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Theoretical and Numerical Local Null Controllability of a Ladyzhenskaya–Smagorinsky Model of Turbulence

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Abstract

This paper deals with the control of a differential turbulence model of the Ladyzhenskaya–Smagorinsky kind. In the equations we find local and nonlocal nonlinearities: the usual transport terms and a turbulent viscosity that depends on the global in space energy dissipated by the mean flow. We prove that the system is locally null-controllable, with distributed controls locally supported in space. The proof relies on rather well known arguments. However, some specific difficulties are found here because of the occurrence of nonlocal nonlinear terms. We also present an iterative algorithm of the quasi-Newton kind that provides a sequence of states and controls that converge towards a solution to the control problem. Finally, we give the details of a numerical approximation and we illustrate the behavior of the algorithm with a numerical experiment.

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References

  1. Alekseev V.M., Tikhomorov V.M., Formin S.V.: Optimal Control. Contemporary Soviet Mathematics. Consultants Bureau, New York (1987)

    Google Scholar 

  2. Argyros I.K.: Convergence and Applications of Newton-Type Iterations. Springer, New York (2008)

    MATH  Google Scholar 

  3. Berselli L.C., Iliescu T., Layton W.J.: Mathematics of Large Eddy Simulation of Turbulent Flows. Scientific Computation. Springer-Verlag, Berlin (2006)

    Google Scholar 

  4. Carreño N., Guerrero S.: Local null controllability of the N-dimensional Navier–Stokes system with N − 1 scalar controls in an arbitrary control domain. J. Math. Fluid Mech. 15(1), 139–153 (2013)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  5. Chacón, T., Lewandowski, R.: Mathematical and Numerical Foundations of Turbulence Models and Applications. Springer, to appear

  6. Clark H., Fernández-Cara E., Limaco J., Medeiros L.A.: Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities. Appl. Math. Comput. 223, 483–505 (2013)

    Article  MathSciNet  Google Scholar 

  7. Coron J.-M.: On the controllability of the 2-D incompressible Navier–Stokes equations with the Navier slip boundary conditions. ESAIM Control Optim. Calc. Var. 1, 35–75 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  8. Coron J.-M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. American Mathematical Society, Rhode Island (2007)

    Google Scholar 

  9. Coron J.-M., Fursikov A.V.: Global exact controllability of the 2D Navier–Stokes equations on a manifold without boundary. Russ. J. Math. Phys. 4(4), 429–448 (1996)

    MATH  MathSciNet  Google Scholar 

  10. Coron, J.-M., Lissy, P.: Local null controllability of the three-dimensional Navier–Stokes system with a distributed control having two vanishing components. Inventiones Math. Phys. 4(4), 429–448 (1996)

  11. Constantin P., Foias C.: Navier–Stokes Equations Chicago Lectures in Mathematics. University of Chicago Press, Chicago (1988)

    Google Scholar 

  12. Ekeland I., Témam R.: Convex analysis and variational problems. Classics in Applied Mathematics, vol. 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1999)

  13. Fabre C., Puel J.-P., Zuazua E.: Approximate controllability of the semilinear heat equation. Proc. R. Soc. Edinb. 125A, 31–61 (1995)

    Article  MathSciNet  Google Scholar 

  14. Fernández-Cara E., Guerrero S., Imanuvilov O.Y., Puel J.-P.: Local exact of the Navier–Stokes systems. J. Math. Pures Appl. 83(12), 1501–1542 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fernández-Cara E., Münch A.: Strong convergent approximations of null controls for the 1D heat equation. SEMA J. 61, 49–78 (2013)

    Article  MATH  Google Scholar 

  16. Fernández-Cara, E., Münch, A.: Numerical null controllability of semi-linear 1D heat equations: fixed point, least squares and Newton methods. Math. Control Relat. Fields 2(3), 217–246 (2012)

  17. Fernández-Cara, E., Münch, A., Souza, D.A.: Numerical null controllability of Stokes and Navier–Stokes equations, in preparation

