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Preprint 23-2004

Distributed Control for a Class of Non-Newtonian Fluids

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Author(s) : Thomas Slawig

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 23-2004

MSC 2000

49J20 Optimal control problems involving partial differential equations
76D55 Flow control and optimization

Abstract :
We consider control problems with a general cost functional where the state equations are the stationary, incompressible Navier-Stokes equations with shear-dependent viscosity. The equations are quasi-linear. The control function is given as the inhomogeneity of the momentum equation. In this paper we study a general class of viscosity functions which correspond to shear-thinning or shear-thickening behavior. The basic results concerning existence, uniqueness, boundedness, and regularity of the solutions of the state equations are reviewed. The main topic of the paper is the proof of G\^{a}teaux differentiability, which extends known results. It is shown that the derivative is the unique solution to a linearized equation. Moreover necessary first order optimality conditions are stated, and the existence of a solution of a class of control problems is shown.

Keywords : optimal control, non-Newtonian fluids, quasilinear elliptic system, optimality conditions

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