Global Regularity for a Class of Generalized Magnetohydrodynamic Equations
Global Regularity for a Class of Generalized Magnetohydrodynamic Equations
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DOI：
10.1007/s00021-009-0017-y
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发表时间：
2011-06
影响因子：
1.3
通讯作者：
Jiahong Wu
中科院分区：
文献类型：
--
作者：
Jiahong Wu
三维不可压磁流体动力学方程的光滑解是否能产生有限时间奇点，目前还不清楚。一个主要的困难是由于拉普拉斯算子给出的耗散不足以控制非线性，因此3D MHD方程有时被视为“超临界”。本文给出了一类具有超耗散的广义MHD方程的整体正则性结果。这一结果是受到了Terence Tao最近关于广义Navier-Stokes方程（T。Tao，Global regularity for a grammically supercritical hyperdissipative Navier-Stokes equations，arXiv：0906.3070v3 [math.AP] 20 June 2009），但MHD方程的结果与Navier-Stokes方程的结果并不完全平行。Besov空间技术被用来建立MHD方程的结果。
It remains unknown whether or not smooth solutions of the 3D incompressible MHD equations can develop finite-time singularities. One major difficulty is due to the fact that the dissipation given by the Laplacian operator is insufficient to control the nonlinearity and for this reason the 3D MHD equations are sometimes regarded as “supercritical”. This paper presents a global regularity result for the generalized MHD equations with a class of hyperdissipation. This result is inspired by a recent work of Terence Tao on a generalized Navier–Stokes equations (T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equations, arXiv: 0906.3070v3 [math.AP] 20 June 2009), but the result for the MHD equations is not completely parallel to that for the Navier–Stokes equations. Besov space techniques are employed to establish the result for the MHD equations.