Consider the Cauchy problem of incompressible Navier–Stokes equations in $$\mathbb {R}^3$$ R 3 with uniformly locally square integrable initial data. If the square integral of the initial datum on a ball vanishes as the ball goes to infinity, the existence of a time-global weak solution has been known. However, such data do not include constants, and the only known global solutions for non-decaying data are either for perturbations of constants, or when the velocity gradients are in $$L^p$$ L p with finite p . In this paper, we construct global weak solutions for non-decaying initial data whose local oscillations decay, no matter how slowly.

中文翻译：

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具有缓慢衰减振荡的非衰减初始数据的全局 Navier-Stokes 流

考虑 $$\mathbb {R}^3$$ R 3 中不可压缩 Navier-Stokes 方程的柯西问题，其中具有均匀局部平方可积初始数据。如果球上的初始数据的平方积分随着球走向无穷大而消失，则已知时间全局弱解的存在。然而，这样的数据不包括常数，并且非衰减数据的唯一已知全局解要么是常数的扰动，要么是速度梯度在 $$L^p$$L p 且有限 p 的情况下。在本文中，我们为非衰减初始数据构建全局弱解，其局部振荡衰减，无论多慢。