  18. Fursikov A.V.: Exact boundary zero controllability of three-dimensional Navier–Stokes equations. J. Dyn. Control Syst. 1(3), 325–350 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fursikov A.V., Imanuvilov O.Y.: On exact boundary zero controllability of two-dimensional Navier–Stokes equations. Acta Appl. Math. 37, 67–76 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fursikov, A.V., Imanuvilov, O.Y.: Controllability of Evolution Equations. Lecture Notes Series, vol. 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996)

  21. Fursikov A.V., Imanuvilov O.Y.: Exact controllability of the Navier–Stokes and Boussinesq equations. Russ. Math. Surv. 54(3), 565–618 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. González-Burgos, M., Guerrero, S., Puel, J.-P.: Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Commun. Pure Appl. Anal. 8(1), 311–333 (2009)

  23. Guermond J.-L., Oden J.T., Prudhomme S.: Mathematical perspectives on large eddy simulation models for turbulent flows. J. Math. Fluid Mech. 6, 194–248 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Imanuvilov O.Y.: Controllability of parabolic equations (Russian). Mat. Sb. 186(6), 109–132 (1995)

    MathSciNet  Google Scholar 

  25. Imanuvilov O.Y., Puel J.-P., Yamamoto M.: Carleman estimates for parabolic equations with nonhomogeneous boundary conditions. Chin. Ann. Math. Ser. B 30(4), 333–378 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Imanuvilov, O.Y., Puel, J.-P., Yamamoto, M.: Carleman estimates for second order non homogeneous parabolic equations, preprint (2010)

  27. Launder, B.E., Reynolds, W.C., Rodi, W.: Turbulence Models and Their Applications, vol. 2. Collection de la Direction des Études et Recherches d’Électricité de France ESE, 56, Éditions Eyrolles, Paris (1984)

  28. Lesieur M.: Turbulence in Fluids, 3rd edn. Kluwer Academic Publishers, Dordrecht (1997)

    Book  MATH  Google Scholar 

  29. Saaty, T.L.: Modern Nonlinear Equations. Corrected reprint of the 1967 original. Dover Books on Advanced Mathematics, Dover Publications, Inc., New York (1981)

  30. Tucsnak M., Weiss G.: Observation and Control for Pperator Semigroups. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel (2009)

    Book  Google Scholar 

  31. Zuazua E.: Exact boundary controllability for the semilinear wave equation. Nonlinear Partial Differ. Equ. Appl. 10, 357–391 (1989)

    Google Scholar 

  32. Zuazua E.: Controllability and observability of partial differential equations: some results and open problems. In: Handbook of Differential Equations: Evolutionary Equations, vol. III, pp. 527–621, Handb. Differ. Equ. Elsevier/North-Holland, Amsterdam (2007)

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Authors and Affiliations

  1. Dpto. EDAN, University of Sevilla, Aptdo. 1160, 41080, Sevilla, Spain

    E. Fernández-Cara

  2. Instituto de Matemática, Universidade Federal Fluminense, Niterói, RJ, Brazil

    J. Límaco

  3. Dpto.de Matemática, Universidade Federal do Ceará, Fortaleza, CE, Brazil

    S. B. de Menezes

Authors
  1. E. Fernández-Cara
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  2. J. Límaco
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  3. S. B. de Menezes
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Corresponding author

Correspondence to E. Fernández-Cara.

Additional information

Communicated by M. Tucsnak

This work has been partially supported by D.G.E.S. (Spain), Grant MTM2010-15592.

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Fernández-Cara, E., Límaco, J. & de Menezes, S.B. Theoretical and Numerical Local Null Controllability of a Ladyzhenskaya–Smagorinsky Model of Turbulence. J. Math. Fluid Mech. 17, 669–698 (2015). https://doi.org/10.1007/s00021-015-0232-7

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  • DOI: https://doi.org/10.1007/s00021-015-0232-7

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