RESEARCH ARTICLE
Insensitizing controls for the micropolar fluids
Qiang Tao,
Zheng-an Yao,
Xuan Yin,
Qiang Tao
School of Mathematical Sciences, Shenzhen University, Shenzhen, China
Contribution: Funding acquisition, Writing - original draft
Search for more papers by this authorZheng-an Yao
School of Mathematics, Sun Yat-sen University, Guangzhou, China
Search for more papers by this authorCorresponding Author
Xuan Yin
School of Mathematics, Sun Yat-sen University, Guangzhou, China
Correspondence
Xuan Yin, School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China.
Email: [email protected]
Communicated by: S. Nicaise
Contribution: Writing - original draft
Search for more papers by this authorQiang Tao,
Zheng-an Yao,
Xuan Yin,
Qiang Tao
School of Mathematical Sciences, Shenzhen University, Shenzhen, China
Contribution: Funding acquisition, Writing - original draft
Search for more papers by this authorZheng-an Yao
School of Mathematics, Sun Yat-sen University, Guangzhou, China
Search for more papers by this authorCorresponding Author
Xuan Yin
School of Mathematics, Sun Yat-sen University, Guangzhou, China
Correspondence
Xuan Yin, School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China.
Email: [email protected]
Communicated by: S. Nicaise
Contribution: Writing - original draft
Search for more papers by this authorFirst published: 22 May 2024
Abstract
In this paper, we investigate the existence of insensitizing controls for the micropolar fluids in a bounded domain with homogeneous Dirichlet boundary conditions and arbitrarily located internal controller. The study of insensitizing controls is essential to solve a stability problem, which means that we look for controls such that some functionals of the velocity fields (the so-called sentinels) are insensitive to the small perturbations of initial data. The problem of insensitizing controls is transformed into a suitable controllability problem for a cascade system. Our proof relies on a new global Carleman inequality and the inverse mapping theorem.
CONFLICT OF INTEREST STATEMENT
This work does not have any conflicts of interest.
REFERENCES
- 1A. C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966), 1–18, DOI 10.1512/iumj.1967.16.16001.
- 2A. C. Eringen, Micropolar fluids with stretch, Int. J. Eng. Sci. 7 (1969), no. 1, 115–127.
- 3J. L. Boldrini, M. Durán, and M. A. Rojas-Medar, Existence and uniqueness of strong solution for the incompressible micropolar fluid equations in domains of
, Ann. Univ. Ferrara Sez. VII Sci. Mat. 56 (2010), no. 1, 37–51, DOI 10.1007/s11565-010-0094-0.
10.1007/s11565-010-0094-0 Google Scholar
- 4J. L. Boldrini, M. A. Rojas-Medar, and E. Fernández-Cara, Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. Math. Pures Appl. 82 (2003), no. 11, 1499–1525, DOI 10.1016/j.matpur.2003.09.005.
- 5P. Braz e Silva and E. G. Santos, Global weak solutions for asymmetric incompressible fluids with variable density, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 575–578, DOI 10.1016/j.crma.2008.03.008.
10.1016/j.crma.2008.03.008 Google Scholar
- 6B.-Q. Dong and Z. Zhang, Global regularity of the 2D micropolar fluid flows with zero angular viscosity, J. Differ. Equ. 249 (2010), no. 1, 200–213, DOI 10.1016/j.jde.2010.03.016.
- 7O. Bodart and C. Fabre, Controls insensitizing the norm of the solution of a semilinear heat equation, J. Math. Anal. Appl. 195 (1995), no. 3, 658–683, DOI 10.1006/jmaa.1995.1382.
- 8J.-L. Lions, Quelques notions dans l'analyse et le contrôle de systèmes à données incomplètes, Proceedings of the XIth Congress on Differential Equations and Applications/First Congress on Applied Mathematics (Spanish) (Málaga, 1989), Univ. Málaga, Málaga, 1990, pp. 43–54.
- 9J.-L. Lions, Sentinelles pour les systèmes distribués à données incomplètes, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 21, Masson, Paris, 1992.
- 10L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm. Partial Differ. Equ. 25 (2000), no. 1-2, 39–72, DOI 10.1080/03605300008821507.
- 11O. Bodart, M. González-Burgos, and R. Pérez-García, Existence of insensitizing controls for a semilinear heat equation with a superlinear nonlinearity, Comm. Partial Differ. Equ. 29 (2004), no. 7-8, 1017–1050, DOI 10.1081/PDE-200033749.
- 12O. Bodart, M. González-Burgos, and R. Pérez-García, A local result on insensitizing controls for a semilinear heat equation with nonlinear boundary Fourier conditions, SIAM J. Control Optim. 43 (2004), no. 3, 955–969, DOI 10.1137/S036301290343161X.
