Energy Bounds for the Two-Dimensional Navier-Stokes Equations in an Infinite Cylinder
Energy Bounds for the Two-Dimensional Navier-Stokes Equations in an Infinite Cylinder
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无限圆柱体中二维纳维-斯托克斯方程的能量界
DOI：
10.1080/03605302.2013.870575
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发表时间：
2013
影响因子：
1.9
通讯作者：
S. Slijepčević
中科院分区：
文献类型：
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作者：
T. Gallay;S. Slijepčević
我们考虑圆柱体中不可压缩的Navier-Stokes方程，在没有外部强迫的情况下，我们研究了仅由有界初始数据引起的解的长时间行为。虽然我们没有证明这些解在任何时候都是一致有界的，但我们证明了它们在适当的意义上收敛于t→∞时的空间齐次均衡族。收敛性在紧子域上是一致的，并且在除正实轴的稀疏子集外的所有时刻保持一致。我们还改进了已知的解的L∞范数的上界，尽管我们在这个方向上的结果不是最优的。我们的方法是基于对系统中局部能量耗散的详细研究，本着最近致力于一类具有形式梯度结构的耗散偏微分方程的工作的精神。
We consider the incompressible Navier-Stokes equations in the cylinder ℝ × ?, with no exterior forcing, and we investigate the long-time behavior of solutions arising from merely bounded initial data. Although we do not prove that such solutions stay uniformly bounded for all times, we show that they converge in an appropriate sense to the family of spatially homogeneous equilibria as t → ∞. Convergence is uniform on compact subdomains, and holds for all times except on a sparse subset of the positive real axis. We also improve the known upper bound on the L ∞ norm of the solutions, although our results in this direction are not optimal. Our approach is based on a detailed study of the local energy dissipation in the system, in the spirit of a recent work devoted to a class of dissipative partial differential equations with a formal gradient structure.