二维柯莫哥洛夫流解的空间对称性不变性
Spatial symmetry invariance of solution of Kolmogorov flow
AI总结：
本文证明了二维柯莫哥洛夫流在Navier-Stokes方程下，其解在时间t>0时保持初始条件的空间对称性，指出该结论对检验数值模拟的可靠性有重要意义，但不支持直接数值模拟结果中对称性的快速丧失。
摘要：
我们证明了一个数学定理，即在周期性边界条件下，由Navier-Stokes方程支配的二维柯莫哥洛夫流，在所有t>0时的解保持其光滑初始条件相同的空间对称性。附录中给出了类似定理在三维NS方程下的证明。这些数学定理可用于检验NS湍流数值模拟的正确性和可靠性。例如，它们支持2D和3D湍流柯莫哥洛夫流的对应CNS（清洁数值模拟）结果在整个模拟时间区间内保持相同的空间对称性，但不支持对应DNS（直接数值模拟）结果中对称性的快速丧失。换句话说，这些DNS结果违反了这些数学定理。因此，这些数学定理严格证实了由DNS给出的NS湍流时空轨迹确实迅速被数值噪声污染。所有这些都表明CNS确实能为深入理解湍流提供有益启示，并且可以接近NS方程的一些数学真理。
英文摘要：
We prove a mathematical theorem that solution for all $t > 0$ of the two-dimensional (2D) Kolmogorov flow governed by Navier-Stokes (NS) equations with periodic boundary condition keeps the same spatial symmetry as its smooth initial condition. The proof of a similar theorem for the three-dimensional NS equations is given in the appendix. These mathematical theorems can be used to check the correctness and reliability of numerical simulations of NS turbulence. For example, they support the corresponding CNS (clean numerical simulation) results of the 2D and 3D turbulent Kolmogorov flows [1-3] that remain the same spatial symmetry in the whole time interval of simulation, but do not support the corresponding DNS (direct numerical simulation) results that lose the spatial symmetry quickly. In other words, these DNS results violate these mathematical theorems. Thus, these mathematical theorems rigorously confirm that the spatiotemporal trajectories of NS turbulence given by DNS are indeed quickly polluted by numerical noises badly. All of these indicate that CNS can indeed provide helpful enlightenments to deepen our understanding about turbulence and besides approach some mathematical truths about NS equations.
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。
