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

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) in 𝐿² space and by Shibata (1999) and Enomoto-Shibata (2005) in 𝐿^𝑝 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 𝐿^𝑟−𝐿^𝑝 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 𝐿^𝑝-norm of solutions. This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008).

1. Introduction

When a boat is sailing with a constant velocity 𝐮_∞, 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: 𝜕()(),𝜕𝑡𝐮−Δ𝐮+𝐮⋅∇𝐮+∇𝑝=𝐟,∇⋅𝐮=0inΩ×0,∞𝐮|_𝑡=0=𝐮₀,𝐮|_𝜕Ω=0,lim_|𝑥|→∞𝐮(𝑥,𝑡)=𝐮_∞, (1.1) 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 constant 𝐮_∞. We set 𝐮=𝐮_∞+𝐯 in (1.1) and have 𝜕𝐮𝜕𝑡𝐯−Δ𝐯+_∞⋅∇𝐯+(𝐯⋅∇)𝐯+∇𝑝₁(),=𝐟,∇⋅𝐯=0,inΩ×0,∞𝐯|_𝑡=0=𝐮₀−𝐮_∞,𝐯|_𝜕Ω=−𝐮_∞,lim_|𝑥|→∞𝐯(𝑥,𝑡)=0. (1.2)

Consider the following linear equations of (1.2): 𝜕𝐮𝜕𝑡𝐮−Δ𝐮+_∞⋅∇𝐮+∇𝑝=0,∇⋅𝐮=0inΩ×(0,∞),𝐮|_𝑡=0=𝐮₀,𝐮|_𝜕Ω=0,lim_|𝑥|→∞𝐮(𝑥,𝑡)=0, (1.3) 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: 𝐿_𝑝(Ω)^𝑛=𝐽_𝑝(Ω)⊕G_𝑝(Ω), (1.4) where 1<𝑝<∞, 𝐿_𝑝(Ω)^𝑛=𝑢=𝑢₁,…,𝑢_𝑛∶𝑢_𝑗∈𝐿_𝑝(Ω),𝐶,𝑗=1,…,𝑛_∞0,𝜎=𝑢=𝑢₁,…,𝑢_𝑛∈𝐶^∞₀(Ω)^𝑛,𝐽∶∇⋅𝑢=0inΩ_𝑝(Ω)=thecompletionof𝐶^∞_0,𝜎(Ω)in𝐿_𝑝(Ω)^𝑛,𝐺_𝑝(Ω)=∇𝜋∈𝐿_𝑝(Ω)^𝑛∶𝜋∈𝐿_𝑝,locΩ. (1.5) The Helmholtz decomposition of 𝐿_𝑝(Ω)^𝑛 was proved by Fujiwara-Morimoto [16], Miyakawa [17], and Simader-Sohr [18]. Let 𝑃 be a continuous projection from 𝐿_𝑝(Ω)^𝑛 onto 𝐽_𝑝(Ω)^𝑛.

By applying 𝑃 into (1.3) and setting 𝒪_𝐮∞=𝑃(−Δ+𝐮_∞⋅∇), one has 𝐮_𝑡+𝒪_𝐮∞𝐮=0,for𝑡>0,𝐮(0)=𝐮₀, (1.6) where the domain of 𝒪_𝐮∞ is given by 𝒟_𝑝𝒪_𝐮∞=𝑢∈𝐽_𝑝(Ω)∩𝑊²_𝑝(Ω)^𝑛∶𝑢∣_𝜕Ω=0. (1.7) Then, Enomoto and Shibata [15] proved that 𝒪_𝐮∞ 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. Let 𝜎₀>0 and assume that |𝐮_∞|≤𝜎₀. Let 1≤𝑟≤𝑞≤∞. Then, ‖𝑇(𝑡)𝑎‖_𝐿^𝑞_(Ω)≤𝐶_{𝑟,𝑞,𝜎0}𝑡^{−3/2(1/𝑟−1/𝑞)}‖𝑎‖_𝐿^𝑟_(Ω),𝑡>0, (1.8) where (𝑟,𝑞)≠(1,1) and (∞,∞), ‖∇𝑇(𝑡)𝑎‖_𝐿^𝑞_(Ω)≤𝐶_{𝑟,𝑞,𝜎0}𝑡^{−3/2(1/𝑟−1/𝑞)−1/2}‖𝑎‖_𝐿^𝑟_(Ω),𝑡>0, (1.9) where 1≤𝑟≤𝑞≤3 and (𝑟,𝑞)≠(1,1).

