The Decay Rate and Higher Approximation of Mild Solutions to the Navier–Stokes Equations

Abstract

In this paper, we consider v(t) = u(t) − etΔu0, where u(t) is the mild solution of the Navier–Stokes equations with the initial data \({u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)}\) . We shall show that the L2 norm of Dβv(t) decays like \({t^{-\frac {|\beta|-1} {2}-\frac n4}}\) for |β| ≥ 0. Moreover, we will find the asymptotic profile u1(t) such that the L2 norm of Dβ (v(t) − u1(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.