American Institute of Mathematical Sciences

March  2015, 35(3): 1359-1385. doi: 10.3934/dcds.2015.35.1359

Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows

 1 School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received  August 2013 Revised  August 2014 Published  October 2014

In this paper we consider the equations of the unsteady viscous, incompressible, and heat conducting magnetohydrodynamic flows in a bounded three-dimensional domain with Lipschitz boundary. By an approximation scheme and a weak convergence method, the existence of a weak solution to the three-dimensional density dependent generalized incompressible magnetohydrodynamic equations with large data is obtained.
Citation: Weiping Yan. Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 1359-1385. doi: 10.3934/dcds.2015.35.1359
References:
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Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629. doi: 10.1007/s00220-006-0052-y.  Google Scholar [12] G. Duvaut and J. L. Lions, Inéquation en thermoélasticité et magnéto-hydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar [13] E. F. El-Shehawey, E. M. E. Elbarbary, N. A. S. Afifi and M. Elshahed, MHD flow of an elastico-viscous fluid under periodic body acceleration, Int. J. Math. Math. Sci., 23 (2000), 795-799. doi: 10.1155/S0161171200002817.  Google Scholar [14] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford: Oxford University Press, 2004.  Google Scholar [15] E. Fernández-Cara, F. Guillén and R. R. 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Phys., 297 (2010), 371-400. doi: 10.1007/s00220-010-0992-0.  Google Scholar [25] S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible Magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal., 42 (2010), 2539-2553. doi: 10.1137/100785168.  Google Scholar [26] A. R. Kantrovits and G. Y. Petchek, Magnitnaya Gidrodinamika (Magnetohydrodynamics), Atomizdat, Moscow, 1958. Google Scholar [27] S. Kawashima and V. V. Shelukhin, Unique global solution with respect to time of initial boundary value problems for one dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  Google Scholar [28] A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Reading, MA: Addison-Wesley, 1965. Google Scholar [29] L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., New York: Pergamon, 1984. Google Scholar [30] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, Vol.3. New York: Oxford University Press, 1996.  Google Scholar [31] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press, 1998.  Google Scholar [32] A. Novotný, I. Straškraba, Introduction to the Theory of Compressible Flow, Oxford: Oxford University Press, 2004. Google Scholar [33] K. Ohkitani and P. Constantin, Two and three dimensional magnetic reconnection observed in the Eulerian Lagrangian analysis of magnetohydrodynamics equations, Phys. Rev. E., 78 (2008), 066315, 11 pp. doi: 10.1103/PhysRevE.78.066315.  Google Scholar [34] T. Sarpkaya, Flow of non-Newtonian fluids in a magnetic field, AIChE. J., 7 (1961), 324-328. doi: 10.1002/aic.690070231.  Google Scholar [35] 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.  Google Scholar [36] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar [37] J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mech., 9 (2007), 104-138. doi: 10.1007/s00021-006-0219-5.  Google Scholar [38] W. P. Yan, Motion of compressible magnetic fluids in $T^3$. Electron, J. Differential Equations., 232 (2013), 29 pp.  Google Scholar [39] W. P. Yan, On weak-strong uniqueness property for full compressible magnetohydrodynamics flows, Cent. Eur. J. Math., 11 (2013), 2005-2019. doi: 10.2478/s11533-013-0297-6.  Google Scholar

