Abstract
We consider optimal control problems of systems governed by quasi-linear, stationary, incompressible Navier–Stokes equations with shear-dependent viscosity in a two-dimensional or three-dimensional domain. We study a general class of viscosity functions with shear-thinning behaviour. Our aim is to prove the existence of a solution for the class of control problems and derive the first order optimality conditions.
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Arada, N.: Optimal control of shear-thinning flows. Technical Report, Centro de Matemática e Aplicações, FCT-UNL, Portugal, Submitted (2011)
Beirão da Veiga H.: On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or nonslip boundary conditions. Comm. Pure App. Math 58(4), 552–577 (2005)
Beirão da Veiga H.: On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem. J. Eur. Math Soc. 11, 127–167 (2009)
Beirão da Veiga H.: On the global regularity of shear thinning flows in smooth domains. J. Math. Anal. Appl. 339, 335–360 (2009)
Beirão da Veiga, H., Kaplický, P., Růžička, M.: Boundary regularity of shear thickening flow. Math. Fluid Mech, 13(3), 387–404, Springer–Base AG (2011)
Berselli L.C., Diening L., Růžička M.: Existence of strong solutions for incompressible fluids with shear dependent viscosities. J. Math. Fluid Mech 12, 101–132 (2010)
Bird R.B., Armstrong R.C., Hassager O.: Dynamic of polymer liquids, 2nd edn. Jonh Wiley, New York (1987)
Casas, E., Fernandez, L.A.: Boundary control of quasilinear elliptic equations. Rapport de Recherche 782, INRIA, (1988)
Casas E., Fernandez L.A: Distributed control of systems governed by a general class of quasilinear elliptic equations. J. Differ. Equ. 35(1033), 20–47
Crispo F., Grisanti C.R.: On the existence, uniqueness and \({C^{1,\gamma}(\bar{\Omega})\cap W^{2,2}(\Omega)}\) regularity for a class of shear-thinning fluids. J. Math. Fluid Mech. 10, 445–487 (2008)
Crispo F., Grisanti C.R.: On the \({C^{1,\gamma}(\bar{\Omega}) \cap W^{2,2} (\Omega)}\) regularity for a class of electro-rheological fluids. J. Math. Anal. App. 356, 119–132 (2009)
Frehse J., Málek J., Steinhauser M.: On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method SIAM. J. Math. Anal. 34(5), 1064–1083 (2003)
Gunzburger M., Trenchea C.: Analysis of an optimal control problem for the three-dimensional coupled modified Navier–Stokes and Maxwell equations. J. Math. Anal. Appl. 333, 295–310 (2007)
Kaplický P., Málek J., Stará J.: C 1,α-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem. Zap. Nauchn. Sem. POMI 259(29), 89–121 (1999)
Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Beach, New York (1969)
Lions J.L.: Quelques méthods de résolution des problèmes aux limites non linéaires. Dunod Gauthier-Villars, Paris (1969)
Málek J., Rajagopal K.L., Růžička M.: Existence and regularity of solutions and the stability of the rest state for fluids with shear dependent viscosity. Math. Models Methods Appl. Sci. 5, 789–812 (1995)
Málek J., Nečas J., Rokyta J., Růžička M.: Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computation, vol. 13. Chapman and Hall, London (1996)
Málek J., Nečas J., Růžička M.: On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2. Adv. Differ. Equ. 6(3), 257–302 (2001)
Málek, J., Rajagopal, K.L.: Mathematical issues concerning the Navier–Stokes equations and some of its generalizations. Handbook Differential Equation, Evolutionary Equations, II, pp. 371–459. Elsevier/ North-Holland, Amsterdam (2005)
Málek J., Roubíček T.: Optimization of steady flows for incompressible viscous fluids. In: Sequiera, A., Beirão da Veiga, H., Videman, J.H. (eds.) Nonlinear Applied Analysis., pp. 355–372. Plenum Press, New York (1999)
Pošta, M., Roubíček T.: Optimal control of Navier–Stokes equations by Oseen approximation, (Preprint2007-013, Nečas Centre, Prague) Computers and Mathematics with Applications, 53, 569–581 (2007)
Roubíček T.: Optimization of steady-state flow of incompressible fluids. In: Barbu, V., Lasiecka, I., Tiba, D., Varsan, C. (eds.) Proceedings of IFIP Conference on Analysis and Optimization of Differential Systems, pp. 357–368. Kluwer Academic Publication, Boston (2003)
Roubíček, T., Tröltzsch, F.: Lipschitz stability of optimal controls for the steady-state Navier–Stokes equations. Technical Report, Institute of Mathematics, TU Berlin, No. 749-2002 Control and Cybernetics, 32, pp. 683–705 (2003)
Slawig T.: Distributed control for a class of non-Newtonian fluids. J. Diff. Equ. 219, 116–143 (2005)
Wachsmuth D., Roubíček T.: Optimal control of incompressible non-Newtonian fluids. Z. Anal. Anwend 29, 351–376 (2010)
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Communicated by H. Beirao da Veiga
The author is very thankful to Professors Tomáš Roubíček and Thomas Slawig for their helpful comments and advices that improved the final version of this paper.
This work was partially supported by The Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) through PEst-OE/MAT/UI0297/2011 and PTDC/MAT109973/2009.
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Guerra, T. Distributed Control for Shear-Thinning Non-Newtonian Fluids. J. Math. Fluid Mech. 14, 771–789 (2012). https://doi.org/10.1007/s00021-012-0101-6
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DOI: https://doi.org/10.1007/s00021-012-0101-6