# Asymptotic Behavior of the Navier-Stokes Equations with Nonzero  Far-Field Velocity

Authors: Jaiok Roh
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) inL2 space and by Shibata (1999) and Enomoto-Shibata (2005) in Lp spaces for p≥3. However, their results did not include enough information to find the spatial decay. So, Bae-Roh (2010) improved Enomoto-Shibata's results in some sense and estimated the spatial decay even though their results are limited. In this paper, we will prove temporal decay with a weighted function by using Lr-Lp decay estimates obtained by Roh (2011). Bae-Roh (2010) proved the temporal rate becomes slower by (1+σ)/2 if a weighted function is |x|σ for 0<σ<1/2. In this paper, we prove that the temporal decay becomes slower by σ, where 0<σ<3/2 if a weighted function is |x|σ. For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted Lp-norm of solutions. This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008).


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## Body

## 1. Introduction

When a boat is sailing with a constant velocityu∞, we may think that the water is flowing around the fixed boat with opposite velocity -u∞ like the water flow around an island. As we have seen, behind the boat the motion of the water is significantly different from other areas, which is called the wake. The motion of nonstationary flow of an incompressible viscous fluid past an isolated rigid body is formulated by the following initial boundary value problem of the Navier-Stokes equations:
(1.1)∂∂tu-Δu+(u⋅∇)u+∇p=f,∇⋅u=0inΩ×(0,∞),u|t=0=u0,u|∂Ω=0,lim⁡|x|→∞u(x,t)=u∞,
where Ω is an exterior domain in R3 with a smooth boundary ∂Ω and u∞ denotes a given constant vector describing the velocity of the fluid at infinity. For u∞=0, the temporal decay and weighted estimates for solutions of (1.1) have been studied in [1–13].In this paper, we consider a nonzero constantu∞. We set u=u∞+v in (1.1) and have (1.2)∂∂tv-Δv+(u∞⋅∇)v+(v⋅∇)v+∇p1=f,∇⋅v=0,inΩ×(0,∞),v|t=0=u0-u∞,v|∂Ω=-u∞,lim⁡|x|→∞v(x,t)=0.Consider the following linear equations of (1.2): (1.3)∂∂tu-Δu+(u∞⋅∇)u+∇p=0,∇⋅u=0inΩ×(0,∞),u|t=0=u0,u|∂Ω=0,lim⁡|x|→∞u(x,t)=0,
which is referred to as the Oseen equations; see [14].In order to formulate the problem (1.3), Enomoto and Shibata [15] used the Helmholtz decomposition: (1.4)Lp(Ω)n=Jp(Ω)⊕Gp(Ω),
where 1<p<∞, (1.5)Lp(Ω)n={u=(u1,…,un):uj∈Lp(Ω),j=1,…,n},C0,σ∞={u=(u1,…,un)∈C0∞(Ω)n:∇⋅u=0inΩ},Jp(Ω)=the completion ofC0,σ∞(Ω)inLp(Ω)n,Gp(Ω)={∇π∈Lp(Ω)n:π∈Lp,loc⁡(Ω̅)}.
The Helmholtz decomposition of Lp(Ω)n was proved by Fujiwara-Morimoto [16], Miyakawa [17], and Simader-Sohr [18]. Let P be a continuous projection from Lp(Ω)n onto Jp(Ω)n.By applyingP into (1.3) and setting  𝒪u∞=P(-Δ+u∞·∇), one has (1.6)ut+Ou∞u=0,fort>0,u(0)=u0,
where the domain of 𝒪u∞ is given by (1.7)Dp(Ou∞)={u∈Jp(Ω)∩Wp2(Ω)n:u∣∂Ω=0}.
Then, Enomoto and Shibata [15] proved that 𝒪u∞ generates an analytic semigroup {T(t)}t≥0 which is called the Oseen semigroup (one can also refer to [17, 19]) and obtained the following properties.Proposition 1.1.
Letσ0>0 and assume that  |u∞|≤σ0. Let  1≤r≤q≤∞. Then,
(1.8)‖T(t)a‖Lq(Ω)≤Cr,q,σ0t-3/2(1/r-1/q)‖a‖Lr(Ω),t>0,
where (r,q)≠(1,1) and (∞,∞),
(1.9)‖∇T(t)a‖Lq(Ω)≤Cr,q,σ0t-3/2(1/r-1/q)-1/2∥a∥Lr(Ω),t>0,
