# ON A VORTICITY MINIMIZATION PROBLEM FOR THE STATIONARY 2D STOKES EQUATIONS

• KIM HONGCHUL (Department of Mathematics Kangnung National University) ;
• KWON OH-KEUN (Department of Mathematics Kangnung National University)
• Published : 2006.01.01

#### Abstract

This paper is concerned with a boundary control problem for the vorticity minimization, in which the flow is governed by the stationary two dimensional Stokes equations. We wish to find a mathematical formulation and a relevant process for an appropriate control along the part of the boundary to minimize the vorticity due to the flow. After showing the existence and uniqueness of an optimal solution, we derive the optimality conditions. The differentiability of the state solution in regard to the control parameter shall be conjunct with the necessary conditions for the optimal solution. For the minimizer, an algorithm based on the conjugate gradient method shall be proposed.

#### 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. R. A. Adams, Sobolev spaces, Academic Press, New York, 1975
3. Jean Cea, Lectures on optimization-theory and algorithms, Tata Institute of Fun-damental Research Lectures on Mathematics and Physics, 53, Tata Institute of Fundamental Research, Bombay, 1978
4. Phillip G. Ciarlet, Introduction to numerical linear algebra and optimization, Cambridge University Press, Cambridge, 1991
5. R. Dautray and J. -L. Lions, Mathematical analysis and numerical methods for science and technology, Vol. 6. Evolution problems. II, Springer-Verlag, New York, 1993
6. Edward J. Dean and R. Glowinsky, On Some Finite Element Methods for the Numerical Simulation of Incompressible Viscous Flow, In : Incompressible Computational Fluid Dynamics Trends and Advances, M. D. Gunzburger and R. A. Nocolaides(Eds.), Cambridge University Press, 1993, 17-66
7. V. Girault and P. Raviart, Finite element methods for Navier-Stokes equations, Theory and algorithms. Springer-Verlag, New York, 1986
8. O. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Gordon and Breach Science Publishers, New York, 1963
9. J. -L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod; Gauthier-Villars, Paris 1969
10. A. Buffa, M. Costabel, and D. Sheen, On traces for H(curl, $\Omega$) in Lipschitz domains, J. Math. Anal. Appl. 276 (2002), no. 2, 845-867 https://doi.org/10.1016/S0022-247X(02)00455-9
11. R. Temam, Navier-Stokes equations. Theory and numerical analysis, North-Holland Publishing Co., Amsterdam, 1984
12. R. E. Showalter, Hilbert space methods for partial differential equations, Electronic reprint of the 1977 original, Electronic Monographs in Differential Equations, San Marcos, TX, 1994

#### Cited by

1. A BOUNDARY CONTROL PROBLEM FOR VORTICITY MINIMIZATION IN TIME-DEPENDENT 2D NAVIER-STOKES EQUATIONS vol.23, pp.2, 2015, https://doi.org/10.11568/kjm.2015.23.2.293