Optimal decay rate for higher-order derivatives of the solution to the Lagrangian-averaged Navier–Stokes-$\alpha$ equation in ${\mathbb{R}}^3$

Abstract

Recently, Bjorland and Schonbek [Ann. Inst. H. Poincaré Anal. Non Linéaire 25 (2008) 907–936] investigated the upper bound of the decay rate for the solution to the Lagrangian-averaged Navier–Stokes-$\alpha$ equation under the condition that the initial data belongs to $L^1({\mathbb{R}}^n)\cap H_{\sigma }^N({\mathbb{R}}^n)$ with $n=2,3,4$. The decay rate can eventually be shown to be optimal if the average of the initial data is nonzero. Thus, the target in this paper is to study the optimal decay rate of the solution when the average of the initial data is zero. If the initial data belongs to $L^1({\mathbb{R}}^3)\cap H_{\sigma }^N({\mathbb{R}}^3)$ and some weighted Sobolev space, we show that the lower and upper bounds of decay rates for the $k$th-order ($k\in [0,N]$) spatial derivatives of the solution tending to zero in $L^2$-norm are $(1+t{)}^{-\frac{5+2k}{4}}$, which implies these decay rates are optimal. As a by-product, we show that the optimal decay rate (including lower and upper bounds) of the time derivative of the solution tending to zero in $L^2$-norm is $(1+t{)}^{-\frac{9}{4}}$.