where 1≤r≤q≤3 and (r,q)≠(1,1).By using Proposition1.1, Bae-Jin [20] considered the spatial stability of stationary solution w of (1.3) and obtained the following: if |x|u0,u0∈Lr(Ω) with ∇·u0=0, then for any t>0,
(1.10)‖|x|u(t)‖p≤Ct-3/2(1/r-1/p)‖|x|u0‖Lr(Ω)+C|u∞|t-3/2(1/r-1/p)+1‖u0‖Lr(Ω),
where p≥3 and 1<r<3.And, for the nonstationary Navier-Stokes equations, we discuss the stability of stationary solutionw of the nonlinear Navier-Stokes equation (1.2), and w satisfies the following equations:
(1.11)-Δw+(u∞⋅∇)w+(w⋅∇)w+∇p2=f,∇⋅w=0,w|∂Ω=-u∞,lim⁡|x|→∞w(x)=0.For suitablef, Shibata [21] proved that for any given 0<δ<1/4 there exists ϵ such that if 0<|u∞|≤ϵ, then one has
(1.12)‖w‖L3/(1+δ1)(Ω)+‖w‖L3/(1-δ2)(Ω)+‖∇w‖L3/(2+δ1)(Ω)+‖∇w‖L3/(2-δ2)(Ω)≤C|u∞|1/2,
for small δ1,δ2, where C is independent of u∞.By settingu=v-w and p=p1-p2 for v,p1,w,p2 in (1.2) and (1.11), we have the following equations in Ω:
(1.13)∂∂tu-Δu+(u∞⋅∇)u+(u⋅∇)w+(w⋅∇)u+(u⋅∇)u+∇p=0,∇⋅u(t,x)=0,u(x,0)=u0(x)forx∈Ω,u(x,t)=0forx∈∂Ω,lim⁡|x|→∞u(x,t)=0.
Here, in fact, the initial data should be u0-u∞-w, but for our convenience we denote by u0 for u0-u∞-w if there is no confusion. Heywood [22, 23], Masuda [24], Shibata [21], Enomoto-Shibata [15], Bae-Roh [25], and Roh [26] have studied the temporal decay for solutions of (1.13), and we have the followings in [26].Proposition 1.2.
There exists smallϵ(p,q,r) such that if 0<|u∞|≤ϵ, and ∥u0∥L3(Ω)<ϵ, then a unique solution u(x,t) of (1.13) has
(1.14)‖u(t)‖Lp(Ω)≤Cϵt-3/2(1/r-1/p)‖u0‖rfor1<r<p≤∞,t>0,‖∇u(t)‖Lq(Ω)≤Cϵt-3/2(1/r-1/q)-(1/2)‖u0‖rfor1<r<q≤3,t>0,
where  u0∈L3(Ω)∩Lr(Ω).Now, we are in the position to introduce our main theorems which are the weighted stability of stationary solutionw.Theorem 1.3.
Let1<r<p<∞ and  (1/r-1/p)>2/3. Then there exists small  ϵ(p,r) such that if  0<|u∞|≤ϵ,  ∥u0∥L3(Ω)<ϵ,  |x|u0∈L3r/(3-2r)(Ω), and  ∇·u0=0, then the solution  u(x,t) of (1.13) satisfies
(1.15)‖|x|u(t)‖Lp(Ω)≤Cϵt-3/2(1/r-1/p)+1‖u0‖r,∀t>0,
where u0∈L3(Ω)∩Lr(Ω).Remark 1.4.
In Theorem1.3, the assumption |x|u0∈L3r/(3-2r)(Ω) is for simple calculations. We also can obtain a similar result where |x|u0∈Lr(Ω).For the proof we have to consider delay solution u(t)=u(t+t0).Then we can follow the method in Bae and Roh [4].Theorem 1.5.
Let1/r-1/p>2σ/3 for  1<σ<3/2 and  1<r<p<∞. Then there exists small  ϵ(p,r) such that if  0<|u∞|≤ϵ,  ∥u0∥L3(Ω)<ϵ,  |x|σu0∈L3r/(3-2r)(Ω), and  ∇·u0=0, then the solution  u(x,t) of (1.13) satisfies
(1.16)‖|x|σu(t)‖Lp(Ω)≤Cϵt-3/2(1/r-1/p)+σ‖u0‖r,∀t≥1,
where  u0∈L3(Ω)∩Lr(Ω).Remark 1.6.
For the exterior Navier-Stokes flows withu∞=0,temporal decay rate with weight function |x|σ becomes slower by σ/2; refer to  [1–4, 8, 13]. However, for u∞≠0,we found out from Theorems 1.3 and 1.5 that temporal decay rate with weight function |x|σ becomes slower by σ for 0≤σ<3/2.In fact, Bae and Roh [25] concluded that it becomes slower by (1+σ)/2 for 0<σ<1/2.Hence, our decay rate is little faster than the one in Bae and Roh [25] for 0<σ<1/2.One of the difficulties for the exterior Navier-Stokes equations is dealing with the boundary ofΩ because a pressure representation in terms of velocity is not a simple problem. So to remove the pressure term, we adapt an indirect method by taking a weight function ϕ vanishing near the boundary. This astonied method for exterior problem was initiated by He and Xin [27] and then developed by Bae and Jin [1, 2, 4, 20].


