The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation
The 3D incompressible magnetohydrodynamic equations with fractional partial dissipation
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DOI：
10.1016/j.jde.2018.07.046
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发表时间：
2019-01
影响因子：
2.4
通讯作者：
Wanrong Yang;Q. Jiu;Jiahong Wu
中科院分区：
文献类型：
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作者：
Wanrong Yang;Q. Jiu;Jiahong Wu
磁流体动力学（MHD）方程在天体物理、天体物理、宇宙学和工程学等领域的许多现象的研究中起着关键的作用。三维MHD方程的经典解是否能产生有限时间奇点的基本问题仍然是一个悬而未决的问题。在数学上，这个问题是超临界的，在这个意义上，三维MHD方程没有足够的耗散。如果我们分别用−（− Δ）α u和−（− Δ）β B代替标准速度耗散Δu和磁扩散ΔB，则所得到的方程在α≥ 5 4和α+ β≥ 5 2时总是有整体经典解.一个直接的问题是是否可以进一步减少过度耗散。本文证明了即使只存在方向速度耗散和水平磁扩散-（-Δ h）5 4 B，其中Δ h=<$1 2+<$2 2，整体正则性仍然成立.
The magnetohydrodynamic (MHD) equations have played pivotal roles in the study of many phenomena in geophysics, astrophysics, cosmology and engineering. The fundamental problem of whether or not classical solutions of the 3D MHD equations can develop finite-time singularities remains an outstanding open problem. Mathematically this problem is supercritical in the sense that the 3D MHD equations do not have enough dissipation. If we replace the standard velocity dissipation Δu and the magnetic diffusion Δb by−(− Δ) α u and−(− Δ) β b, respectively, the resulting equations with α≥ 5 4 and α+ β≥ 5 2 then always have global classical solutions. An immediate issue is whether or not the hyperdissipation can be further reduced. This paper shows that the global regularity still holds even if there is only directional velocity dissipation and horizontal magnetic diffusion−(− Δ h) 5 4 b, where Δ h=∂ 1 2+∂ 2 2.