By using Proposition 1.1, Bae-Jin [20] considered the spatial stability of stationary solution 𝐰 of (1.3) and obtained the following: if |𝑥|𝐮₀,𝐮₀∈𝐿^𝑟(Ω) with ∇⋅𝐮₀=0, then for any 𝑡>0, ‖|𝑥|𝐮(𝑡)‖_𝑝≤𝐶𝑡^{−3/2(1/𝑟−1/𝑝)}|𝑥|𝐮_{0𝐿𝑟(Ω)}+𝐶𝐮_∞𝑡^{−3/2(1/𝑟−1/𝑝)+1}𝐮_{0𝐿𝑟(Ω)}, (1.10) where 𝑝≥3 and 1<𝑟<3.

And, for the nonstationary Navier-Stokes equations, we discuss the stability of stationary solution 𝐰 of the nonlinear Navier-Stokes equation (1.2), and 𝐰 satisfies the following equations: −Δ𝐰+𝐮_∞⋅∇𝐰+(𝐰⋅∇)𝐰+∇𝑝₂=𝐟,∇⋅𝐰=0,𝐰|_𝜕Ω=−𝐮_∞,lim_|𝑥|→∞𝐰(𝑥)=0. (1.11)

For suitable 𝐟, Shibata [21] proved that for any given 0<𝛿<1/4 there exists 𝜖 such that if 0<|𝐮_∞|≤𝜖, then one has ‖𝐰‖_𝐿1^)3/(1+𝛿_(Ω)+‖𝐰‖_𝐿2^)3/(1−𝛿_(Ω)+‖∇𝐰‖_𝐿1^)3/(2+𝛿_(Ω)+‖∇𝐰‖_𝐿2^)3/(2−𝛿_(Ω)≤𝐶𝐮_∞^1/2, (1.12) for small 𝛿₁,𝛿₂, where 𝐶 is independent of 𝐮_∞.

By setting 𝐮=𝐯−𝐰 and 𝑝=𝑝₁−𝑝₂ for 𝐯,𝑝₁,𝐰,𝑝₂ in (1.2) and (1.11), we have the following equations in Ω: 𝜕𝐮𝜕𝑡𝐮−Δ𝐮+_∞()()()()()⋅∇𝐮+𝐮⋅∇𝐰+𝐰⋅∇𝐮+𝐮⋅∇𝐮+∇𝑝=0,∇⋅𝐮𝑡,𝑥=0,𝐮𝑥,0=𝐮₀(𝑥)𝐮()for𝑥∈Ω,𝑥,𝑡=0for𝑥∈𝜕Ω,lim_|𝑥|→∞𝐮(𝑥,𝑡)=0. (1.13) 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 small 𝜖(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 ‖𝐮(𝑡)‖_𝐿^𝑝_(Ω)≤𝐶_𝜖𝑡^{−3/2(1/𝑟−1/𝑝)}𝐮_0r‖‖for1<𝑟<𝑝≤∞,𝑡>0,∇𝐮(𝑡)_𝐿^𝑞_(Ω)≤𝐶_𝜖𝑡^{−3/2(1/𝑟−1/𝑞)−(1/2)}𝐮_0rfor1<𝑟<𝑞≤3,𝑡>0, (1.14) where 𝐮₀∈𝐿³(Ω)∩𝐿^𝑟(Ω).

