Relations Between Energy and Enstrophy on the Global Attractor of the 2-D Navier-Stokes Equations
Relations Between Energy and Enstrophy on the Global Attractor of the 2-D Navier-Stokes Equations
复制标题
二维纳维-斯托克斯方程整体吸引子上能量与熵的关系
DOI：
10.1007/s10884-005-8269-6
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发表时间：
2005
影响因子：
1.3
通讯作者：
Michael S. Jolly
中科院分区：
文献类型：
--
作者：
R. Dascaliuc;C. gname;C. gname;Michael S. Jolly
摘要我们研究了全局吸引子
$$\mathcal {A}$$的二维周期Navier-Stokes方程投影在归一化的无量纲能量熵平面（e, e）上。我们用理解吸引子如何依赖于力的性质的观点来看待与时间无关的力。首先我们证明，对于任何力，
$$\mathcal {A}$$以抛物线E = e1/2为界
直线E= E。然后我们来证明
$$\mathcal {A}$$要有足够靠近抛物线的点，力必须接近Stokes算子A的特征向量；只有当力正好是这样一个特征向量时，它才能与抛物线相交，并且在与这个力平行的稳定状态下这样做。我们沿着抛物线构造一个薄区域，在这样的稳定状态下被压缩，以至于吸引子永远无法进入。我们证明0不可能在吸引子上，除非所有m的力都在Hm中。对能量和熵的不同下界估计
$$\mathcal {A}$$是为光滑和非光滑力导出的，就像远离0和E = E线附近的不变集的边界一样。特别注意抛物线附近和0附近区域的动机来自湍流理论，如引言中所解释的那样。
AbstractWe examine how the global attractor
$$\mathcal {A}$$ of the 2-D periodic Navier–Stokes equations projects in the normalized, dimensionless energy–enstrophy plane (e, E). We treat time independent forces, with the view of understanding how the attractor depends on the nature of the force. First we show that for any force,
$$\mathcal {A}$$ is bounded by the parabola E = e1/2
and the line E=e. We then show that for
$$\mathcal {A}$$ to have points near enough to the parabola, the force must be close to an eigenvector of the Stokes operator A; it can intersect the parabola only when the force is precisely such an eigenvector, and does so at a steady state parallel to this force. We construct a thin region along the parabola, pinched at such steady states, that the attractor can never enter. We show that 0 cannot be on the attractor unless the force is in Hm for all m. Different lower bound estimates on the energy and enstrophy on
$$\mathcal {A}$$ are derived for both smooth and nonsmooth forces, as are bounds on invariant sets away from 0 and near the line E = e. Motivation for the particular attention to the regions near the parabola and near 0 comes from turbulence theory, as explained in the introduction.