arXiv 1507.011422015-07-04math-phmath.MP

具有常能量和拟能的二维纳维-斯托克斯方程的解

On solutions of the 2D Navier-Stokes equations with constant energy and enstrophy

Jing Tian, Bingsheng Zhang

备注：: 4 figures, 33 pages

摘要：

空间周期二维纳维-斯托克斯方程的全局吸引子是否包含非定常解$u(x,t)$，使得对于每个$t \in (-\infty, \infty)$，其能量和单位质量拟能都是常数，目前还不清楚.这种解决方案的性质的研究开始于\cite{CMM13}，在那里，由于这种解决方案的假设存在，他们被称为“幽灵解决方案”。在这项工作中，我们介绍和研究的几何结构共享的所有鬼解决方案。这项研究使我们考虑一个子类的鬼的解决方案，这些几何结构有一个补充的稳定性。特别是，我们表明，这个子类的鬼解决方案的活动模式的波矢量必须满足一定的补充约束。我们还找到了一种计算方法来检查这些幽灵解决方案的存在。

英文摘要：

It is not yet known if the global attractor of the space periodic 2D Navier-Stokes equations contains nonstationary solutions $u(x,t)$ such that their energy and enstrophy per unit mass are constant for every $t \in (-\infty, \infty)$. The study of the properties of such solutions was initiated in \cite{CMM13}, where, due to the hypothetical existence of such solutions, they were called "ghost solutions". In this work, we introduce and study geometric structures shared by all ghost solutions. This study led us to consider a subclass of ghost solutions for which those geometric structures have a supplementary stability property. In particular, we show that the wave vectors of the active modes of this subclass of ghost solutions must satisfy certain supplementary constraints. We also found a computational way to check for the existence of these ghost solutions.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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