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March  2012, 11(2): 763-783. doi: 10.3934/cpaa.2012.11.763

Global solutions to the incompressible magnetohydrodynamic equations

Xiaoli Li 1, and  Dehua Wang 2,

1. 

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260

Received  August 2010 Revised  June 2011 Published  October 2011

An initial-boundary value problem of the three-dimensional incompressible magnetohydrodynamic (MHD) equations is considered in a bounded domain. The homogeneous Dirichlet boundary condition is prescribed on the velocity, and the perfectly conducting wall condition is prescribed on the magnetic field. The existence and uniqueness is established for both the local strong solution with large initial data and the global strong solution with small initial data. Furthermore, the weak-strong uniqueness of solutions is also proved, which shows that the weak solution is equal to the strong solution with certain initial data.
Keywords: incompressible flow, strong solutions, Magnetohydrodynamic (MHD), existence and uniqueness, weak-strong uniqueness..
Mathematics Subject Classification: Primary: 35Q36, 35D05; Secondary: 76W0.
Citation: Xiaoli Li, Dehua Wang. Global solutions to the incompressible magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 763-783. doi: 10.3934/cpaa.2012.11.763
References:
[1]

H. Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Birkhúser Boston, Inc., Boston, 1995.  Google Scholar

[2]

J. Bergh, J. Löfström, "Interpolation spaces. An introduction," Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[3]

T. G. Cowling, "Magnetohydrodynamics," Interscience Tracts on Physics and Astronomy, New York, 1957.  Google Scholar

[4]

R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381.  Google Scholar

[5]

B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Comm. Partial Differential Equations, 22 (1997), 977-1008.  Google Scholar

[6]

J. I. Díaz and M. B. Lerena, On the inviscid and non-resistive limit for the equations of incompressible magnetohydrodynamics, Mathematical Models and Methods in Applied Science, 12 (2002), 1401-1419.  Google Scholar

[7]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar

[8]

J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Analysis, 69 (2008), 3637-3660.  Google Scholar

[9]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.  Google Scholar

[10]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I., Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[11]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I. Linearized Steady Problems," Springer-Verlag, New York, 1994.  Google Scholar

[12]

C. Guillope, Long-time behavior of the solutions of the time-dependent Navier-Stokes equations and property of functional invariant (or attractor) sets, C. R. Acad. Sci. Paris Sér. I Math., 294 (1982), 221-224.  Google Scholar

[13]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equation, 213 (2005), 235-254.  Google Scholar

[14]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, Journal of Functional Analysis, 227 (2005), 113-152.  Google Scholar

[15]

P. Houston, D. Schötzau and X. Wei, A mixed DG method for linearized incompressible magnetohydrodynamics, J Sci. Comput., 40 (2009), 281-314.  Google Scholar

[16]

X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Commun. Math. Phys., 283 (2008), 255-284.  Google Scholar

[17]

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.  Google Scholar

[18]

A. G. Kulikovskiy and G. A. Lyubimov, "Magnetohydrodynamics," Addison-Wesley, Reading, Massachusetts, 1965.  Google Scholar

[19]

L. Landau and E. Lifchitz, "Electrodynamics of Continuous Media," Theoretical physics, Vol. 8, MIR, Moscow, 1990.  Google Scholar

[20]

P. L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[21]

P. L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[23]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  Google Scholar

[24]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  Google Scholar

[25]

A. Novotný and I. Stravskraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Series in Mathematics and its Applications, 27. Oxford University Press, Oxford, 2004.  Google Scholar

[26]

P. Schmidt, On a magnetohydrodynamic problem of Euler type, J. Differential Equation, 74 (1988), 318-335.  Google Scholar

[27]

M. E. Schonbek, T. P. Schonbek and E. Süli, Large-time behaviour of solutions to the magnetohydrodynamics equations, Math. Ann., 304 (1996), 717-756.  Google Scholar

[28]

P. Secchi, On the equations of ideal incompressible magnetohydrodynamics, Rend. Sem. Ma. Univ. Padova, 90 (1993), 103-119.  Google Scholar

[29]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  Google Scholar

[30]

C. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$ spaces for bounded and exterior domains, in "Mathematical Problems Relating to the Navier-Stokes Equation," 1-35, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992.  Google Scholar

[31]

C. Sulem, Quelques résultats de régularité pour les équations de la magnétohydrodynamique, C. R. Acad. Sci. Paris Sér. A-B, 285 (1977), A365-A368.  Google Scholar

show all references

References:
[1]

H. Amann, "Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory," Birkhúser Boston, Inc., Boston, 1995.  Google Scholar

[2]

J. Bergh, J. Löfström, "Interpolation spaces. An introduction," Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[3]

T. G. Cowling, "Magnetohydrodynamics," Interscience Tracts on Physics and Astronomy, New York, 1957.  Google Scholar

[4]

R. Danchin, Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8 (2006), 333-381.  Google Scholar

[5]

B. Desjardins, Regularity of weak solutions of the compressible isentropic Navier-Stokes equations, Comm. Partial Differential Equations, 22 (1997), 977-1008.  Google Scholar

[6]

J. I. Díaz and M. B. Lerena, On the inviscid and non-resistive limit for the equations of incompressible magnetohydrodynamics, Mathematical Models and Methods in Applied Science, 12 (2002), 1401-1419.  Google Scholar

[7]

G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279.  Google Scholar

[8]

J. Fan and W. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Analysis, 69 (2008), 3637-3660.  Google Scholar

[9]

E. Feireisl, "Dynamics of Viscous Compressible Fluids," Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.  Google Scholar

[10]

H. Fujita and T. Kato, On the Navier-Stokes initial value problem. I., Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[11]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I. Linearized Steady Problems," Springer-Verlag, New York, 1994.  Google Scholar

[12]

C. Guillope, Long-time behavior of the solutions of the time-dependent Navier-Stokes equations and property of functional invariant (or attractor) sets, C. R. Acad. Sci. Paris Sér. I Math., 294 (1982), 221-224.  Google Scholar

[13]

C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equation, 213 (2005), 235-254.  Google Scholar

[14]

C. He and Z. Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, Journal of Functional Analysis, 227 (2005), 113-152.  Google Scholar

[15]

P. Houston, D. Schötzau and X. Wei, A mixed DG method for linearized incompressible magnetohydrodynamics, J Sci. Comput., 40 (2009), 281-314.  Google Scholar

[16]

X. Hu and D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Commun. Math. Phys., 283 (2008), 255-284.  Google Scholar

[17]

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.  Google Scholar

[18]

A. G. Kulikovskiy and G. A. Lyubimov, "Magnetohydrodynamics," Addison-Wesley, Reading, Massachusetts, 1965.  Google Scholar

[19]

L. Landau and E. Lifchitz, "Electrodynamics of Continuous Media," Theoretical physics, Vol. 8, MIR, Moscow, 1990.  Google Scholar

[20]

P. L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models," Oxford Lecture Series in Mathematics and its Applications, 3. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[21]

P. L. Lions, "Mathematical Topics in Fluid Mechanics. Vol. 2. Compressible Models," Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.  Google Scholar

[23]

A. Matsumura and T. Nishida, Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids, Comm. Math. Phys., 89 (1983), 445-464.  Google Scholar

[24]

C. Miao and B. Yuan, On the well-posedness of the Cauchy problem for an MHD system in Besov spaces, Math. Methods Appl. Sci., 32 (2009), 53-76.  Google Scholar

[25]

A. Novotný and I. Stravskraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Series in Mathematics and its Applications, 27. Oxford University Press, Oxford, 2004.  Google Scholar

[26]

P. Schmidt, On a magnetohydrodynamic problem of Euler type, J. Differential Equation, 74 (1988), 318-335.  Google Scholar

[27]

M. E. Schonbek, T. P. Schonbek and E. Süli, Large-time behaviour of solutions to the magnetohydrodynamics equations, Math. Ann., 304 (1996), 717-756.  Google Scholar

[28]

P. Secchi, On the equations of ideal incompressible magnetohydrodynamics, Rend. Sem. Ma. Univ. Padova, 90 (1993), 103-119.  Google Scholar

[29]

M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.  Google Scholar

[30]

C. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in $L^q$ spaces for bounded and exterior domains, in "Mathematical Problems Relating to the Navier-Stokes Equation," 1-35, Ser. Adv. Math. Appl. Sci., 11, World Sci. Publ., River Edge, NJ, 1992.  Google Scholar

[31]

C. Sulem, Quelques résultats de régularité pour les équations de la magnétohydrodynamique, C. R. Acad. Sci. Paris Sér. A-B, 285 (1977), A365-A368.  Google Scholar

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