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.