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On $ L^p $ estimates for a simplified Ericksen-Leslie system
Unique strong solutions and V-attractor of a three dimensional globally modified magnetohydrodynamic equations
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 |
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.
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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