On global attractors of the 3D Navier-Stokes equations
On global attractors of the 3D Navier-Stokes equations
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DOI：
10.1016/j.jde.2006.08.021
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发表时间：
2006-08
影响因子：
2.4
通讯作者：
A. Cheskidov;C. Foias
中科院分区：
文献类型：
--
作者：
A. Cheskidov;C. Foias
鉴于三维Navier-Stokes方程（NSE）不一定有正则解的可能性，我们引入了一个抽象的框架来研究多值耗散演化系统关于两种拓扑--弱拓扑和强拓扑的渐近行为.每个这样的系统在弱拓扑中具有全局吸引子，但不一定在强拓扑中。当后者存在且弱闭时，它与弱全局吸引子一致。我们给出了强整体吸引子存在的一个充分条件，当弱整体吸引子上的所有解都是强连续的时，证明了这一充分条件。我们还介绍和研究的Navier-Stokes方程的两个参数的家庭模型，具有类似的属性和开放的问题。这些模型总是具有弱的全局吸引子，但在其中一些模型上，每个解都在有限时间内爆破（在比标准能量更强的范数下）。
In view of the possibility that the 3D Navier–Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier–Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time.