Now, we are in the position to introduce our main theorems which are the weighted stability of stationary solution 𝐰.

Theorem 1.3. Let 1<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∈𝐿^{3𝑟/(3−2𝑟)}(Ω), and ∇⋅𝐮₀=0, then the solution 𝐮(𝑥,𝑡) of (1.13) satisfies ‖|𝑥|𝐮(𝑡)‖_𝐿^𝑝_(Ω)≤𝐶_𝜖𝑡^{−3/2(1/𝑟−1/𝑝)+1}𝐮_0𝑟,∀𝑡>0, (1.15) where 𝐮₀∈𝐿³(Ω)∩𝐿^𝑟(Ω).

Remark 1.4. In Theorem 1.3, the assumption |𝑥|𝐮₀∈𝐿^{3𝑟/(3−2𝑟)}(Ω) is for simple calculations. We also can obtain a similar result where |𝑥|𝐮₀∈𝐿^𝑟(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. Let 1/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∈𝐿^{3𝑟/(3−2𝑟)}(Ω), and ∇⋅𝐮₀=0, then the solution 𝐮(𝑥,𝑡) of (1.13) satisfies |𝑥|^𝜎𝐮(𝑡)_𝐿𝑝(Ω)≤𝐶_𝜖𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎}𝐮_0𝑟,∀𝑡≥1, (1.16) where 𝐮₀∈𝐿³(Ω)∩𝐿^𝑟(Ω).

Remark 1.6. For the exterior Navier-Stokes flows with 𝐮_∞=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 of Ω 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 Theorem 1.3

We define 𝜙_𝑅(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 𝜙_𝑅 for convenience.

As in [1], we set 𝐯(𝑥)≡_ℝ3𝑁𝑥−𝑦𝜙𝑦(∇×𝐮)𝑦𝑑𝑦, (2.1) 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, ∇×𝐯=_Ω𝑁𝑥−𝑦∇×𝜙(∇×𝐮)𝑦𝑑𝑦=𝜙𝐮+𝐑₀, (2.2) where 𝐑₀∶=∇𝑁∗(𝐮⋅∇)𝜙−∇×𝑁∗∇𝜙×𝐮. (2.3)

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 𝑉^𝑖_𝑡(𝑥)=𝑉^𝑖(𝑥,𝑡)=Γ_𝑡(𝑥)𝐞^𝑖𝜕+∇𝜕𝑥_𝑖𝑁∗Γ_𝑡(𝑥), (2.4) where Γ_𝑡(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 𝜔^𝑖_𝑡(𝑥)=𝜔^𝑖(𝑥,𝑡)=(𝑁∗Γ_𝑡)(𝑥)𝐞^𝑖, 𝑖=1,2,3, where 𝐞^𝑖 is the standard unit vector of which the 𝑖th term is 1. Then, we have ∇×∇×𝜔^𝑖=−Δ𝜔^𝑖+∇div𝜔^𝑖=𝑉^𝑖. (2.5) Hence, we have the identity ∇_𝑦×𝜙𝑦∇_𝑦×𝜔^𝑖𝑥−𝑦,𝑡−𝜏=𝜙𝑦𝑉^𝑖𝑥−𝑦,𝑡−𝜏+𝑅^𝑖₁𝑥,𝑦,𝑡−𝜏, (2.6) where 𝑅^𝑖₁𝑥,𝑦,𝑡−𝜏=∇𝜙𝑦×∇_𝑦×𝜔^𝑖𝑥−𝑦,𝑡−𝜏. (2.7)

From straightforward calculations we have that for 1≤𝑠≤∞, 𝜕^𝛽Γ_{𝑡−𝜏𝑠}≤𝑐(𝑡−𝜏)^{−3/2(1−1/𝑠)−(|𝛽|/2)}. (2.8)

