# Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

## TL;DR

This paper proves global existence, regularity, and uniqueness of local energy solutions to the Navier-Stokes equations with initial data small in certain Morrey-type spaces, including critical cases.

## Contribution

It establishes new results on global existence, regularity, and uniqueness for Navier-Stokes solutions with initial data in critical Morrey spaces, extending previous understanding.

## Key findings

- Global existence for data in critical Morrey spaces
- Initial and eventual regularity for small-scale data
- Uniqueness results including local generalized Von Wahl criteria

## Abstract

This paper addresses several problems associated to local energy solutions (in the sense of Lemari\'e-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical $L^2$-based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical $L^2$-based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.