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

Abstract

This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-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 L²-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 L²-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.

Keywords:

• Eventual regularity
• existence
• local energy solution
• Navier-Stokes
• uniqueness

Acknowledgments

The research of Tsai was partially supported by NSERC grant RGPIN-2018-04137. The research of Bradshaw was supported in part by a grant from the Simons Foundation (635438, ZB). We thank the anonymous referees for many helpful suggestions, including the approach in Remark 4.1.

Notes

1 The constant $c_{x_0,R}(t)$ can depend on $T\prime$ in principle. This does not matter in practice and we omit this dependence.

2 That $u\in L_{\text{uloc}}^4({\mathbb{R}}^3\times [0,T])$ can also be proven using the embedding $m^{2,1}\subset B_{\infty ,\infty }^{-1}$ and [33, (2)].