One might note that we may sometimes use ‖𝑉^𝑖‖_𝑠≤‖Γ_𝑡‖_{3𝑠/(3+𝑠)}<𝑐𝑡^−1+3/2𝑠 instead of ‖𝑉^𝑖‖_𝑠≤‖Γ_𝑡‖_𝑠<𝑐𝑡^{−3/2(1−1/𝑠)} because of technical reason. By the definition of 𝑉^𝑖, both inequalities hold for any 𝑠≥3/2. We multiply (1.13) by ∇_𝑦×[𝜙(𝑦)∇_𝑦×𝜔^𝑖(𝑥−𝑦,𝑡−𝜏)] and integrate over Ω×(0,𝑡−𝜖), and then we have ₀^𝑡−𝜖_Ω𝜕𝐮𝜕𝜏−Δ_𝑦𝐮+𝐮_∞⋅∇_𝑦𝐮+(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰+(𝐮⋅∇)𝐮⋅∇_𝑦×𝜙𝑦∇_𝑦×𝜔^𝑖𝑥−𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏=−₀^𝑡−𝜖_Ω∇𝑝𝑦⋅∇_𝑦×𝜙𝑦∇_𝑦×𝜔^𝑖𝑥−𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏=0. (2.9)

We finally get the following integral representation for ∇×𝐯 (refer to [2, 3] for the detail): ∇_𝑥×𝐯_𝑖=∇_𝑥×𝐯₀∗Γ_𝑡−^𝑡_0Ω𝐮⋅𝜕_𝜏+Δ_𝑦+𝐮_∞⋅∇_𝑦𝑅₁−𝑥,𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏_𝑡0Ω𝐮⋅𝑅^𝑖₂−𝑥,𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏_𝑡0Ω𝐮⋅𝑉^𝑖𝑥−𝑦,𝑡−𝜏𝐮_∞⋅∇_𝑦𝜙𝑦−𝑑𝑦𝑑𝜏_𝑡0Ω𝑤_𝑘𝐮+𝑢_𝑘𝐰⋅𝜕_𝑦𝑘𝜙𝑦𝑉^𝑖𝑥−𝑦,𝑡−𝜏+𝜕_𝑦𝑘𝑅^𝑖₁−𝑥,𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏_{0𝑡−𝜖Ω}𝑢_𝑘𝐮⋅𝜕_𝑦𝑘𝜙𝑦𝑉^𝑖𝑥−𝑦,𝑡−𝜏+𝜕_𝑦𝑘𝑅^𝑖₁𝑥,𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏=𝐼+𝐼𝐼+𝐼𝐼𝐼+𝐼𝑉+𝑉+𝑉𝐼, (2.10) where 𝑅^𝑖₂𝑥,𝑦,𝑡−𝜏=2∇_𝑦𝜙𝑦⋅∇_𝑦𝑉^𝑖𝑥−𝑦,𝑡−𝜏+Δ_𝑦𝜙𝑦𝑉^𝑖𝑥−𝑦,𝑡−𝜏. (2.11)

Applying Young’s convolution and the Calderon-Zygmund inequalities, we obtain ‖𝐼‖_𝑝=(∇×𝐯₀)∗Γ_𝑡‖_𝑝≤‖𝐮₀𝜙∗Γ_𝑡‖_𝑝+‖∇𝑁∗[𝐮₀∇𝜙]∗Γ_𝑡𝑝≤‖𝐮₀𝜙‖_{3𝑟/(3−2𝑟)}‖Γ_𝑡‖_{3𝑝𝑟/(5𝑝𝑟+3𝑟−3𝑝)}+‖∇𝑁∗𝐮₀∇𝜙‖_{3𝑟/(3−2𝑟)}‖Γ_𝑡‖_{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, (2.12) if 𝜙𝐮₀∈𝐿^{3𝑟/(3−2𝑟)} and 𝐮₀∈𝐿^𝑟.