## 2. Proof of Main Theorems

In this section, we will prove the weighted stability of stationary solutions of the Navier-Stokes equations with nonzero far-field velocity. We first consider|x| for a weight function and then |x|σ for σ<3/2. Our method can be applied to the Oseen equations. As a result, we note that we can improve the result of Bae-Jin [1] by the same method.


### 2.1. Proof of Theorem1.3

We defineϕR(x)=|x|χ(|x|)(1-χ(|x|/R)) for large R>0, where χ is a nonnegative cutoff function with χ∈C∞[0,∞), χ(s)=0 for s≤1, and χ(s)=1 for s≥2. When there is no confusion, we use the same notation ϕ instead of ϕR for convenience.As in [1], we set (2.1)v(x)≡∫R3N(x-y)[ϕ(y)(∇×u)(y)]dy,
where N is the fundamental function of -Δ, that is, N=N(x-y)=1/(4π|x-y|). By the definition of v, we have -Δv=ϕ∇×u. Moreover, (2.2)∇×v=∫ΩN(x-y)∇×[ϕ(∇×u)](y)dy=ϕu+R0,
where (2.3)R0∶=∇N*[(u⋅∇)ϕ]-∇×N*[(∇ϕ)×u].We first estimate∥∇×v(t)∥p and then obtain the estimate of ∥ϕu(t)∥p=∥|x|u(t)∥p.Now, we consider the fundamental solutions for the nonstationary Oseen equations, written as(2.4)Vti(x)=Vi(x,t)=Γt(x)ei+∇∂∂xi(N*Γt)(x),
where  Γt(x)=Γ(x,t)=(4πt)-3/2e-|x-tu∞|2/4t (refer to [15, 28]). In fact, Γ is a translation in the direction of x by tu∞ of the heat kernel K(x,t)=(4πt)-3/2e-|x|2/4t, that is, Γ(x,t)=K(x-tu∞,t). Set ωti(x)=ωi(x,t)=(N*Γt)(x)ei, i=1,2,3, where ei is the standard unit vector of which the ith term is 1. Then, we have (2.5)∇×∇×ωi=-Δωi+∇div⁡ωi=Vi.
Hence, we have the identity (2.6)∇y×[ϕ(y)∇y×ωi(x-y,t-τ)]=ϕ(y)Vi(x-y,t-τ)+R1i(x,y,t-τ),
where (2.7)R1i(x,y,t-τ)=∇ϕ(y)×∇y×ωi(x-y,t-τ).From straightforward calculations we have that for1≤s≤∞, (2.8)‖∂βΓt-τ‖s≤c(t-τ)-3/2(1-1/s)-(|β|/2).One might note that we may sometimes use∥Vi∥s≤∥Γt∥3s/(3+s)<ct-1+3/2s instead of ∥Vi∥s≤∥Γt∥s<ct-3/2(1-1/s) because of technical reason. By the definition of Vi, both inequalities hold for any s≥3/2. We multiply (1.13) by ∇y×[ϕ(y)∇y×ωi(x-y,t-τ)] and integrate over Ω×(0,t-ϵ), and then we have (2.9)∫0t-ϵ∫Ω(∂u∂τ-Δyu+(u∞⋅∇y)u+(w⋅∇)u+(u⋅∇)w+(u⋅∇)u)⋅∇y×[ϕ(y)∇y×ωi(x-y,t-τ)]dydτ=-∫0t-ϵ∫Ω∇p(y)⋅∇y×[ϕ(y)∇y×ωi(x-y,t-τ)]dydτ=0.We finally get the following integral representation for∇×v (refer to  [2, 3] for the detail): (2.10)(∇x×v)i=(∇x×v0)*Γt-∫0t∫Ωu⋅[∂τ+Δy+(u∞⋅∇y)]R1(x,y,t-τ)dydτ-∫0t∫Ωu⋅[R2i(x,y,t-τ)]dydτ-∫0t∫Ωu⋅Vi(x-y,t-τ)(u∞⋅∇y)ϕ(y)dydτ-∫0t∫Ω(wku+ukw)⋅[∂yk(ϕ(y)Vi(x-y,t-τ))+∂ykR1i(x,y,t-τ)]dydτ-∫0t-ϵ∫Ωuku⋅[∂yk(ϕ(y)Vi(x-y,t-τ))+∂ykR1i(x,y,t-τ)]dydτ=I+II+III+IV+V+VI,
where (2.11)R2i(x,y,t-τ)=2(∇yϕ(y)⋅∇y)Vi(x-y,t-τ)+Δyϕ(y)Vi(x-y,t-τ).Applying Young’s convolution and the Calderon-Zygmund inequalities, we obtain(2.12)‖I‖p=‖(∇×v0)*Γt∥p≤∥u0ϕ*Γt∥p+∥∇N*[u0∇ϕ]*Γt‖p≤∥u0ϕ∥3r/(3-2r)∥Γt∥3pr/(5pr+3r-3p)+∥∇N*u0∇ϕ∥3r/(3-2r)∥Γt∥3pr/(5pr+3r-3p)≤Ct-3/2(1/r-1/p)+1∥ϕu0∥3r/3-2r+Ct-3/2(1/r-1/p)+1∥u0∥r,∀t>0,
if ϕu0∈L3r/(3-2r) and u0∈Lr.AndII is bounded by as follows:(2.13)∥II∥p≤c∫0t‖u‖s1‖∇2ϕ‖∞‖∂k∇×ωt-τi‖s2+‖u‖s3‖∇Δϕ‖3‖∇×ωt-τi‖s4+|u∞|‖|x|-1u‖s5‖∇×ωt-τi‖s6dτ=II1+II2+II3,
where 1/s1+1/s2=1+1/p, 1/s3+1/s4=1+1/p-1/3 and 1/s5+1/s6=1+1/p.We have(2.14)II1≤C‖u0‖r∫0tτ-3/2(1/r-1/s1)(t-τ)-3/2(1-1/s2)dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r-1/s1<2/3 and s2<3. Also, we obtain (2.15)II2≤C‖u0‖r∫0tτ-3/2(1/r-1/s3)(t-τ)-1+3/2s4dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r-1/s3<2/3. Finally, we get (2.16)II3≤C∫0t‖∇u‖s5‖∇×ωt-τi‖s6dτ≤C‖u0‖r∫0tτ-3/2(1/r-1/s5)-1/2(t-τ)-1+3/2s6dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r-1/s5<1/3. Hence, for any t>0, we have (2.17)‖I‖p+‖II‖p≤Ct-3/2(1/r-1/p)+1‖ϕu0‖3r/(3-2r)+C‖u0‖rt-3/2(1/r-1/p)+1.Also, we obtain(2.18)‖III‖p≤∫0t∥(u∂jϕ)*∂jVi∥p+∥(uΔϕ)*Vi∥pdτ≤∫0t∥u∥s∥∂jVi∥ps/(ps+s-p)+∥u∥s1∥∇2ϕ∥∞∥∂k∇×ωt-τi∥s2dτ≤C∥u0∥rt-3/2(1/r-1/p)+1,∀t>0,
