2024/09/20 by Zachary Bradshaw, Bradshaw, Zachary, P. Phelps +1
Engineering · #Fluid Dynamics and Turbulent Flows #Hydraulic flow and structures #Hydraulic Fracturing and Reservoir Analysis
paper · pdf · doi:10.48550/arxiv.2409.13586
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 Lqloc(ℝ3∖\0\) for q∈ (3,∞]. The optimal rate occurs when q=∞ 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=∞ decay rate -- i.e.~have asymptotics like (|x|+√ t)-1 -- and a term with the q<∞ decay rate multiplied by a prefactor which can be taken to be arbitrarily small. This smallness property is new and implies the q<∞ 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 L3loc(ℝ3∖\0\) can be expanded into a term satisfying the (|x|+√ 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.