We study the three-dimensional Navier–Stokes equations of rotating incompressible viscous fluids with periodic boundary conditions. The asymptotic expansions, as time goes to infinity, are derived in all Gevrey spaces for any Leray–Hopf weak solutions in terms of oscillating, exponentially decaying functions. The results are established for all non-zero rotation speeds, and for both cases with and without the zero spatial average of the solutions. Our method makes use of the Poincaré waves to rewrite the equations, and then implements the Gevrey norm techniques to deal with the resulting time-dependent bi-linear form. Special solutions are also found which form infinite dimensional invariant linear manifolds.

我们研究了具有周期性边界条件的旋转不可压缩粘性流体的三维Navier-Stokes方程。随着时间的推移，渐近展开是在所有Gevrey空间中针对任何Leray-Hopf弱解的振荡，指数衰减函数得出的。对于所有非零转速，以及在有和没有零空间平均值的情况下，都建立了结果。我们的方法利用庞加莱波重写方程，然后实施Gevrey范数技术来处理所得的时间相关双线性形式。还发现了形成无限维不变线性流形的特殊解决方案。