Blow-up in finite time for the dyadic model of the Navier-Stokes equations
Blow-up in finite time for the dyadic model of the Navier-Stokes equations
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DOI：
10.1090/s0002-9947-08-04494-2
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发表时间：
2006-01
影响因子：
1.3
通讯作者：
A. Cheskidov
中科院分区：
文献类型：
--
作者：
A. Cheskidov
研究了Katz和Pavlovic提出的Navier-Stokes方程的并矢模型。当耗散度小于1/4时，它们表现出有限时间爆破.本文证明了方程组的弱解的存在性，从任意时刻开始的非负初值的弱解的能量不等式，当λ> 1/3时的局部正则性，当λ 1/2时的整体正则性.此外，我们还证明了在ε < 1/3的情形下的有限时间爆破。值得注意的是，当ε = 1/3时，模型在非线性项上具有与4D Navier-Stokes方程相同的估计。最后，我们讨论了一个弱全局吸引子，它与一个极大有界不变集相一致，并且对λ 1/2成为一个强全局吸引子。
We study the dyadic model of the Navier-Stokes equations introduced by Katz and Pavlovic. They showed a finite time blow-up in the ca se where the dissipation degreeis less than 1/4. In this paper we prove the existence of weak solutions for all �, energy inequality for every weak solution with nonnegative initial data starting from any time, local regularity for � > 1/3, and global regularity for � � 1/2. In addition, we prove a finite time blow-up in the case where � < 1/3. It is remarkable that the model with � = 1/3 enjoys the same estimates on the nonlinear term as the 4D Navier-Stokes equations. Finally, we discuss a weak global attractor, which coincides with a maximal bounded invariant set for alland becomes a strong global attractor for � � 1/2.