And 𝐼𝐼 is bounded by as follows:‖𝐼𝐼‖_𝑝≤𝑐^𝑡₀‖𝐮‖_𝑠1∇²𝜙_∞𝜕_𝑘∇×𝜔^𝑖_{𝑡−𝜏𝑠2}+‖𝐮‖_𝑠3∇Δ𝜙₃∇×𝜔^𝑖_{𝑡−𝜏𝑠4}+𝐮_∞|𝑥|⁻¹𝐮_𝑠5∇×𝜔^𝑖_{𝑡−𝜏𝑠6}𝑑𝜏=𝐼𝐼₁+𝐼𝐼₂+𝐼𝐼₃, (2.13) 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 𝐼𝐼₁≤𝐶𝐮_0𝑟^𝑡₀𝜏^{−3/2(1/𝑟−1/𝑠1)}(𝑡−𝜏)^{−3/2(1−1/𝑠2)}𝑑𝜏≤𝐶𝐮_0r𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.14) where 1/𝑟−1/𝑠₁<2/3 and 𝑠₂<3. Also, we obtain 𝐼𝐼₂≤𝐶𝐮_0r^𝑡₀𝜏^{−3/2(1/𝑟−1/𝑠3)}(𝑡−𝜏)^{−1+3/2𝑠4}𝑑𝜏≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.15) where 1/𝑟−1/𝑠₃<2/3. Finally, we get 𝐼𝐼₃≤𝐶^𝑡₀‖∇𝐮‖_𝑠5∇×𝜔^𝑖_{𝑡−𝜏𝑠6}𝑑𝜏≤𝐶𝐮_0𝑟^𝑡₀𝜏^{−3/2(1/𝑟−1/𝑠5)−1/2}(𝑡−𝜏)^{−1+3/2𝑠6}𝐮𝑑𝜏≤𝐶_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.16) where 1/𝑟−1/𝑠₅<1/3. Hence, for any 𝑡>0, we have ‖𝐼‖_𝑝+‖𝐼𝐼‖_𝑝≤𝐶𝑡^{−3/2(1/𝑟−1/𝑝)+1}𝜙𝐮_{03𝑟/(3−2𝑟)}+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}. (2.17)

Also, we obtain ‖𝐼𝐼𝐼‖_𝑝≤^𝑡₀‖𝐮𝜕_𝑗𝜙∗𝜕_𝑗𝑉^𝑖‖_𝑝+‖𝐮Δ𝜙∗𝑉^𝑖‖_𝑝≤𝑑𝜏_𝑡0‖𝐮‖_𝑠‖𝜕_𝑗𝑉^𝑖‖_{𝑝𝑠/(𝑝𝑠+𝑠−𝑝)}+‖𝐮‖_𝑠1‖∇²𝜙‖_∞‖𝜕_𝑘∇×𝜔^𝑖_𝑡−𝜏‖_𝑠2𝑑𝜏≤𝐶‖𝐮₀‖_𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.18) where 1/𝑠₁+1/𝑠₂=1+1/𝑝, 1/𝑟−1/𝑠₁<2/3 and 𝑠₂<3. In the above calculation, we used ‖𝜕𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−(3/2)(1−1/𝑞)} instead of ‖𝜕𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)−1/2} because of simplicity of calculations.

And we have ‖𝐼𝑉‖_𝑝≤𝑐𝐮_∞^𝑡₀‖𝐮‖_𝑠7∇𝜙_∞𝑉^𝑖_𝑠8𝑑𝜏≤𝐶𝐮_0𝑟𝑡^{−(3/2)(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.19) where 1/𝑠₇+1/𝑠₈=1+1/𝑝,1/𝑟−1/𝑠₇<2/3 and 𝑠₈<3.

