Korean J. Math.  Vol 23, No 2 (2015)  pp.293-312
DOI: https://doi.org/10.11568/kjm.2015.23.2.293

A boundary control problem for vorticity minimization in time-dependent 2D Navier-Stokes equations

Hongchul Kim


We deal with a boundary control problem for the vorticity minimization,   in which the flow 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 flow 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.


Navier-Stokes equations, boundary control, vorticity minimization.

Subject classification

35Q30, 49J20, 49J50, 76F70.


Full Text:



F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoretical and Computational Fluid Dynamics, 1 (1990), 303–325. (Google Scholar)

A. Buffa, M. Costabel, and D. Sheen, On traces for H(curl,Ω) in Lipschitz domains, J. Math. Anal. Appl., 276 (2002), 845–867. (Google Scholar)

P. Constantin and I. Foias, Navier-Stokes equations, The University of Chicago Press, Chicago (1989). (Google Scholar)

R. Dautray and J.-L. Lions, Analysis and Numerical Methods for Science and Technology, Vol 2, Springer-Verlag, New York (1993). (Google Scholar)

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. (Google Scholar)

V. Girault and P. Raviart, The Finite Element Method for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, New York (1986). (Google Scholar)

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. (Google Scholar)

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. (Google Scholar)

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. (Google Scholar)

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Goldon and Breach, New York (1963). (Google Scholar)

J.-L. Lions, Quelques m ́ethodes de r ́esolution des probl`emes aux limites non lin ́eaires, Dunod, Paris (1968). (Google Scholar)

R. E. Showalter, Hilbert space methods for partial differential equations, Elec- tronic Monographs in Differential Equations, San Marcos, Texas (1994). (Google Scholar)

R. Temam, Navier-Stokes equation, Theory and Numerical Analysis, Elsevier Science Publishes B.V., Amsterdam (1984). (Google Scholar)


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