where 1/s1+1/s2=1+1/p, 1/r-1/s1<2/3 and s2<3. In the above calculation, we used ∥∂Vi(t)∥q≤t-(3/2)(1-1/q) instead of ∥∂Vi(t)∥q≤t-3/2(1-1/q)-1/2 because of simplicity of calculations.And we have(2.19)‖IV‖p≤c|u∞|∫0t‖u‖s7‖∇ϕ‖∞‖Vi‖s8dτ≤C‖u0‖rt-(3/2)(1/r-1/p)+1,∀t>0,
where 1/s7+1/s8=1+1/p,1/r-1/s7<2/3 and s8<3.Next, forV, we have (2.20)V=-∫0t∫Ω(wku+ukw)⋅[(∂ykϕ(y))Vi(x-y,t-τ)+ϕ(y)∂ykVi(x-y,t-τ)+∂ykR1i(x-y,t-τ)]dydτ≤V1+V2+V3.We get(2.21)∥V1∥p≤c∫0t‖u‖r1‖w‖3‖∇ϕ‖∞‖Vi‖r2dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r1+1/r2=2/3+1/p,1/r-1/r1<2/3 and r2<3. In the above calculation, we used ∥Vi(t)∥q≤t-3/2(1-1/q)+1/2 instead of ∥Vi(t)∥q≤t-3/2(1-1/q) because of simplicity of calculations.Since∥|x|w∥∞<C (see [21]), we have (2.22)‖V2‖p≤c∫0t‖u‖r3‖ϕw‖∞‖∇Vi‖r4dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r3+1/r4=1+1/p, 1/r-1/r3<2/3 and r4<3. In the above calculation, we used  ∥∂Vi(t)∥q≤t-3/2(1-1/q)instead of ∥∂Vi(t)∥q≤t-3/2(1-1/q)-1/2 because of simplicity of calculations.Next, for anyt>0, we have (2.23)‖V3‖p≤∫0t‖u‖r5‖w‖3‖∇2ϕ‖∞‖∇×ωi‖r6+‖u‖r7‖w‖∞‖∇ϕ‖∞‖∂k∇×ωi‖r8dτ≤‖u0‖rt-3/2(1/r-1/p)+1,
where 1/r5+1/r6=1/r7+1/r8=2/3+1/p,1/r-1/r5<2/3, 1/r-1/r7<2/3 and r8<3.Hence, we have(2.24)‖V‖p≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0.ConsiderVIas follows:(2.25)VI=-∫0t∫Ωuku⋅[(∂ykϕ(y))Vi(x-y,t-τ)+ϕ(y)∂ykVi(x-y,t-τ)+∂ykR1i(x-y,t-τ)]dydτ≤VI1+VI2+VI3.We have, for anyt>0, (2.26)‖VI1‖p≤∫0t‖u‖s1‖u‖s2‖∇ϕ‖∞‖Vi‖s3dτ≤C2‖u0‖rt-3/2(1/r-1/p)+1,
where 1/s1+1/s2+1/s3=1+1/p,  1/r-1/s1+1/s2<1/3, and ∥u(t)∥s2≤ct-1/2+3/2s2. In the above calculation, we used ∥Vi(t)∥q≤t-3/2(1-1/q)+1/2 instead of  ∥Vi(t)∥q≤t-3/2(1-1/q) because of technical reason.Similar toVI1, we get (2.27)‖VI3‖p≤∫0t‖u‖r1‖u‖r2‖∇2ϕ‖∞‖∇×ωi‖r3+‖u‖s1‖u‖s2‖∇ϕ‖∞‖∂k∇×ωi‖s3dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r1+1/r2+1/r3=1+1/p=1/s1+1/s2+1/s3, 1/r-1/r1+1/r2<1/3, ∥u(t)∥r2≤ct-1/2+3/2r2, and 1/r-(1/s1+1/s2)<1/3,∥u(t)∥s2≤ct-1/2+3/2s2.Note that(2.28)‖ϕu(t)‖p≤‖∇×v(t)‖p+‖∇N*u(t)∇ϕ‖p≤‖∇×v(t)‖p+‖u(t)‖3p/(3+2p)≤‖∇×v(t)‖p+C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0.Since fort>0, ∥u(t)∥9≤ϵt-1/3, from Shibata [21], we have (2.29)‖VI2‖p≤∫0t‖ϕu(τ)‖p‖u(τ)‖9‖∇Vi(t-τ)‖9/8dτ≤ϵ∫0t‖∇×v(τ)‖pτ-1/3(t-τ)-2/3dτ+C‖u0‖rt-3/2(1/r-1/p)+1.Hence, we have(2.30)‖VI‖p≤ϵ∫0t‖∇×v(τ)‖pτ-1/3(t-τ)-2/3dτ+C‖u0‖rt-3/2(1/r-1/p)+1.Thus, by (2.17)–(2.19), (2.24), (2.28), and (2.30), for all t>0, we obtain (2.31)‖∇×v(t)‖p≤ϵ∫0t‖∇×v(τ)‖pτ-1/3(t-τ)-2/3dτ+C‖u0‖rt-3/2(1/r-1/p)+1.Now, we use the following lemma (refer to  [25]).Lemma 2.1.
Let a functionS(t) satisfy the inequality, for some α<2/3,
(2.32)S(t)≤ct-α+ε∫0tS(τ)τ-1/3(t-τ)-2/3dτ∀t>0.
One also assumes that
(2.33)lim⁡t→0+t-ε∫0tτ-1/3S(τ)dτ=0.
Then, there is ε0 so that if ε≤ε0, then one has
(2.34)S(t)≤ct-α
for some c independent of t.Since(2.35)‖∇×v(t)‖p≤‖ϕu(t)‖p+‖∇N*u(t)∇ϕ‖p≤R‖u(t)‖p+‖u(t)‖3p/(3+p)≤CR‖u0‖3t-1/2+3/2p+C2‖u0‖3t-3/2(1/3-1/p)+1/2,∀t>0,condition (2.33) satisfies
(2.36)lim⁡t→0+t-ε∫0tτ-1/3‖∇×v(τ)‖pdτ=lim⁡t→0+t-ε∫0tτ-1/3(CR‖u0‖3τ-1/2+3/2p+C2‖u0‖3τ3/2p)dτ=lim⁡t→0+(CR‖u0‖3τ-ϵ+1/3+3/2p+C2‖u0‖3τ-ϵ+1+3/2p)=0,
for ϵ<(1/3+3/2p).So, by Lemma2.1, we have
(2.37)‖∇×v(t)‖p≤C‖u0‖rt-3/2(1/r-1/p)+1.Hence, by (2.28), for any t>0, we have
(2.38)‖ϕu(t)‖p≤C‖u0‖rt-3/2(1/r-1/p)+1,
and by taking R→∞, we complete the proof of Theorem 1.3.


