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doi: 10.3934/mcrf.2022005
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Carleman estimates for a magnetohydrodynamics system and application to inverse source problems

1. 

Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

2. 

JSPS Postdoctoral Fellowships for research in Japan

3. 

Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti, Romania

4. 

Correspondence member of Accademia Peloritana dei Pericolanti, Palazzo Università, Piazza S. Pugliatti 1 98122 Messina, Italy

*Corresponding author: Xinchi Huang

Received  July 2021 Revised  November 2021 Early access March 2022

Fund Project: The first author is supported by JSPS grant 20F20319, the second author is supported by JSPS grant 20H00117, NSFC grant 11771270, 91730303

In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We first prove two kinds of Carleman estimates. This is done by combining the Carleman estimates for the parabolic and the elliptic equations. Then we apply the Carleman estimates to prove Hölder type stability results for some inverse source problems.

Citation: Xinchi Huang, Masahiro Yamamoto. Carleman estimates for a magnetohydrodynamics system and application to inverse source problems. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022005
References:
[1] R. A. Adams and J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Pure and Applied Mathematics Series Vol. 140, Academic Press, 2003. 
[2]

M. BellassouedO. Y. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal., 40 (2008), 238-265.  doi: 10.1137/070679971.

[3]

M. Bellassoued, O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for the Navier-Stokes equations and an application to a lateral Cauchy problem, Inverse Problems, 32 (2016), 025001, 23 pp. doi: 10.1088/0266-5611/32/2/025001.

[4]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer-Japan, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[5]

A. L. Bukhgeim and M. V. Klibanov, Global Uniqueness of a class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247. 

[6]

P. Cannarsa, G. Floridia, F. Gölgeleyen and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 105013, 22 pp. doi: 10.1088/1361-6420/ab1c69.

[7]

P. Cannarsa, G. Floridia and M. Yamamoto, Observability inequalities for transport equations through Carleman estimates, In Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser., Springer, 32 (2019), 69–87. doi: 10.1007/978-3-030-17949-6_4.

[8]

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

[9]

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

[10]

T. HavârneanuC. Popa and S. S. Sritharan, Exact internal controllability for the magnetohydrodynamic equations in multi-connected domains, Adv. Differential Equations, 11 (2006), 893-929. 

[11]

T. HavârneanuC. Popa and S. S. Sritharan, Exact internal controllability for the two-dimensional magnetohydrodynamic equations, SIAM J. Control Optim., 46 (2007), 1802-1830.  doi: 10.1137/040611884.

[12]

X. Huang, Inverse coefficient problem for a magnetohydrodynamics system by Carleman estimates, Appl. Anal., 100 (2021), 1010-1038.  doi: 10.1080/00036811.2019.1632440.

[13]

X. Huang, O. Y. Imanuvilov and M. Yamamoto, Stability for inverse source problems by Carleman estimates, Inverse Problems, 36 (2020), 125006, 20 pp. doi: 10.1088/1361-6420/aba892.

[14]

O. Y. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.

[15]

O. Y. Imanuvilov, L. Lorenzi and M. Yamamoto, Carleman estimate for the Navier-Stokes equations and applications, to appear in Inverse Problems, 2022, arXiv: 2107.04495 [math.AP]. doi: 10.1088/1361-6420/ac4c33.

[16]

O. Y. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, IMRN, 16 (2003), 883-913.  doi: 10.1155/S107379280321117X.

[17]

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

[18]

O. Y. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717-728.  doi: 10.1088/0266-5611/17/4/310.

[19]

O. Y. Imanuvilov and M. Yamamoto, Global Lipschitz stability for an inverse source problem for the Navier-Stokes equations, Applicable Analysis, online published, 2021. doi: 10.1080/00036811.2021.2021189.

[20]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[21]

O. A. Ladyzhenskaya and V. A. Solonnikov, Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids, Journal of Soviet Mathematics, 9 (1978), 697-749. 

[22]

H. Li, Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity, Math. Methods Appl. Sci., 41 (2018), 3062-3092.  doi: 10.1002/mma.4801.

