This paper is concerned with the lower and upper bounds of rates of decay for $n$-dimensional Navier–Stokes equations with fractional hyperviscosity ${(-\Delta )}^{\alpha }$ when $0<\alpha <\frac{n+2}{4}$. Taking advantage of the spectral representation technique and delicate energy estimates, we show the non-uniform decay and the upper bound of rate of optimal decay for the weak solutions. Furthermore, the lower bound of rate of decay of weak solutions is also established by employing the solution of the linearized equations of the system considered here to approximate the solution of it. Moreover, we make use of the method of bootstrap argument and the properties of generalized heat operator to obtain the lower and upper bounds of rates of optimal decay for the small data global solutions and its derivatives when initial data are in a negative Sobolev space.

本文关注的是衰减速率的上限和下限 $ñ$分数高粘度的三维Navier–Stokes方程 ${（-\Delta ）}^{\alpha }$ 什么时候 $0<\alpha <\frac{ñ+\mathrm{2个}}{4}$。利用频谱表示技术和精细的能量估计，我们显示了弱解的非均匀衰减和最佳衰减率的上限。此外，还通过采用此处考虑的系统线性化方程的解来近似求弱解的衰减率的下界。此外，当初始数据在负Sobolev空间中时，我们利用引导参数方法和广义热算子的性质来获得小数据全局解及其导数的最优衰减率的上下限。