### 2.2. Proof of Theorem1.5

By using the results in previous section, for any0<α<1, we have small β>0 such that (2.39)‖|x|αu(t)‖s≤‖|x|αuα‖3/(α-3β)‖u1-α‖1/(1-α-β)≤[Ct-3/2(1/r-(α-3β)/3α)+1]α[Ct-3/2(1/r-(1-α-β)/(1-α))]1-α≤Ct-3/2(1/r-1/s)+α,
where  1-2α/3-2β=1/s.Now, in this section, we considerϕ(x)=|x|σχ(|x|), where 1<σ<3/2.Similar to previous section, for∥I∥p, II1, and II2, we obtain the same decay rate with previous section. And for any t>0, we have (2.40)II3≤C∫0t‖|x|σ-2‖3/(2-σ)2‖u‖s1‖∇×ωt-τi‖s2dτ≤C‖u0‖r∫0tτ-3/2(1/r-1/s1)(t-τ)-1+3/2s2dτ≤C‖u0‖rt-3/2(1/r-1/p)+3/2-(2-σ)2/2,
where 1/s1+1/s2=1+1/p and 1/r-1/s1<2/3. Also, for ∥III∥p, we obtain (2.41)‖III‖p≤∫0t‖(u∂jϕ)*∂jVi‖p+‖(uΔϕ)*Vi‖pdτ≤∫0t‖|x|σ-1u‖s‖∂jVi‖ps/(ps+s-p)+‖u‖s1‖∇2ϕ‖∞‖∂k∇×ωt-τi‖s2dτ≤C‖u0‖rt-3/2(1/r-1/p)+σ-1/2+C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/s1+1/s2=1+1/p, 1/r-1/s1<2/3 and s2<3.Next, we have(2.42)‖IV‖p≤c|u∞|∫0t‖|x|σ-1u‖s1‖Vi‖s2dτ≤C|u∞|‖u0‖rt-3/2(1/r-1/p)+σ,∀t>0,
where 1/s1+1/s2=1+1/p, s2<3, and 1/r-1/s1<2σ/3. Also, since ∥|x|w∥∞<C, we get (2.43)‖V1‖p≤c∫0t‖u‖r1‖|x|σ-1w‖∞‖Vi‖r2dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r1+1/r2=1+1/p, 1/r-1/r1<2/3, and r2<3.And we obtain(2.44)‖V2‖p≤c∫0t‖|x|σ-1u‖r3‖|x|w‖∞‖∇Vi‖r4dτ≤C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,∀t>0,
where 1/r3+1/r4=1+1/p, 1/r-1/r3<2σ/3, and r4<3/2.Next, for anyt>0, we have (2.45)‖V3‖p≤∫0t‖u‖r5‖w‖3‖∇2ϕ‖∞‖∇×ωi‖r6+‖|x|σ-1u‖r7‖w‖3‖∂k∇×ωi‖r8dτ≤C‖u0‖rt-3/2(1/r-1/p)+1+C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,
where 1/r5+1/r6=1/r7+1/r8=2/3+1/p, 1/r-1/r5<2/3, 1/r-1/r7<2σ/3, and r8<3.Hence, we have(2.46)‖V‖p≤C‖u0‖rt-3/2(1/r-1/p)+1+C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,∀t>0.ConsiderVI  as follows:(2.47)VI=-∫0t∫Ωuku⋅[(∂ykϕ(y))Vi(x-y,t-τ)+ϕ(y)∂ykVi(x-y,t-τ)+∂ykR1i(x-y,t-τ)]dydτ≤VI1+VI2+VI3.We have, for anyt>0, (2.48)‖VI1‖p≤∫0t‖|x|σ-1u‖s1‖u‖s2‖Vi‖s3dτ≤C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,
where 1/s1+1/s2+1/s3=1+1/p, s3<3, 1/r-1/s1<2σ/3, and ∥u(t)∥s2≤ct-1/2+3/2s2.Similar toVI1, we get (2.49)‖VI3‖p≤∫0t‖u‖r1‖u‖r2‖∇2ϕ‖∞‖∇×ωi‖r3+‖|x|σ-1u‖s1‖u‖s2‖∂k∇×ωi‖s3dτ≤C‖u0‖rt-3/2(1/r-1/p)+1/2+C‖u0‖rt-3/2(1/r-1/p)+σ,∀t>0,
where 1/r1+1/r2+1/r3=1+1/p=1/s1+1/s2+1/s3, 1/r-1/r1<2/3, 1/r-(1/r1+1/r2)<1/3, 1/r-1/s1<2σ/3, 1/r-1/s2<2/3, 1/r-(1/s1+1/s2)<(2σ-1)/3, ∥u(t)∥s2≤ct-1/2+3/2s2, and ∥u(t)∥r2≤ct-1/2+3/2r2. In the above calculation, we used ∥Vi(t)∥q≤t-3/2(1-1/q)+1/2 instead of ∥Vi(t)∥q≤t-3/2(1-1/q) because of technical reason. Now, we have (2.50)‖VI2‖p≤∫0t‖|x|u(τ)‖s1‖|x|σ-1u(τ)‖s2‖∇Vi(t-τ)‖s3dτ≤C‖u0‖rt-3/2(1/r-1/p)+σ+1/2-3/2r1,
where 1/s1+1/s2+1/s3=1+1/p, s2<3/2, ∥|x|σ-1u(τ)∥s2<Ct-3/2(1/r1-1/s2)+σ-1, and r1<3(≈3).So, we obtain(2.51)‖ϕu(t)‖p≤‖∇×v(t)‖p+‖∇N*u(t)∇ϕ‖p≤‖∇×v(t)‖p+‖|x|σ-1u(t)‖3p/(3+p)≤‖∇×v(t)‖p+C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,∀t>0≤C‖u0‖rt-3/2(1/r-1/p)+σ,∀t≥1,
which completes the proof.


