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Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison

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Abstract

We consider two magnetohydrodynamic-α (MHDα) models with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). More precisely, we consider the Navier–Stokes-α-MHD and the Modified Leray-α-MHD models. Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations. In both cases, the global existence of the solution and of a global attractor can be shown. We provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the long-time dynamics of the involved system and gives information about the numerical stability of the model. We get the same bound that holds for the Simplified Bardina-MHD model, considered in a previous paper (this result provides, in some sense, an intermediate bound between the number of degrees of freedom for the Simplified Bardina model and the Navier–Stokes-α equation in the nonmagnetic case). However, the Navier–Stokes-α-MHD system is preferable since, in the ideal case, it conserves more quadratic invariants derived from the standard MHD model.

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References

  1. Cao Y., Lunasin E.M., Titi E.S.: Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun. Math. Sci. 4, 823–848 (2006)

    MathSciNet  MATH  Google Scholar 

  2. Casella E., Secchi P., Trebeschi P.: Global classical solutions for MHD system. J. Math. Fluid Mech. 5, 70–91 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Catania D.: Global existence for a regularized magnetohydrodynamic-α model. Ann. Univ. Ferrara 56(1), 1–20 (2010). doi:1007/s11565-009-0069-1 (2009)

  4. Catania D., Secchi P.: Global existence and finite dimensional global attractor for a 3D double viscous MHD-α model. Commun. Math. Sci. 8(4), 1021–1040 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Catania, D., Secchi, P.: Global existence for two regularized MHD models in three space-dimension. Portugal Math. 68(1), 37–99 (2011)

    Google Scholar 

  6. Cheskidov A., Holm D.D., Olson E., Titi E.S.: On a Leray-α model of turbulence. Proc. R. Soc. A 461, 629–649 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Constantin, P., Foias, C.: Navier–Stokes Equations. The University of Chicago Press (1988)

  8. Fan J., Ozawa T.: Global Cauchy problem for the 2-D magnetohydrodynamic-α models with partial viscous terms. J. Math. Fluid Mech. 12(2), 306–319 (2010). doi:10.1007/s00021-008-0289-7 (2008)

  9. Foias C., Holm D.D., Titi E.S.: The three dimensional viscous Camassa–Holm equations and their relation to the Navier–Stokes equations and turbulence theory. J. Dyn. Differ. Equ. 14, 1–35 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Linearised Steady Problems, vol. I. In: Springer Tracts in Natural Philosophy, vol. 38. Springer, New York (1994). Second printing, 1998

  11. Ilyin A., Lunasin E., Titi E.S.: A modified-Leray-α subgrid scale model of turbulence. Nonlinearity 19, 879–897 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Jiu Q., Niu D.: Mathematical results related to a two-dimensional magnetohydrodynamic equations. Acta Math. Scientia 26B, 744–756 (2006)

    Article  MathSciNet  Google Scholar 

  13. Kozono H.: Weak and classical solutions of the 2-D MHD equations. Tohoku Math. J. 41, 471–488 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lieb, E., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schringer Hamiltonian and their relation to Sobolev inequalities. In: Lieb, E., Simon, B., Wightman, A.S. (eds.) Studies in Mathematical Physics: Essays in Honor of V. Bargman, pp. 226–303. Princeton University Press, Princeton (1976)

  15. Linshiz, J.S., Titi, E.S.: Analytical study of certain magnetohydrodynamic-α models. J. Math. Phys. 48, 065504 (2007)

    Google Scholar 

  16. Sermange M., Temam R.: Some mathematical questions related to the MHD equations. Commun. Pure Appl. Math. 36, 635–664 (1983)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Temam, R.: Infinite-dimensional dynamical systems in mechanics and physics. In: Applied Mathematical Sciences, vol. 68. Springer-Verlag, New York (1988)

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  1. Department of Mathematics, Faculty of Engineering, Brescia University, Via Valotti 9, 25133, Brescia, Italy

    Davide Catania

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  1. Davide Catania
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Correspondence to Davide Catania.

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Communicated by G.P. Galdi

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Catania, D. Finite Dimensional Global Attractor for 3D MHD-α Models: A Comparison. J. Math. Fluid Mech. 14, 95–115 (2012). https://doi.org/10.1007/s00021-010-0041-y

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  • DOI: https://doi.org/10.1007/s00021-010-0041-y

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