Next, for 𝑉, we have 𝑉=−^𝑡_0Ω𝑤_𝑘𝐮+𝑢_𝑘𝐰⋅𝜕_𝑦𝑘𝜙𝑦𝑉^𝑖𝑦𝜕𝑥−𝑦,𝑡−𝜏+𝜙_𝑦𝑘𝑉^𝑖𝑥−𝑦,𝑡−𝜏+𝜕_𝑦𝑘𝑅^𝑖₁𝑥−𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏≤𝑉₁+𝑉₂+𝑉₃. (2.20)

We get ‖𝑉₁‖_𝑝≤𝑐^𝑡₀‖𝐮‖_𝑟1‖𝐰‖_𝟑∇𝜙_∞𝑉^𝑖_𝑟2𝑑𝜏≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.21) where 1/𝑟₁+1/𝑟₂=2/3+1/𝑝,1/𝑟−1/𝑟₁<2/3 and 𝑟₂<3. In the above calculation, we used ‖𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)+1/2} instead of ‖𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)} because of simplicity of calculations.

Since ‖|𝑥|𝐰‖_∞<𝐶 (see [21]), we have 𝑉_2𝑝≤𝑐^𝑡₀‖𝐮‖_𝑟3𝜙𝐰_∞∇𝑉^𝑖_𝑟4𝑑𝜏≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.22) where 1/𝑟₃+1/𝑟₄=1+1/𝑝, 1/𝑟−1/𝑟₃<2/3 and 𝑟₄<3. In the above calculation, we used ‖𝜕𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)}instead of ‖𝜕𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)−1/2} because of simplicity of calculations.

Next, for any 𝑡>0, we have 𝑉_3𝑝≤^𝑡₀‖𝐮‖_𝑟5‖𝐰‖_𝟑∇²𝜙_∞∇×𝜔^𝑖_𝑟6+‖𝐮‖_𝑟7‖𝐰‖_∞∇𝜙_∞𝜕_𝑘∇×𝜔^𝑖_𝑟8≤𝐮𝑑𝜏_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}, (2.23) 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 ‖𝑉‖_𝑝≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0. (2.24)

Consider 𝑉𝐼as follows:𝑉𝐼=−^𝑡_0Ω𝑢_𝑘𝐮⋅𝜕_𝑦𝑘𝜙𝑦𝑉^𝑖𝑦𝜕𝑥−𝑦,𝑡−𝜏+𝜙_𝑦𝑘𝑉^𝑖𝑥−𝑦,𝑡−𝜏+𝜕_𝑦𝑘𝑅^𝑖₁𝑥−𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏≤𝑉𝐼₁+𝑉𝐼₂+𝑉𝐼₃. (2.25)

We have, for any 𝑡>0, 𝑉𝐼_1𝑝≤^𝑡₀‖𝐮‖_𝑠1‖𝐮‖_𝑠2∇𝜙_∞𝑉^𝑖_𝑠3𝑑𝜏≤𝐶₂𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}, (2.26) 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 ‖𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)+1/2} instead of ‖𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)} because of technical reason.

Similar to 𝑉𝐼₁, we get 𝑉𝐼_3𝑝≤^𝑡₀‖𝐮‖_𝑟1‖𝐮‖_𝑟2∇²𝜙_∞∇×𝜔^𝑖_𝑟3+‖𝐮‖_𝑠1‖𝐮‖_𝑠2∇𝜙_∞𝜕_𝑘∇×𝜔^𝑖_𝑠3𝐮𝑑𝜏≤𝐶_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.27) 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 𝜙𝐮(𝑡)_𝑝≤‖∇×𝐯(𝑡)‖_𝑝+∇𝑁∗𝐮(𝑡)∇𝜙_𝑝≤‖∇×𝐯(𝑡)‖_𝑝+‖𝐮(𝑡)‖_{3𝑝/(3+2𝑝)}≤‖∇×𝐯(𝑡)‖_𝑝+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0. (2.28)

