Global Navier-Stokes flows for non-decaying initial data with slowly decaying oscillation

Abstract

Consider the Cauchy problem of incompressible Navier-Stokes equations in ${\mathbb{R}}^3$ 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 $L^p$ with finite $p$. In this paper, we construct global weak solutions for non-decaying initial data whose local oscillations decay, no matter how slowly.

We added reference to Lemarie-Rieusset [21] on page 2, Example 1.2 after Theorem 1.1, and Theorem 6.1 in a new Section 6 of 2 pages. This version is accepted by the Communications in Mathematical Physics