On the Liouville Theorems for 3D Stationary Magnetohydrodynamic Equations
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摘要: 研究了三维稳态磁流体动力学方程的Liouville定理. 首先由能量估计建立了一个Caccioppoli型不等式,再结合Sobolev嵌入得到了Liouville定理成立的3个充分条件,其中一个充分条件表明:若三维稳态磁流体动力学方程的光滑解( u , b )∈Lp,3/2 < p < 3,则 u = b ≡ 0 . 该结果在不需要有限Dirichlet积分的条件下,将Lebesgue空间中可积指标的下界从2扩展至3/2,改进和推广了已有关于磁流体动力学方程Liouville定理的一些结论.
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关键词:
- 磁流体动力学方程 /
- Liouville定理 /
- Caccioppoli型不等式
Abstract: The Liouville theorems for 3D stationary magnetohydrodynamic equations were studied. First, a Caccioppoli type inequality was obtained with the energy method, then 3 sufficient conditions for the Liouville theorems were obtained based on the Sobolev embedding theorems, of which 1 sufficient condition indicates that, given a smooth solution to the 3D stationary magnetohydrodynamic equation satisfying( u , b )∈Lp, 3/2 < p < 3, equality u = b ≡ 0 will be tenable. This work extends the lower bound of the integrable index in the Lebesgue space from 2 to 3/2 without the finite Dirichlet integral condition, which improves and generalizes some conclusions about the Liouville theorems for stationary magnetohydrodynamic equations. -
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