Decay character in some function spaces and applications to decay rates of generalized Navier–Stokes equations

Abstract

The aim of this paper is twofold. First, we develop the notion of decay characters and evaluate these characters in important data spaces that often appear in problems related to fluid mechanics. Next, we apply our evaluation for the decay character of the initial datum $r^{\ast }(u_0)$ to study the decay of solution and its derivatives to $n$-dimensional generalized Navier–Stokes equations with fractional-order Laplace operator $(-\Delta {)}^{\alpha }$ when $0<\alpha <\frac{n+2}{4}$. We establish the optimal uniform decay estimates for the $L^2$-norm of solution $u$, as well as decay estimates for $\Vert {\Lambda }^{m}u{\Vert }_{L^2}$ and $\Vert {\partial }_t^k{\Lambda }^{m}u{\Vert }_{L^2}$ for positive integer $k$ and non-negative real number $m$.