Existence of global weak solutions to the Navier-Stokes equations in weighted spaces

TL;DR

This paper proves the global existence of weak solutions to the 3D Navier-Stokes equations for a broad class of initial data in weighted spaces, including self-similar data, and identifies conditions for eventual regularity.

Contribution

It introduces a new weighted space framework for initial data, extending global existence results to larger classes of solutions for the Navier-Stokes equations.

Findings

Global weak solutions exist for initial data in the new weighted space.

The class includes all locally square integrable discretely self-similar data.

Certain solutions become regular after some time in a specific space-time region.

Abstract

We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of data allowing growth at spatial infinity. Namely, we show the global existence of suitable weak solutions when the initial data belongs to the weighted space $\mathring M^{2,2}_{\mathcal C}$ introduced in [Z. Bradshaw and I. Kukavica, Existence of suitable weak solutions to the Navier-Stokes equations for intermittent data, J. Math. Fluid Mech. to appear]. This class is strictly larger than currently available spaces of initial data for global existence and includes all locally square integrable discretely self-similar data. We also identify a sub-class of data for which solutions exhibit eventual regularity on a parabolic set in space-time.