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March  2020, 19(3): 1509-1535. doi: 10.3934/cpaa.2020076

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

G. Deugoué 1,2,, , J. K. Djoko 3, , A. C. Fouape 1, and  A. Ndongmo Ngana 1,

1. 

Department of Mathematics and Computer Science, University of Dschang, P. O. BOX 67, Dschang, Cameroon
2. 

Department of Mathematics and Statistics, Florida International University, MMC, Miami, FL 33199, USA
3. 

School of Computational and Applied Mathematics, University of the Witwatersrand, 1 Jan Smuts Avenue, Braamfontein 2000, Johannesburg, South Africa

* Corresponding author

Received  May 2019 Revised  August 2019 Published  November 2019

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

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.

Keywords: Magnetohydrodynamic equations, globally modified, strong solutions, global attractor.
Mathematics Subject Classification: 76D05, 35Q35, 76D03, 35B41, 35D40.
Citation: G. Deugoué, J. K. Djoko, A. C. Fouape, A. Ndongmo Ngana. Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations. Communications on Pure and Applied Analysis, 2020, 19 (3) : 1509-1535. doi: 10.3934/cpaa.2020076
References:
[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. Caraballo, J. 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. Caraballo, P. 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. Caraballo, J. 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. Foias, Navier-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. Kloeden, J. 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. Kloeden, P. 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-Rubio, J. 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. Ma, S. 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.

show all references

References:
[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. Caraballo, J. 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. Caraballo, P. 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. Caraballo, J. 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. Foias, Navier-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. Kloeden, J. 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. Kloeden, P. 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-Rubio, J. 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. Ma, S. 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.

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