Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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A BOUNDARY CONTROL PROBLEM FOR VORTICITY MINIMIZATION IN TIME-DEPENDENT 2D NAVIER-STOKES EQUATIONS

  • KIM, HONGCHUL (Department of Mathematics Gangneung-Wonju National University)
  • Received : 2015.01.21
  • Accepted : 2015.06.09
  • Published : 2015.06.30
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Abstract

We deal with a boundary control problem for the vorticity minimization, in which the ow is governed by the time-dependent two dimensional incompressible Navier-Stokes equations. We derive a mathematical formulation and a process for an appropriate control along the portion of the boundary to minimize the vorticity motion due to the ow in the fluid domain. After showing the existence of an optimal solution, we derive the optimality system for which optimal solutions may be determined. The differentiability of the state solution in regard to the control parameter shall be conjunct with the necessary conditions for the optimal solutions.

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References

  1. F. ABERGEL, AND R. TEMAM, On some control problems in fluid mechanics, Theoretical and Computational Fluid Dynamics, 1 (1990), 303-325. https://doi.org/10.1007/BF00271794
  2. A. BUFFA, M. COSTABEL, AND D. SHEEN, On traces for H(curl, $\Omega) in Lips-chitz domains, J. Math. Anal. Appl., 276 (2002), 845-867. https://doi.org/10.1016/S0022-247X(02)00455-9
  3. P. CONSTANTIN AND I. FOIAS, Navier-Stokes equations, The University of Chicago Press, Chicago (1989).
  4. R. DAUTRAY AND J.-L. LIONS, Analysis and Numerical Methods for Science and Technology, Vol 2, Springer-Verlag, New York (1993).
  5. A. V. FURSIKOV, M. D. GUNZBURGER, AND L. S. HOU, Boundary vlue problems and optimal boundary control for the Navier-Stokes system: The two-dimensional case, SIAM J. Cont. Optim., 36 (3) (1998), 852-894. https://doi.org/10.1137/S0363012994273374
  6. V. GIRAULT AND P. RAVIART, The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, New York (1986).
  7. M. GUNZBURGER AND S. MANSERVISI, The velocity tracking problem for Navier-Stokes flows with boundary control, SIAM J. Cont. Optim., 39 (2) (2000), 594-634. https://doi.org/10.1137/S0363012999353771
  8. HONGCHUL KIM AND OH-KEUN KWON, On a vorticity minimization problem for the stationaary 2D Stokes equations, J. Korean Math. Soc., 43 (1) (2006), 45-63. https://doi.org/10.4134/JKMS.2006.43.1.045
  9. HONGCHUL KIM AND SEON-KYU KIM, A bondary control problem for the time-dependent 2D Navier-Stokes equations, Korean J. Math., 16 (1) (2008), 57-84.
  10. O. LADYZHENSKAYA, The Mathematical Theory of Viscous Incompressible Flow, Goldon and Breach, New York (1963).
  11. J.-L. LIONS, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod, Paris (1968).
  12. R. E. SHOWALTER, Hilbert space methods for partial differential equations, Electronic Monographs in Differential Equations, San Marcos, Texas (1994).
  13. R. TEMAM, Navier-Stokes equation, Theory and Numerical Analysis, Elsevier Science Publishes B.V., Amsterdam (1984).