Asymptotic properties of discretely self-similar Navier-Stokes solutions with rough data

Abstract

In this paper we explore the extent to which discretely self-similar (DSS) solutions to the 3D Navier-Stokes equations with rough data almost have the same asymptotics as DSS flows with smoother data. In a previous work, we established algebraic spatial decay rates for data in $L^q_{loc}(\mathbb{R}^3\setminus\{0\})$ for $q\in (3,\infty]$. The optimal rate occurs when $q=\infty$ and rates degrade as $q$ decreases. In this paper, we show that these solutions can be further decomposed into a term satisfying the optimal $q=\infty$ decay rate -- i.e.~have asymptotics like $(|x|+\sqrt t)^{-1}$ -- and a term with the $q<\infty$ decay rate multiplied by a prefactor which can be taken to be arbitrarily small. This smallness property is new and implies the $q<\infty$ asymptotics should be understood in a little-o sense. The decay rates in our previous work broke down when $q=3$, in which case spatial asymptotics have not been explored. The second result of this paper shows that DSS solutions with data in $L^3_{loc}(\mathbb{R}^3\setminus\{0\})$ can be expanded into a term satisfying the $(|x|+\sqrt t)^{-1}$ decay rate and a term that can be taken to be arbitrarily small in a scaling invariant class. A Besov space version of this result is also included.