Blow-up criteria for smooth solutions to the generalized 3D MHD equations
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RESEARCH
Open Access
Blow-up criteria for smooth solutions to the generalized 3D MHD equations Liping Hu1* and Yinxia Wang2 *
Correspondence: [email protected] 1 College of Information and Management Sciences, Henan Agricultural University, Zhengzhou, 450011, China Full list of author information is available at the end of the article
Abstract In this paper, we focus on the generalized 3D magnetohydrodynamic equations. Two logarithmically blow-up criteria of smooth solutions are established. MSC: 76D03; 76W05 Keywords: generalized MHD equations; blow-up criteria
1 Introduction We study blow up criteria of smooth solutions to the incompressible generalized magnetohydrodynamics (GMHD) equations in R ⎧ α ⎪ ⎪ ⎨∂t u + u · ∇u – B · ∇B + (–) u + ∇(p + |B| ) = , ⎪ ⎪ ⎩
∂t B + u · ∇B – B · ∇u + (–)β B = , ∇ · u = ,
(.)
∇ ·B=
with the initial condition t=:
u = u (x), B = B (x),
x ∈ R .
(.)
Here u = (u , u , u ), B = (B , B , B ) and P = p + |B| are non-dimensional quantities corresponding to the flow velocity, the magnetic field and the total kinetic pressure at the point (x, t), while u (x) and B (x) are the given initial velocity and initial magnetic field with ∇ · u = and ∇ · B = , respectively. The GMHD equations is a generalized model of MHD equations. It has important physical background. Therefore, the GMHD equations are also mathematically significant. For D Navier-Stokes equations, whether there exists a global smooth solution to D impressible GMHD equations is still an open problem. In the absence of global well-posedness, the development of blow-up/ non blow-up theory is of major importance for both theoretical and practical purposes. Fundamental mathematical issues such as the global regularity of their solutions have generated extensive research and many interesting results have been established (see [–]). When α = β = , (.) reduces to MHD equations. There are numerous important progresses on the fundamental issue of the regularity for the weak solution to (.), (.) (see [–]). A criterion for the breakdown of classical solutions to (.) with zero viscosity and positive resistivity in R was derived in []. Some sufficient integrability conditions on two © 2013 Hu and Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Hu and Wang Boundary Value Problems 2013, 2013:187 http://www.boundaryvalueproblems.com/content/2013/1/187
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components or the gradient of two components of u + B and u – B in Morrey-Campanato spaces were obtained in []. A logarithmal improved blow-up criterion of smooth solutions in an appropriate homogeneous Besov space was obtained by Wang et al. []. Zhou and Fan [] established various logarithmically improved regularity criteria for the D MHD equations in terms of the velocit
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