## 2.1. Proof of Theorem1.3

We defineϕR(x)=|x|χ(|x|)(1-χ(|x|/R)) for large R>0, where χ is a nonnegative cutoff function with χ∈C∞[0,∞), χ(s)=0 for s≤1, and χ(s)=1 for s≥2. When there is no confusion, we use the same notation ϕ instead of ϕR for convenience.As in [1], we set (2.1)v(x)≡∫R3N(x-y)[ϕ(y)(∇×u)(y)]dy,
where N is the fundamental function of -Δ, that is, N=N(x-y)=1/(4π|x-y|). By the definition of v, we have -Δv=ϕ∇×u. Moreover, (2.2)∇×v=∫ΩN(x-y)∇×[ϕ(∇×u)](y)dy=ϕu+R0,
where (2.3)R0∶=∇N*[(u⋅∇)ϕ]-∇×N*[(∇ϕ)×u].We first estimate∥∇×v(t)∥p and then obtain the estimate of ∥ϕu(t)∥p=∥|x|u(t)∥p.Now, we consider the fundamental solutions for the nonstationary Oseen equations, written as(2.4)Vti(x)=Vi(x,t)=Γt(x)ei+∇∂∂xi(N*Γt)(x),
where  Γt(x)=Γ(x,t)=(4πt)-3/2e-|x-tu∞|2/4t (refer to [15, 28]). In fact, Γ is a translation in the direction of x by tu∞ of the heat kernel K(x,t)=(4πt)-3/2e-|x|2/4t, that is, Γ(x,t)=K(x-tu∞,t). Set ωti(x)=ωi(x,t)=(N*Γt)(x)ei, i=1,2,3, where ei is the standard unit vector of which the ith term is 1. Then, we have (2.5)∇×∇×ωi=-Δωi+∇div⁡ωi=Vi.
Hence, we have the identity (2.6)∇y×[ϕ(y)∇y×ωi(x-y,t-τ)]=ϕ(y)Vi(x-y,t-τ)+R1i(x,y,t-τ),
where (2.7)R1i(x,y,t-τ)=∇ϕ(y)×∇y×ωi(x-y,t-τ).From straightforward calculations we have that for1≤s≤∞, (2.8)‖∂βΓt-τ‖s≤c(t-τ)-3/2(1-1/s)-(|β|/2).One might note that we may sometimes use∥Vi∥s≤∥Γt∥3s/(3+s)<ct-1+3/2s instead of ∥Vi∥s≤∥Γt∥s<ct-3/2(1-1/s) because of technical reason. By the definition of Vi, both inequalities hold for any s≥3/2. We multiply (1.13) by ∇y×[ϕ(y)∇y×ωi(x-y,t-τ)] and integrate over Ω×(0,t-ϵ), and then we have (2.9)∫0t-ϵ∫Ω(∂u∂τ-Δyu+(u∞⋅∇y)u+(w⋅∇)u+(u⋅∇)w+(u⋅∇)u)⋅∇y×[ϕ(y)∇y×ωi(x-y,t-τ)]dydτ=-∫0t-ϵ∫Ω∇p(y)⋅∇y×[ϕ(y)∇y×ωi(x-y,t-τ)]dydτ=0.We finally get the following integral representation for∇×v (refer to  [2, 3] for the detail): (2.10)(∇x×v)i=(∇x×v0)*Γt-∫0t∫Ωu⋅[∂τ+Δy+(u∞⋅∇y)]R1(x,y,t-τ)dydτ-∫0t∫Ωu⋅[R2i(x,y,t-τ)]dydτ-∫0t∫Ωu⋅Vi(x-y,t-τ)(u∞⋅∇y)ϕ(y)dydτ-∫0t∫Ω(wku+ukw)⋅[∂yk(ϕ(y)Vi(x-y,t-τ))+∂ykR1i(x,y,t-τ)]dydτ-∫0t-ϵ∫Ωuku⋅[∂yk(ϕ(y)Vi(x-y,t-τ))+∂ykR1i(x,y,t-τ)]dydτ=I+II+III+IV+V+VI,
where (2.11)R2i(x,y,t-τ)=2(∇yϕ(y)⋅∇y)Vi(x-y,t-τ)+Δyϕ(y)Vi(x-y,t-τ).Applying Young’s convolution and the Calderon-Zygmund inequalities, we obtain(2.12)‖I‖p=‖(∇×v0)*Γt∥p≤∥u0ϕ*Γt∥p+∥∇N*[u0∇ϕ]*Γt‖p≤∥u0ϕ∥3r/(3-2r)∥Γt∥3pr/(5pr+3r-3p)+∥∇N*u0∇ϕ∥3r/(3-2r)∥Γt∥3pr/(5pr+3r-3p)≤Ct-3/2(1/r-1/p)+1∥ϕu0∥3r/3-2r+Ct-3/2(1/r-1/p)+1∥u0∥r,∀t>0,
if ϕu0∈L3r/(3-2r) and u0∈Lr.AndII is bounded by as follows:(2.13)∥II∥p≤c∫0t‖u‖s1‖∇2ϕ‖∞‖∂k∇×ωt-τi‖s2+‖u‖s3‖∇Δϕ‖3‖∇×ωt-τi‖s4+|u∞|‖|x|-1u‖s5‖∇×ωt-τi‖s6dτ=II1+II2+II3,
where 1/s1+1/s2=1+1/p, 1/s3+1/s4=1+1/p-1/3 and 1/s5+1/s6=1+1/p.We have(2.14)II1≤C‖u0‖r∫0tτ-3/2(1/r-1/s1)(t-τ)-3/2(1-1/s2)dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r-1/s1<2/3 and s2<3. Also, we obtain (2.15)II2≤C‖u0‖r∫0tτ-3/2(1/r-1/s3)(t-τ)-1+3/2s4dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r-1/s3<2/3. Finally, we get (2.16)II3≤C∫0t‖∇u‖s5‖∇×ωt-τi‖s6dτ≤C‖u0‖r∫0tτ-3/2(1/r-1/s5)-1/2(t-τ)-1+3/2s6dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r-1/s5<1/3. Hence, for any t>0, we have (2.17)‖I‖p+‖II‖p≤Ct-3/2(1/r-1/p)+1‖ϕu0‖3r/(3-2r)+C‖u0‖rt-3/2(1/r-1/p)+1.Also, we obtain(2.18)‖III‖p≤∫0t∥(u∂jϕ)*∂jVi∥p+∥(uΔϕ)*Vi∥pdτ≤∫0t∥u∥s∥∂jVi∥ps/(ps+s-p)+∥u∥s1∥∇2ϕ∥∞∥∂k∇×ωt-τi∥s2dτ≤C∥u0∥rt-3/2(1/r-1/p)+1,∀t>0,
