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

Liouville theorem for MHD system and its applications

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we construct Liouville theorem for the MHD system and apply it to study the potential singularities of its weak solution. And we mainly study weak axi-symmetric solutions of MHD system in $\mathbb{R}^3× (0, T)$.

    Mathematics Subject Classification: 35Q35, 76W05, 35B65.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] H. Beirão da Veiga, A new regularity class for the Navier-Stokes equations in Rn, Chinese Ann. Math. Ser. B, 16 (1995), 407-412. 
    [2] L. CaffarelliR. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35 (1982), 771-831. 
    [3] Dongho Chae, Pierre Degond and Jian-Guo Liu, Well-posedness for Hall-magnetohydrodynamics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 555-565. 
    [4] G. Duvaut and J.-L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. 
    [5] Giovanni P. Galdi. An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. II, Springer-Verlag, New York, 39(1994), xii+323.
    [6] Yoshikazu GigaKatsuya Inui and Shin'ya Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, Dept. Math., Seconda Univ. Napoli, Caserta, 4 (1999), 27-68. 
    [7] Cheng He and Zhouping Xin, Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations, J. Funct. Anal., 227 (2005), 113-152. 
    [8] Cheng He and Zhouping Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254. 
    [9] L. IskauriazaG. A. Serëgin and V. ShverakL3, ∞-solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk, 58 (2003), 3-44. 
    [10] Tosio Kato, Strong Lp-solutions of the avier-tokes equation in Rm, with applications to weak solutions, Math. Z., 187 (1984), 471-480. 
    [11] Gabriel KochNikolai NadirashviliGregory A. Seregin and Vladimir Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math., 203 (2009), 83-105. 
    [12] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa. Izdat. Nauka, Moscow, (1967), 736.
    [13] Zhen Lei, On axially symmetric incompressible magnetohydrodynamics in three dimensions, J. Differential Equations, 259 (2015), 3202-3215. 
    [14] A. MahalovB. Nicolaenko and T. ShilkinL3, ∞-solutions to the MHD equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 336 (2006), 112-276. 
    [15] J. NečasM. Růžička and V. Šverák, On Leray's self-similar solutions of the Navier-Stokes equations, Acta Math., 176 (1996), 283-294. 
    [16] Vladimir Scheffer, Partial regularity of solutions to the Navier-Stokes equations, Pacific J. Math., 66 (1976), 535-552. 
    [17] Michel Sermange and Roger Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664. 
    [18] James Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 9 (1962), 187-195. 
    [19] Michael Struwe, On partial regularity results for the Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 437-458. 
    [20] Roger Temam. Navier-Stokes Equations, AMS Chelsea Publishing, Providence, RI, 2001, xiv+408.
    [21] Gang Tian and Zhouping Xin, Gradient estimation on Navier-Stokes equations, Comm. Anal. Geom., 7 (1999), 221-257. 
    [22] Tai-Peng Tsai, On Leray's self-similar solutions of the Navier-Stokes equations satisfying local energy estimates, Arch. Rational Mech. Anal., 143 (1998), 29-51. 
    [23] Zujin ZhangXian Yang and Shulin Qiu, Remarks on Liouville type result for the 3D Hall-MHD system, J. Partial Differ. Equ., 28 (2015), 286-290. 
    [24] Yong Zhou and Milan Pokorny, On the regularity of the solutions of the Navier-Stokes equations via one velocity component, Nonlinearity, 23 (2010), 1097-1107. 
  • 加载中
SHARE

Article Metrics

HTML views(591) PDF downloads(304) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return