Spatial decay of discretely self-similar solutions to the Navier-Stokes equations

TL;DR

This paper derives new spatial decay rates for discretely self-similar solutions to the Navier-Stokes equations, improving understanding of their regularity and behavior at large distances, especially with rough initial data.

Contribution

It lowers regularity requirements for decay rates, establishes decay without logarithmic corrections, and provides bounds on solution separation from the origin.

Findings

Established optimal decay rates for nonlinear flow components.

Derived decay rates for solutions with rough initial data.

Provided bounds on solution separation from the origin.

Abstract

Forward self-similar and discretely self-similar weak solutions of the Navier-Stokes equations are known to exist globally in time for large self-similar and discretely self-similar initial data and are known to be regular outside of a space-time paraboloid. In this paper, we establish spatial decay rates for such solutions which hold in the region of regularity provided the initial data has locally sub-critical regularity away from the origin. In particular, we (1) lower the H\"older regularity of the data required to obtain an optimal decay rate for the nonlinear part of the flow compared to the existing literature, (2) establish new decay rates without logarithmic corrections for some smooth data, (3) provide new decay rates for solutions with rough data, and, as an application of our decay rates, (4) provide new upper bounds on how rapidly potentially {non-unique}, scaling invariant local energy solutions can separate away from the origin.