where 1/s1+1/s2=1+1/p, 1/r-1/s1<2/3 and s2<3. In the above calculation, we used ∥∂Vi(t)∥q≤t-(3/2)(1-1/q) instead of ∥∂Vi(t)∥q≤t-3/2(1-1/q)-1/2 because of simplicity of calculations.And we have(2.19)‖IV‖p≤c|u∞|∫0t‖u‖s7‖∇ϕ‖∞‖Vi‖s8dτ≤C‖u0‖rt-(3/2)(1/r-1/p)+1,∀t>0,
where 1/s7+1/s8=1+1/p,1/r-1/s7<2/3 and s8<3.Next, forV, we have (2.20)V=-∫0t∫Ω(wku+ukw)⋅[(∂ykϕ(y))Vi(x-y,t-τ)+ϕ(y)∂ykVi(x-y,t-τ)+∂ykR1i(x-y,t-τ)]dydτ≤V1+V2+V3.We get(2.21)∥V1∥p≤c∫0t‖u‖r1‖w‖3‖∇ϕ‖∞‖Vi‖r2dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r1+1/r2=2/3+1/p,1/r-1/r1<2/3 and r2<3. In the above calculation, we used ∥Vi(t)∥q≤t-3/2(1-1/q)+1/2 instead of ∥Vi(t)∥q≤t-3/2(1-1/q) because of simplicity of calculations.Since∥|x|w∥∞<C (see [21]), we have (2.22)‖V2‖p≤c∫0t‖u‖r3‖ϕw‖∞‖∇Vi‖r4dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r3+1/r4=1+1/p, 1/r-1/r3<2/3 and r4<3. In the above calculation, we used  ∥∂Vi(t)∥q≤t-3/2(1-1/q)instead of ∥∂Vi(t)∥q≤t-3/2(1-1/q)-1/2 because of simplicity of calculations.Next, for anyt>0, we have (2.23)‖V3‖p≤∫0t‖u‖r5‖w‖3‖∇2ϕ‖∞‖∇×ωi‖r6+‖u‖r7‖w‖∞‖∇ϕ‖∞‖∂k∇×ωi‖r8dτ≤‖u0‖rt-3/2(1/r-1/p)+1,
where 1/r5+1/r6=1/r7+1/r8=2/3+1/p,1/r-1/r5<2/3, 1/r-1/r7<2/3 and r8<3.Hence, we have(2.24)‖V‖p≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0.ConsiderVIas follows:(2.25)VI=-∫0t∫Ωuku⋅[(∂ykϕ(y))Vi(x-y,t-τ)+ϕ(y)∂ykVi(x-y,t-τ)+∂ykR1i(x-y,t-τ)]dydτ≤VI1+VI2+VI3.We have, for anyt>0, (2.26)‖VI1‖p≤∫0t‖u‖s1‖u‖s2‖∇ϕ‖∞‖Vi‖s3dτ≤C2‖u0‖rt-3/2(1/r-1/p)+1,
where 1/s1+1/s2+1/s3=1+1/p,  1/r-1/s1+1/s2<1/3, and ∥u(t)∥s2≤ct-1/2+3/2s2. In the above calculation, we used ∥Vi(t)∥q≤t-3/2(1-1/q)+1/2 instead of  ∥Vi(t)∥q≤t-3/2(1-1/q) because of technical reason.Similar toVI1, we get (2.27)‖VI3‖p≤∫0t‖u‖r1‖u‖r2‖∇2ϕ‖∞‖∇×ωi‖r3+‖u‖s1‖u‖s2‖∇ϕ‖∞‖∂k∇×ωi‖s3dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r1+1/r2+1/r3=1+1/p=1/s1+1/s2+1/s3, 1/r-1/r1+1/r2<1/3, ∥u(t)∥r2≤ct-1/2+3/2r2, and 1/r-(1/s1+1/s2)<1/3,∥u(t)∥s2≤ct-1/2+3/2s2.Note that(2.28)‖ϕu(t)‖p≤‖∇×v(t)‖p+‖∇N*u(t)∇ϕ‖p≤‖∇×v(t)‖p+‖u(t)‖3p/(3+2p)≤‖∇×v(t)‖p+C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0.Since fort>0, ∥u(t)∥9≤ϵt-1/3, from Shibata [21], we have (2.29)‖VI2‖p≤∫0t‖ϕu(τ)‖p‖u(τ)‖9‖∇Vi(t-τ)‖9/8dτ≤ϵ∫0t‖∇×v(τ)‖pτ-1/3(t-τ)-2/3dτ+C‖u0‖rt-3/2(1/r-1/p)+1.Hence, we have(2.30)‖VI‖p≤ϵ∫0t‖∇×v(τ)‖pτ-1/3(t-τ)-2/3dτ+C‖u0‖rt-3/2(1/r-1/p)+1.Thus, by (2.17)–(2.19), (2.24), (2.28), and (2.30), for all t>0, we obtain (2.31)‖∇×v(t)‖p≤ϵ∫0t‖∇×v(τ)‖pτ-1/3(t-τ)-2/3dτ+C‖u0‖rt-3/2(1/r-1/p)+1.Now, we use the following lemma (refer to  [25]).Lemma 2.1.
Let a functionS(t) satisfy the inequality, for some α<2/3,
(2.32)S(t)≤ct-α+ε∫0tS(τ)τ-1/3(t-τ)-2/3dτ∀t>0.
One also assumes that
(2.33)lim⁡t→0+t-ε∫0tτ-1/3S(τ)dτ=0.
Then, there is ε0 so that if ε≤ε0, then one has
(2.34)S(t)≤ct-α
for some c independent of t.Since(2.35)‖∇×v(t)‖p≤‖ϕu(t)‖p+‖∇N*u(t)∇ϕ‖p≤R‖u(t)‖p+‖u(t)‖3p/(3+p)≤CR‖u0‖3t-1/2+3/2p+C2‖u0‖3t-3/2(1/3-1/p)+1/2,∀t>0,condition (2.33) satisfies
(2.36)lim⁡t→0+t-ε∫0tτ-1/3‖∇×v(τ)‖pdτ=lim⁡t→0+t-ε∫0tτ-1/3(CR‖u0‖3τ-1/2+3/2p+C2‖u0‖3τ3/2p)dτ=lim⁡t→0+(CR‖u0‖3τ-ϵ+1/3+3/2p+C2‖u0‖3τ-ϵ+1+3/2p)=0,
for ϵ<(1/3+3/2p).So, by Lemma2.1, we have
(2.37)‖∇×v(t)‖p≤C‖u0‖rt-3/2(1/r-1/p)+1.Hence, by (2.28), for any t>0, we have
(2.38)‖ϕu(t)‖p≤C‖u0‖rt-3/2(1/r-1/p)+1,
and by taking R→∞, we complete the proof of Theorem 1.3.


