  Google Scholar

[23]

L. B. Jin and J. S. Fan, Uniform regularity for the 2D Boussinesq system with a slip boundary condition, Journal of Mathematical Analysis and Applications, 400 (2013), 96-99.  doi: 10.1016/j.jmaa.2012.10.051.  Google Scholar

[24]

L. B. Jin, J. S. Fan, G. Nakamura and Y. Zhou, Partial vanishing viscosity limit for the 2D Boussinesq system with a slip boundary condition, Boundary Value Problems, 2012 (2012), 5 pp. doi: 10.1186/1687-2770-2012-20.  Google Scholar

[25]

M. V. Klibanov and M. Yamamoto, Lipschitz stability of an inverse problem for an acoustic equation, Applicable Analysis, 85 (2006), 515-538.  doi: 10.1080/00036810500474788.  Google Scholar

[26]

H. P. LiR. H. Pan and W. Z. Zhang, Initial boundary value problem for 2D Boussinesq equations with temperature-dependent diffusion, Journal of Hyperbolic Differential Equations, 12 (2015), 469-488.  doi: 10.1142/S0219891615500137.  Google Scholar

[27]

S. A. Lorca and J. L. Boldrini, The initial value problem for a generalized Boussinesq model, Nonlinear Analysis, Ser. A: Methods and Applications, 36 (1999), 457-480.  doi: 10.1016/S0362-546X(97)00635-4.  Google Scholar

[28]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[29]

Y. M. QinX. SuY. Wang and J. L. Zhang, Global regularity for a two-dimensional nonlinear Boussinesq system, Mathematical Methods in the Applied Sciences, 40 (2017), 2042-2056.  doi: 10.1002/mma.4118.  Google Scholar

[30]

Y. Z. Sun and Z. F. Zhang, Global regularity for the initial boundary value problem of the 2-D Boussinesq system with variable viscosity and thermal diffusivity, J. Differential Equations, 255 (2013), 1069-1085.  doi: 10.1016/j.jde.2013.04.032.  Google Scholar

[31]

C. Wang and Z. F. Zhang, Global well-posedness for the 2-D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity, Advances in Mathematics, 228 (2011), 43-62.  doi: 10.1016/j.aim.2011.05.008.  Google Scholar

[32]

B. Wu and J. J. Liu, Conditional stability and uniqueness for determining two coefficients in a hyperbolic-parabolic system, Inverse Problems, 27 (2011), 075013, 18 pp. doi: 10.1088/0266-5611/27/7/075013.  Google Scholar

[33]

M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013, 75 pp. doi: 10.1088/0266-5611/25/12/123013.  Google Scholar

[34]

G. H. Yuan and M. Yamamoto, Lipschitz stability in the determination of the principal part of a parabolic equation, ESAIM Control Optim. Calc. Var., 15 (2009), 525-554.  doi: 10.1051/cocv:2008043.  Google Scholar

[35]

K. Zhao, 2D inviscid heat conductive Boussinesq equations on a bounded domain, Michigan Mathematical Journal, 59 (2010), 329-352.  doi: 10.1307/mmj/1281531460.  Google Scholar

[36]

D. G. Zhou, Global regularity of the two-dimensional Boussinesq equations without diffusivity in bounded domains, Nonlinear Analysis: Real World Applications, 43 (2018), 144-154.  doi: 10.1016/j.nonrwa.2018.02.009.  Google Scholar

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