Asymptotic Properties of Discretely Self-Similar Navier–Stokes Solutions with Rough Data

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 [ Bradshaw and Phelps, Pure Appl. Anal., 5 (2023), pp. 377–407 ], we established algebraic spatial decay rates for data in [Formula: see text] for [Formula: see text]. The optimal rate occurs when [Formula: see text] and rates degrade as [Formula: see text] decreases. In this paper, we show that these solutions can be further decomposed into a term satisfying the optimal [Formula: see text] decay rate—i.e., have asymptotics like [Formula: see text]—and a term with the [Formula: see text] decay rate multiplied by a prefactor which can be taken to be arbitrarily small. This smallness property is new and implies the [Formula: see text] asymptotics should be understood in a little-o sense. The decay rates in Bradshaw and Phelps break down when [Formula: see text], in which case spatial asymptotics have not been explored. The second result of this paper shows that DSS solutions with data in [Formula: see text] can be expanded into a term satisfying the [Formula: see text] 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.