## 2.2. Proof of Theorem1.5

By using the results in previous section, for any0<α<1, we have small β>0 such that (2.39)‖|x|αu(t)‖s≤‖|x|αuα‖3/(α-3β)‖u1-α‖1/(1-α-β)≤[Ct-3/2(1/r-(α-3β)/3α)+1]α[Ct-3/2(1/r-(1-α-β)/(1-α))]1-α≤Ct-3/2(1/r-1/s)+α,
where  1-2α/3-2β=1/s.Now, in this section, we considerϕ(x)=|x|σχ(|x|), where 1<σ<3/2.Similar to previous section, for∥I∥p, II1, and II2, we obtain the same decay rate with previous section. And for any t>0, we have (2.40)II3≤C∫0t‖|x|σ-2‖3/(2-σ)2‖u‖s1‖∇×ωt-τi‖s2dτ≤C‖u0‖r∫0tτ-3/2(1/r-1/s1)(t-τ)-1+3/2s2dτ≤C‖u0‖rt-3/2(1/r-1/p)+3/2-(2-σ)2/2,
where 1/s1+1/s2=1+1/p and 1/r-1/s1<2/3. Also, for ∥III∥p, we obtain (2.41)‖III‖p≤∫0t‖(u∂jϕ)*∂jVi‖p+‖(uΔϕ)*Vi‖pdτ≤∫0t‖|x|σ-1u‖s‖∂jVi‖ps/(ps+s-p)+‖u‖s1‖∇2ϕ‖∞‖∂k∇×ωt-τi‖s2dτ≤C‖u0‖rt-3/2(1/r-1/p)+σ-1/2+C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/s1+1/s2=1+1/p, 1/r-1/s1<2/3 and s2<3.Next, we have(2.42)‖IV‖p≤c|u∞|∫0t‖|x|σ-1u‖s1‖Vi‖s2dτ≤C|u∞|‖u0‖rt-3/2(1/r-1/p)+σ,∀t>0,
where 1/s1+1/s2=1+1/p, s2<3, and 1/r-1/s1<2σ/3. Also, since ∥|x|w∥∞<C, we get (2.43)‖V1‖p≤c∫0t‖u‖r1‖|x|σ-1w‖∞‖Vi‖r2dτ≤C‖u0‖rt-3/2(1/r-1/p)+1,∀t>0,
where 1/r1+1/r2=1+1/p, 1/r-1/r1<2/3, and r2<3.And we obtain(2.44)‖V2‖p≤c∫0t‖|x|σ-1u‖r3‖|x|w‖∞‖∇Vi‖r4dτ≤C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,∀t>0,
where 1/r3+1/r4=1+1/p, 1/r-1/r3<2σ/3, and r4<3/2.Next, for anyt>0, we have (2.45)‖V3‖p≤∫0t‖u‖r5‖w‖3‖∇2ϕ‖∞‖∇×ωi‖r6+‖|x|σ-1u‖r7‖w‖3‖∂k∇×ωi‖r8dτ≤C‖u0‖rt-3/2(1/r-1/p)+1+C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,
where 1/r5+1/r6=1/r7+1/r8=2/3+1/p, 1/r-1/r5<2/3, 1/r-1/r7<2σ/3, and r8<3.Hence, we have(2.46)‖V‖p≤C‖u0‖rt-3/2(1/r-1/p)+1+C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,∀t>0.ConsiderVI  as follows:(2.47)VI=-∫0t∫Ωuku⋅[(∂ykϕ(y))Vi(x-y,t-τ)+ϕ(y)∂ykVi(x-y,t-τ)+∂ykR1i(x-y,t-τ)]dydτ≤VI1+VI2+VI3.We have, for anyt>0, (2.48)‖VI1‖p≤∫0t‖|x|σ-1u‖s1‖u‖s2‖Vi‖s3dτ≤C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,
where 1/s1+1/s2+1/s3=1+1/p, s3<3, 1/r-1/s1<2σ/3, and ∥u(t)∥s2≤ct-1/2+3/2s2.Similar toVI1, we get (2.49)‖VI3‖p≤∫0t‖u‖r1‖u‖r2‖∇2ϕ‖∞‖∇×ωi‖r3+‖|x|σ-1u‖s1‖u‖s2‖∂k∇×ωi‖s3dτ≤C‖u0‖rt-3/2(1/r-1/p)+1/2+C‖u0‖rt-3/2(1/r-1/p)+σ,∀t>0,
where 1/r1+1/r2+1/r3=1+1/p=1/s1+1/s2+1/s3, 1/r-1/r1<2/3, 1/r-(1/r1+1/r2)<1/3, 1/r-1/s1<2σ/3, 1/r-1/s2<2/3, 1/r-(1/s1+1/s2)<(2σ-1)/3, ∥u(t)∥s2≤ct-1/2+3/2s2, and ∥u(t)∥r2≤ct-1/2+3/2r2. In the above calculation, we used ∥Vi(t)∥q≤t-3/2(1-1/q)+1/2 instead of ∥Vi(t)∥q≤t-3/2(1-1/q) because of technical reason. Now, we have (2.50)‖VI2‖p≤∫0t‖|x|u(τ)‖s1‖|x|σ-1u(τ)‖s2‖∇Vi(t-τ)‖s3dτ≤C‖u0‖rt-3/2(1/r-1/p)+σ+1/2-3/2r1,
where 1/s1+1/s2+1/s3=1+1/p, s2<3/2, ∥|x|σ-1u(τ)∥s2<Ct-3/2(1/r1-1/s2)+σ-1, and r1<3(≈3).So, we obtain(2.51)‖ϕu(t)‖p≤‖∇×v(t)‖p+‖∇N*u(t)∇ϕ‖p≤‖∇×v(t)‖p+‖|x|σ-1u(t)‖3p/(3+p)≤‖∇×v(t)‖p+C‖u0‖rt-3/2(1/r-1/p)+σ-1/2,∀t>0≤C‖u0‖rt-3/2(1/r-1/p)+σ,∀t≥1,
which completes the proof.



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*Source: 369571-2011-11-30.xml*