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References:
 [1] W. Andra and H. Nowak, Magnetism in Medicine, Wiley VCH, Berlin, 2007. doi: 10.1002/9783527610174.  Google Scholar [2] M. F. Barnothy (Ed.), Biological Effects of Magnetic Fields, Plenum Press, New York, 1964. Google Scholar [3] R. Bhargava, H. S. Sugandha and O. A. Takhar, Computational simulation of biomagnetic micropolar blood flow in porous media, J. Biomech., 39 (2006), S648-S649. doi: 10.1016/S0021-9290(06)85704-0.  Google Scholar [4] R. M. Brown and Z. Shen, Estimates for the Stokes problem operator in Lipschitz domains, Indiana Univ. Math. J., 44 (1995), 1183-1206. doi: 10.1512/iumj.1995.44.2025.  Google Scholar [5] M. Bulíček, E. Feireisl and J. Málek, Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Anal. Real World Appl., 10 (2009), 992-1015. doi: 10.1016/j.nonrwa.2007.11.018.  Google Scholar [6] M. Bulíček, J. Málek and K. R. Rajagopal, Navier's slip and evolutionary Navier-Stokes like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., 56 (2007), 51-85. doi: 10.1512/iumj.2007.56.2997.  Google Scholar [7] M. Bulíček, J. Málek and K. R. Rajagopal, Mathematical analysis of unsteady flows of fluids with pressure, shear-rate and temperature dependent material moduli, that slip at solid boundaries, SIAM J. Math. Anal., 41 (2009), 665-707. doi: 10.1137/07069540X.  Google Scholar [8] H. Cabannes, Theoretical Magnetofluiddynamics, New York: Academic Press, 1970. Google Scholar [9] L. Diening, M. Ružička and J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids, Annali della Scuola Normale Superiore di Pisa IX., 9 (2010), 1-46.  Google Scholar [10] R. J. DiPerna and P. L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar [11] B. Ducomet and E. Feireisl, The equations of Magnetohydrodynamics: On the interaction between matter and radiation in the evolution of gaseous stars, Commun. Math. Phys., 226 (2006), 595-629. doi: 10.1007/s00220-006-0052-y.  Google Scholar [12] G. Duvaut and J. L. Lions, Inéquation en thermoélasticité et magnéto-hydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar [13] E. F. El-Shehawey, E. M. E. Elbarbary, N. A. S. Afifi and M. Elshahed, MHD flow of an elastico-viscous fluid under periodic body acceleration, Int. J. Math. Math. Sci., 23 (2000), 795-799. doi: 10.1155/S0161171200002817.  Google Scholar [14] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford: Oxford University Press, 2004.  Google Scholar [15] E. Fernández-Cara, F. Guillén and R. R. Ortega, Some theoretical results for viscoplastic and dilatant fluids with variable desity, Nonlinear Anal., 28 (1997), 1079-1100. doi: 10.1016/S0362-546X(97)82861-1.  Google Scholar [16] J. Frehse and M. Ružička, Non-homogeneous generalized newtonian fluids, Math. Z., 260 (2008), 355-375. doi: 10.1007/s00209-007-0278-1.  Google Scholar [17] J. Frehse, J. Málek and M. Ružička, Large data existence result for unsteady flows of inhomogeneous shear thickening heat conducting incompressible fluids, Comm. PDE., 35 (2010), 1891-1919. doi: 10.1080/03605300903380746.  Google Scholar [18] G. P. Galdi, C. G. Simader and H. Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl., 167 (1994), 147-163. doi: 10.1007/BF01760332.  Google Scholar [19] F. Guillén-González, Density dependent incompressible fluids with non-Newtonian viscosity, Czechoslovak Math. J., 54 (2004), 637-656. doi: 10.1007/s10587-004-6414-8.  Google Scholar [20] X. Hu and D. Wang, Global solutions to the three dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys., 283 (2008), 255-284. doi: 10.1007/s00220-008-0497-2.  Google Scholar [21] X. Hu and D. Wang, Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations, J. Differential Equations., 245 (2008), 2176-2198. doi: 10.1016/j.jde.2008.07.019.  Google Scholar [22] X. Hu and D. Wang, Global existence and large time behavior of solutions to the three dimensional equations of compressible magnetohydrodynamic flows, Arch. Rational Mech. Anal., 197 (2010), 203-238. doi: 10.1007/s00205-010-0295-9.  Google Scholar [23] X. Hu and D. Wang, Low mach number limit of viscous compressible magnetohydrodynamic flows, SIAM J. Math. Anal., 41 (2009), 1272-1294. doi: 10.1137/080723983.  Google Scholar [24] S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible Magnetohydrodynamic equations with periodic boundary conditions, Comm. Math. Phys., 297 (2010), 371-400. doi: 10.1007/s00220-010-0992-0.  Google Scholar [25] S. Jiang, Q. C. Ju and F. C. Li, Incompressible limit of the compressible Magnetohydrodynamic equations with vanishing viscosity coefficients, SIAM J. Math. Anal., 42 (2010), 2539-2553. doi: 10.1137/100785168.  Google Scholar [26] A. R. Kantrovits and G. Y. Petchek, Magnitnaya Gidrodinamika (Magnetohydrodynamics), Atomizdat, Moscow, 1958. Google Scholar [27] S. Kawashima and V. V. Shelukhin, Unique global solution with respect to time of initial boundary value problems for one dimensional equations of a viscous gas, J. Appl. Math. Mech., 41 (1977), 273-282.  Google Scholar [28] A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Reading, MA: Addison-Wesley, 1965. Google Scholar [29] L. D. Laudau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., New York: Pergamon, 1984. Google Scholar [30] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1. Incompressible models. Oxford Lecture Series in Mathematics and its Applications, Vol.3. New York: Oxford University Press, 1996.  Google Scholar [31] P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. New York: The Clarendon Press, Oxford University Press, 1998.  Google Scholar [32] A. Novotný, I. Straškraba, Introduction to the Theory of Compressible Flow, Oxford: Oxford University Press, 2004. Google Scholar [33] K. Ohkitani and P. Constantin, Two and three dimensional magnetic reconnection observed in the Eulerian Lagrangian analysis of magnetohydrodynamics equations, Phys. Rev. E., 78 (2008), 066315, 11 pp. doi: 10.1103/PhysRevE.78.066315.  Google Scholar [34] T. Sarpkaya, Flow of non-Newtonian fluids in a magnetic field, AIChE. J., 7 (1961), 324-328. doi: 10.1002/aic.690070231.  Google Scholar [35] 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.  Google Scholar [36] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar [37] J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mech., 9 (2007), 104-138. doi: 10.1007/s00021-006-0219-5.  Google Scholar [38] W. P. Yan, Motion of compressible magnetic fluids in $T^3$. Electron, J. Differential Equations., 232 (2013), 29 pp.  Google Scholar [39] W. P. Yan, On weak-strong uniqueness property for full compressible magnetohydrodynamics flows, Cent. Eur. J. Math., 11 (2013), 2005-2019. doi: 10.2478/s11533-013-0297-6.  Google Scholar
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