Advanced Search
Article Contents
Article Contents

Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations

  • * Corresponding author

    * Corresponding author 

The first author is supported by the Fulbright Scholar Program Advanced Research and the Florida International University, 2019

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we provide a detailed investigation of the problem of existence and uniqueness of strong solutions of a three-dimensional system of globally modified magnetohydrodynamic equations which describe the motion of turbulent particles of fluids in a magnetic field. We use the flattening property to establish the existence of the global $ V $-attractor and a limit argument to obtain the existence of bounded entire weak solutions of the three-dimensional magnetohydrodynamic equations with time independent forcing.

    Mathematics Subject Classification: 76D05, 35Q35, 76D03, 35B41, 35D40.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] D. Biskamp, Nonlinear Magnetohydrodynamics, in Cambridge Monographs on Plasma Physics, 1, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511599965.
    [2] T. G. Cowling, Magnetohydrodynamics, in Interscience Tracts on Physics and Astronomy, 4, Interscience Publishers, Inc., New York, 1957, Interscience Publishers, Ltd., London.
    [3] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, in The International Series of Monographs on Physics, Clarendon Press, Oxford, 1961.
    [4] T. CaraballoJ. Real and P. E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436.  doi: 10.1515/ans-2006-0304.
    [5] T. Caraballo and J. Real, Navier-Stokes equations with delays, R. Soc. Lond. Proc. Ser., A Math. Phys. Eng. Sci., 457 (2001), 2441-2453.  doi: 10.1098/rspa.2001.0807.
    [6] T. CaraballoP. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Continuous Dyn. Syst. Ser-B, 10 (2008), 761-781.  doi: 10.3934/dcdsb.2008.10.761.
    [7] T. CaraballoJ. Real and A. M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Int. J. Bifurcat. Chaos Appl. Sci. Eng., 20 (2010), 2869-2883.  doi: 10.1142/S0218127410027428.
    [8] T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: recent developments, In Recent Trends in Dynamical Systems, Proceedings of a Conference in Honor of Jürgen Scheurle (A. Johann, H. P. Kruse, F. Rupp eds.), pp. 473–492, Springer Proceedings in Mathematics and Statistics, 35 (2013). doi: 10.1007/978-3-0348-0451-6_18.
    [9] P. Constantin and  C. FoiasNavier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988. 
    [10] G. Deugoué and J. K. Djoko, On the time discretization for the globally modified three-dimensional Navier-Stokes equations, J. Comput. Appl. Math., 235 (2011), 2015-2029.  doi: 10.1016/j.cam.2010.10.003.
    [11] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics (Transl. from the French by C. W. John), in, Grundlehren der Mathematischen Wissenschaften, 219, Springer-Verlag, Berlin, New York, 1976.
    [12] P. E. Kloeden and J. A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181.  doi: 10.1098/rspa.2006.1753.
    [13] P. E. KloedenJ. A. Langa and J. Real, Pullback $V$-attractors of the three-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955.  doi: 10.3934/cpaa.2007.6.937.
    [14] P. E. KloedenP. Marn-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802.  doi: 10.3934/cpaa.2009.8.785.
    [15] P. E. Kloeden and J. Valero, The weak connectnedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys Engs. Sci., 463 (2007), 1491-1508.  doi: 10.1098/rspa.2007.1831.
    [16] J. L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires, Dunod, Paris, 1969.
    [17] O. A. Ladyzenskaya, The Mathematical Theory of Viscous Incompressible Flow, in, Mathematics and its Applucations, 2, Gordon and Breach, Science Publishers, New York, Lndon, Paris, 1969.
    [18] A. M. Márquez, Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay, SeMA J., 51 (2010), 117-124.  doi: 10.1007/bf03322562.
    [19] P. Marín-RubioJ. Real and A. M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927.  doi: 10.1515/ans-2011-0409.
    [20] Q. MaS. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559.  doi: 10.1512/iumj.2002.51.2255.
    [21] M. Romito, The uniqueness of weak solutions of the Globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427.  doi: 10.1515/ans-2009-0209.
    [22] M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  doi: 10.1002/cpa.3160360506.
    [23] Shih.-I. Pai, Magnetogasdynamics and Plasma Dynamics, Springer-Verlag, Vienna, 1962, Prentice-Hall, Inc., Englewood Cliffs, NJ.
    [24] T. Tachim Medjo, Unique strong and $\mathbb{V}$-attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Applicable Analysis, 96 (2017). doi: 10.1080/00036811.2016.1236924.
    [25] T. Tachim Medjo, Unique strong and $\mathbb{V}$-attractor of a three-dimensional globally modified two-phase flow model, Annali di Mathematica, 197 (2018), 843-868.  doi: 10.1007/s10231-017-0706-8.
    [26] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, in, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam, New York, Oxford, 1977.
    [27] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, Second Edition, SIAM, Philadelphia, 1995. doi: 10.1137/1.9781611970050.
    [28] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., 68, Applied mathematics at science: Springer-Verlag, New York; 1997. doi: 10.1007/978-1-4612-0645-3.
    [29] R. Temam, Navier-Stokes equations, Theory and Numerical Analysis, AMS-Chelsea Series. AMS, Providence 2001. doi: 10.1090/chel/343.
    [30] M. I. Vishik, A. I. Komech and A. V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk., 209 (1979), 135–210.
    [31] C. Zhao and T. Caraballo, Asymptotic regularity of trajectory attractor and trajectory statistical solution for 27 the 3D globally modified Navier-Stokes equations, J. Diff. Equations, 266 (2019), 7205-7229.  doi: 10.1016/j.jde.2018.11.032.
  • 加载中

Article Metrics

HTML views(223) PDF downloads(293) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint