\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Global existence of strong solutions to incompressible MHD

Abstract Related Papers Cited by
  • We establish the global existence and uniqueness of strong solutions to the initial boundary value problem for the incompressible MHD equations in bounded smooth domains of $\mathbb R^3$ under some suitable smallness conditions. The initial density is allowed to have vacuum, in particular, it can vanish in a set of positive Lebessgue measure. More precisely, under the assumption that the production of the quantities $\|\sqrt\rho_0u_0\|_{L^2(\Omega)}^2+\|H_0\|_{L^2(\Omega)}^2$ and $\|\nabla u_0\|_{L^2(\Omega)}^2+\|\nabla H_0\|_{L^2(\Omega)}^2$ is suitably small, with the smallness depending only on the bound of the initial density and the domain, we prove that there is a unique strong solution to the Dirichlet problem of the incompressible MHD system.
    Mathematics Subject Classification: Primary: 35Q35, 35B65; Secondary: 76D05.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    A. G. Kulikovskiy and G. A. Lyubimov, Magnetohydrodynamics, Addison–Wesley, Reading, MA, 1965.

    [2]

    L. D. Landau and E. M. Lifchitz, Electrodynamics of Continuous Media, 2nd ed., Pergamon, New York, 1984.

    [3]

    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.

    [4]

    G. Duvaut and J. L. Lions, Inequations en thermoelasticite et magnetohydrodynamique, Ach.Rational Mech. Anal., 46 (1972), 241-279.

    [5]

    J. F. Gerbeau and C. Le Bris, Existence of solution for a density-dependent magnetohydrodynamic equation, Adv. Differential Equations, 2 (1997), 427-452.

    [6]

    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.

    [7]

    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.

    [8]

    X. P. Hu and D. H. 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.

    [9]

    X. P. Hu and D. H. Wang, Global solutions to the three-dimensional full compressible Magnetohydrodynamic flows, Commun. Math. Phys., 283 (2008), 255-284.doi: 10.1007/s00220-008-0497-2.

    [10]

    J. S. Fan and W. H. Yu, Global variational solutions to the compressible magnetohydrodynamic equations, Nonlinear Analysis, 69 (2008), 3637-3660.doi: 10.1016/j.na.2007.10.005.

    [11]

    E. Feireisl, Dynamics of viscous compressible fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.

    [12]

    E. Feireisl, A. Novotný and H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations, J. Math. Fluid Mech., 3 (2001), 358-392.doi: 10.1007/PL00000976.

    [13]

    B. Ducomet and E. Feireisl, The equation of Magnetohydrodynamics: on the interaction between matter and ration in the evolution of gaseous stars, Commun. Math. Phys., 266 (2006), 595-629.doi: 10.1007/s00220-006-0052-y.

    [14]

    Q. Chen, Z. Tan and Y. J. Wang, Strong solutions to the incompressible magnetohydrodynamic equations, Math. Methods Appl. Sci., 34 (2011), 94-107.doi: 10.1002/mma.1338.

    [15]

    H. W. Wu, Strong solutions to the incompressible magnetohydrodynamic equations with vacuum, Comput. Math. Appl., 61 (2011), 2742-2753.doi: 10.1016/j.camwa.2011.03.033.

    [16]

    X. D. Huang and Y. Wang, Global strong solution to the 2D nonhomogeneous incompressible MHD system, J. Differential Equations, 254 (2013), 511-527.doi: 10.1016/j.jde.2012.08.029.

    [17]

    J. S. Fan and W. H. Yu, Strong solution to the compressible magnetohydrodynamic equations with vacuum, Nonlinear Anal. Real World Appl., 10 (2009), 392-409.doi: 10.1016/j.nonrwa.2007.10.001.

    [18]

    X. L. Li, N. Su, and D. H. Wang, Local strong solution to the compressible magnetohydrodynamic flow with large data, J. Hyperbolic Differ. Equ., 8 (2011), 415-436.doi: 10.1142/S0219891611002457.

    [19]

    X. L. Li and D. H. Wang, Global strong solution to the three-dimensional density-dependent incompressible magnetohydrodynamic flows, J. Differential Equations, 251 (2011), 1580-1615.doi: 10.1016/j.jde.2011.06.004.

    [20]

    W. Von Wahl, Estimating $\nabla u$ by $\text{div} u$ and $\text{curl}u$, Math. Methods Appl. Sci., 15 (1992), 123-143.doi: 10.1002/mma.1670150206.

    [21]

    Y. Zhou, Remarks on regularities for the 3D MHD equations, Discrete Contin. Dyn. Syst., 12 (2005), 881-886.doi: 10.3934/dcds.2005.12.881.

    [22]

    Y. Zhou, Regularity criteria for the 3D MHD equations in terms of the pressure, Internat. J. Non-Linear Mech., 41 (2006), 1174-1180.doi: 10.1016/j.ijnonlinmec.2006.12.001.

    [23]

    Y. Zhou, Regularity criteria for the generalized viscous MHD equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 491-505.doi: 10.1016/j.anihpc.2006.03.014.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(105) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return