Exact boundary controllability of Galerkin approximations of Navier-Stokes system for soret convection
DOI:
https://doi.org/10.11121/ijocta.01.2015.00253Keywords:
Exact boundary controllability, Galerkin approximation, doubly diffusive convection with Soret effectAbstract
We study the exact controllability of finite dimensional Galerkin approximation of a Navier-Stokes type system describing doubly diffusive convection with Soret effect in a bounded smooth domain in Rd (d = 2, 3) with controls on the boundary. The doubly diffusive convection system with Soret effect involves a difficult coupling through second order terms. The Galerkin approximations are introduced undercertain assumptions on the Galerkin basis related to the linear independence of suitable traces of its elements over the boundary. By Using Hilbert uniqueness method in combination with a fixed point argument, we prove that the finite dimensional Galerkin approximations are exactly controllable.Downloads
References
R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
J. Singer, H.H. Bau, Active control of convection, Physics of Fluids A, 3(12) (1991), pp. 2859-2865. Crossref
L. Bergman and M. T. Hyun, Simulation of two dimensional thermosolutal convection in liquid metals induced by horizontal temperature and species gradients, Int. J. Heat Mass Transfer, 39 (1996), pp. 2883. Crossref
J.-M. Coron, On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier slip boundary conditions, ESAIM Control Optim. Calc. Var., 1 (1995/96), pp. 35-75. Crossref
J.-M. Coron and A. V. Fursikov, Global exact controllability of the 2D Navier-Stokes equations on a manifold without boundary, Russian J. Math. Phys., 4 (4) (1996), pp. 429-448.
E. Fernandez-Cara and S. Guerrero, Local exact controllability of micropolar fluids, J. Math. Fluid Mech., 9 (2007), pp. 419-453. Crossref
E. Fernandez-Cara, S. Guerrero, O. Y. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system, J. Math. Pures Appl., 83 (12) (2004), pp. 1501-1542. Crossref
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 Center
E. Gagliardo, Proprieta di alcune classi di funzioni in piu variabili, Ricerche. Mat., 7 (1958), pp. 102-137
V. Girault and P.A. Raviart, Finite Element Method for Navier-Stokes Equations, Springer, Berlin, 1986. Crossref
M. Gad-el-Hak, A. Pollard, and J. P. Bonnet, Flow Control, Fundamentals and Practices, Lecture Notes in Physics, Springer, Berlin, 1998.
O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), pp. 39-72. Crossref
M.D. Gunzburger, Editor, Flow Control, IMA 68, Springer-Verlag, New York, 1995. Crossref
D.T.J. Hurle and E. Jakeman, Soret driven thermo-solutal convection, J. Fluid Mech. 47 (1971), pp. 667–687. Crossref
K. Ito and S. S. Ravindran, Optimal control of thermally convected fluid flows, SIAM J. Scientific Computing, 19(6) (1998), pp. 1847–1869. Crossref
J.L. Lions, Controlabilite exacte, stabilisation et perturbations de systemes distribues. Tomes 1 & 2, Masson, RMA 8&9, Paris, 1988
J.-L. Lions and E. Zuazua, Controlabilite exacte des approximations de Galerkin des equations de Navier-Stokes, C. R. Acad. Sci. Paris, Ser. I, 234 (1997), pp. 1015-1021.
J.-L. Lions and E. Zuazua, Exact boundary controllability of Galerkins approximations of Navier- Stokes equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XXVI (4) (1998), pp. 605-621.
J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: the case of boundary controls, J. Anal. Math., 73 (1997), pp. 225-249. Crossref
I. Mercader, O. Batiste, A. Alonso and E. Knoblock, Convections, anti-convections and multi- convections in binary fluid convection, Journal of Fluid Mechanics, 667 (2011), pp 586-606. Crossref
S.S. Ravindran, Dirichlet control of unsteady Navier-Stokes type system related to Soret convection by boundary penalty method, ESAIM Control, Optimization and Calculus of Variations, 20 (2014), no. 3, pp. 840-863.
V.K. Andreev and I.I. Ryzhkov, Symmetry classification and exact solutions of the thermal diffusion equations, Differential Equations, 41 (4) (2005), pp. 538-547. Crossref
R.T. Rockafellar, Extension of Fenchel's duality theorem for convex functions, Duke Mathematical Journal, 33 (1966), pp. 81-89. Crossref
V. M. Shevtsova, D. E. Melnikov and J. C. Legros, Onset of convection in Soret-driven instability, Physical Review E, 73 (2006), 047302. Crossref
J. Simon, Compact sets in the space Lp(0, T;B), Annali di Matematika Pura ed Applicata (IV), 146 (1987), pp. 65-96.
B.L. Smorodin, Convection of a binary mixture under conditions of thermal diffusion and variable temperature gradient, J. Appl. Mech. Tech. Phys., 43 (2) (2002), pp. 217-223. Crossref
S.S. Sritharan, Editor, Optimal Control of Viscous Flows, SIAM, Philadelphia, 1998. Crossref
G. Yang and N. Zabaras, The adjoint method for an inverse design problem in the directional solidification of binary alloys, Journal of Computational Physics, 40 (1998), pp. 432-452. Crossref
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