[23] T. Li and T. Qin, Physics and Partial Differential Equations, Volume 1, Higher Education Press, Beijing, 2012. 
[24]

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

[25]

X. Zhou, On uniqueness theorem of a vector function, Progress In Electromagnetics Research, 65 (2006), 93-102.  doi: 10.2528/PIER06081202.

show all references

References:
[1] R. A. Adams and J. F. Fournier, Sobolev Spaces, 2$^{nd}$ edition, Pure and Applied Mathematics Series Vol. 140, Academic Press, 2003. 
[2]

M. BellassouedO. Y. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal., 40 (2008), 238-265.  doi: 10.1137/070679971.

[3]

M. Bellassoued, O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for the Navier-Stokes equations and an application to a lateral Cauchy problem, Inverse Problems, 32 (2016), 025001, 23 pp. doi: 10.1088/0266-5611/32/2/025001.

[4]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer-Japan, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[5]

A. L. Bukhgeim and M. V. Klibanov, Global Uniqueness of a class of multidimensional inverse problems, Sov. Math. Dokl., 24 (1981), 244-247. 

[6]

P. Cannarsa, G. Floridia, F. Gölgeleyen and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 105013, 22 pp. doi: 10.1088/1361-6420/ab1c69.

[7]

P. Cannarsa, G. Floridia and M. Yamamoto, Observability inequalities for transport equations through Carleman estimates, In Trends in Control Theory and Partial Differential Equations, Springer INdAM Ser., Springer, 32 (2019), 69–87. doi: 10.1007/978-3-030-17949-6_4.

[8]

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

[9]

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

[10]

T. HavârneanuC. Popa and S. S. Sritharan, Exact internal controllability for the magnetohydrodynamic equations in multi-connected domains, Adv. Differential Equations, 11 (2006), 893-929. 

[11]

T. HavârneanuC. Popa and S. S. Sritharan, Exact internal controllability for the two-dimensional magnetohydrodynamic equations, SIAM J. Control Optim., 46 (2007), 1802-1830.  doi: 10.1137/040611884.

[12]

X. Huang, Inverse coefficient problem for a magnetohydrodynamics system by Carleman estimates, Appl. Anal., 100 (2021), 1010-1038.  doi: 10.1080/00036811.2019.1632440.

[13]

X. Huang, O. Y. Imanuvilov and M. Yamamoto, Stability for inverse source problems by Carleman estimates, Inverse Problems, 36 (2020), 125006, 20 pp. doi: 10.1088/1361-6420/aba892.

[14]

O. Y. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.

[15]

O. Y. Imanuvilov, L. Lorenzi and M. Yamamoto, Carleman estimate for the Navier-Stokes equations and applications, to appear in Inverse Problems, 2022, arXiv: 2107.04495 [math.AP]. doi: 10.1088/1361-6420/ac4c33.

[16]

O. Y. Imanuvilov and J.-P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems, IMRN, 16 (2003), 883-913.  doi: 10.1155/S107379280321117X.

[17]

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

[18]

O. Y. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717-728.  doi: 10.1088/0266-5611/17/4/310.

[19]

O. Y. Imanuvilov and M. Yamamoto, Global Lipschitz stability for an inverse source problem for the Navier-Stokes equations, Applicable Analysis, online published, 2021. doi: 10.1080/00036811.2021.2021189.

[20]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[21]

O. A. Ladyzhenskaya and V. A. Solonnikov, Unique solvability of an initial- and boundary-value problem for viscous incompressible nonhomogeneous fluids, Journal of Soviet Mathematics, 9 (1978), 697-749. 

[22]

H. Li, Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density-dependent viscosity and resistivity, Math. Methods Appl. Sci., 41 (2018), 3062-3092.  doi: 10.1002/mma.4801.

[23] T. Li and T. Qin, Physics and Partial Differential Equations, Volume 1, Higher Education Press, Beijing, 2012. 
[24]

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

[25]

X. Zhou, On uniqueness theorem of a vector function, Progress In Electromagnetics Research, 65 (2006), 93-102.  doi: 10.2528/PIER06081202.

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