- 13S. Guerrero, Null controllability of some systems of two parabolic equations with one control force, SIAM J. Control Optim. 46 (2007), no. 2, 379–394, DOI 10.1137/060653135.
- 14X. Liu, Insensitizing controls for a class of quasilinear parabolic equations, J. Differ. Equ. 253 (2012), no. 5, 1287–1316, DOI 10.1016/j.jde.2012.05.018.
- 15X. Liu, Global Carleman estimate for stochastic parabolic equations, and its application, ESAIM Control Optim. Calc. Var. 20 (2014), no. 3, 823–839, DOI 10.1051/cocv/2013085.
- 16Y. Yan and F. Sun, Insensitizing controls for a forward stochastic heat equation, J. Math. Anal. Appl. 384 (2011), no. 1, 138–150, DOI 10.1016/j.jmaa.2011.05.058.
- 17S. Guerrero, Controllability of systems of Stokes equations with one control force: existence of insensitizing controls, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 6, 1029–1054, DOI 10.1016/j.anihpc.2006.11.001.
- 18M. Gueye, Insensitizing controls for the Navier–Stokes equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013), no. 5, 825–844, DOI 10.1016/j.anihpc.2012.09.005.
- 19N. Carreño and M. Gueye, Insensitizing controls with one vanishing component for the Navier–Stokes system, J. Math. Pures Appl. (9) 101 (2014), no. 1, 27–53, DOI 10.1016/j.matpur.2013.03.007.
- 20N. Carreño and J. Prada, Existence of controls insensitizing the rotational of the solution of the Navier–Stokes system having a vanishing component, Appl. Math. Optim. 88 (2023), no. 2, 37, DOI:10.1007/s00245-023-10011-7.
- 21N. Carreño, S. Guerrero, and M. Gueye, Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system, ESAIM Control Optim. Calc. Var. 21 (2015), no. 1, 73–100, DOI 10.1051/cocv/2014020.
- 22S. Guerrero and P. Cornilleau, On the local exact controllability of micropolar fluids with few controls, ESAIM Control Optim. Calc. Var. 23 (2017), no. 2, 637–662, DOI 10.1051/cocv/2016010.
- 23O. Kavian and L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var. 16 (2010), no. 2, 247–274, DOI 10.1051/cocv/2008077.
- 24S. Micu, J. H. Ortega, and L. de Teresa, An example of -insensitizing controls for the heat equation with no intersecting observation and control regions, Appl. Math. Lett. 17 (2004), no. 8, 927–932, DOI 10.1016/j.aml.2003.10.006.
- 25A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, Lecture Notes Series, vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
- 26N. Carreño, Local controllability of the -dimensional Boussinesq system with scalar controls in an arbitrary control domain, Math. Control Relat. Fields 2 (2012), no. 4, 361–382, DOI 10.3934/mcrf.2012.2.361.
- 27O. Y. Imanuvilov, Remarks on exact controllability for the Navier–Stokes equations, ESAIM Control Optim. Calc. Var. 6 (2001), 39–72, DOI 10.1051/cocv:2001103.
- 28O. Y. Imanuvilov, J. P. Puel, and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chinese Ann. Math. Ser. B 30 (2009), no. 4, 333–378, DOI 10.1007/s11401-008-0280-x.
- 29L. C. Evans, Partial differential equations, Second, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010, DOI:10.1090/gsm/019.
10.1090/gsm/019 Google Scholar
- 30E. Fernández-Cara, S. Guerrero, O. Y. Imanuvilov, and J.-P. Puel, Some controllability results for the -dimensional Navier–Stokes and Boussinesq systems with scalar controls, SIAM J. Control Optim. 45 (2006), no. 1, 146–173, DOI 10.1137/04061965X.
- 31V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal control, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1987. Translated from the Russian by V. M. Volosov.
10.1007/978-1-4615-7551-1 Google Scholar
- 32J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, vol. 2, Travaux et Recherches Mathématiques, vol. 18, Dunod, Paris, 1968.
- 33J.-M. Coron and S. Guerrero, Null controllability of the -dimensional Stokes system with scalar controls, J. Differ. Equ. 246 (2009), no. 7, 2908–2921, DOI 10.1016/j.jde.2008.10.019.
- 34O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1968. Translated from the Russian by S. Smith.
10.1090/mmono/023 Google Scholar
- 35R. Temam, Navier-Stokes equations. Theory and numerical analysis, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
- 36O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, English, Gordon and Breach Science Publishers, New York-London, 1963. Translated from the Russian by Richard A. Silverman.