Since for 𝑡>0, ‖𝐮(𝑡)‖₉≤√𝜖𝑡^−1/3, from Shibata [21], we have 𝑉𝐼_2𝑝≤^𝑡₀𝜙𝐮(𝜏)_𝑝‖𝐮(𝜏)‖_𝟗∇𝑉^𝑖(𝑡−𝜏)_9/8𝑑𝜏≤𝜖^𝑡₀‖∇×𝐯(𝜏)‖_𝑝𝜏^−1/3(𝑡−𝜏)^−2/3𝑑𝜏+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}. (2.29)

Hence, we have ‖𝑉𝐼‖_𝑝≤𝜖^𝑡₀‖∇×𝐯(𝜏)‖_𝑝𝜏^−1/3(𝑡−𝜏)^−2/3𝑑𝜏+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}. (2.30)

Thus, by (2.17)–(2.19), (2.24), (2.28), and (2.30), for all 𝑡>0, we obtain ‖∇×𝐯(𝑡)‖_𝑝≤𝜖^𝑡₀‖∇×𝐯(𝜏)‖_𝑝𝜏^−1/3(𝑡−𝜏)^−2/3𝑑𝜏+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}. (2.31)

Now, we use the following lemma (refer to [25]).

Lemma 2.1. Let a function 𝑆(𝑡) satisfy the inequality, for some 𝛼<2/3, 𝑆(𝑡)≤𝑐𝑡^−𝛼+𝜀^𝑡₀𝑆(𝜏)𝜏^−1/3(𝑡−𝜏)^−2/3𝑑𝜏∀𝑡>0. (2.32) One also assumes that lim_𝑡→0+𝑡^−𝜀𝑡₀𝜏^−1/3𝑆(𝜏)𝑑𝜏=0. (2.33) Then, there is 𝜀₀ so that if 𝜀≤𝜀₀, then one has 𝑆(𝑡)≤𝑐𝑡^−𝛼 (2.34) for some 𝑐 independent of 𝑡.

Since ‖∇×𝐯(𝑡)‖_𝑝≤𝜙𝐮(𝑡)_𝑝+∇𝑁∗𝐮(𝑡)∇𝜙_𝑝≤𝑅‖𝐮(𝑡)‖_𝑝+‖𝐮(𝑡)‖_{3𝑝/(3+𝑝)}≤𝐶𝑅𝐮₀₃𝑡^{−1/2+3/2𝑝}+𝐶₂𝐮₀₃𝑡^{−3/2(1/3−1/𝑝)+1/2},∀𝑡>0, (2.35)

condition (2.33) satisfies lim_𝑡→0+𝑡^−𝜀𝑡₀𝜏^−1/3‖∇×𝐯(𝜏)‖_𝑝𝑑𝜏=lim_𝑡→0+𝑡^−𝜀𝑡₀𝜏^−1/3𝐶𝑅𝐮₀₃𝜏^{−1/2+3/2𝑝}+𝐶₂𝐮₀₃𝜏^3/2𝑝𝑑𝜏=lim_𝑡→0+𝐶𝑅𝐮₀₃𝜏^{−𝜖+1/3+3/2𝑝}+𝐶₂𝐮₀₃𝜏^{−𝜖+1+3/2𝑝}=0, (2.36) for 𝜖<(1/3+3/2𝑝).

So, by Lemma 2.1, we have ‖∇×𝐯(𝑡)‖_𝑝≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}. (2.37)

Hence, by (2.28), for any 𝑡>0, we have 𝜙𝐮(𝑡)_𝑝≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}, (2.38) and by taking 𝑅→∞, we complete the proof of Theorem 1.3.

2.2. Proof of Theorem 1.5

By using the results in previous section, for any 0<𝛼<1, we have small 𝛽>0 such that |𝑥|^𝛼𝐮(𝑡)_𝑠≤|𝑥|^𝛼𝐮^𝛼_{3/(𝛼−3𝛽)}𝐮^1−𝛼_{1/(1−𝛼−𝛽)}≤𝐶𝑡^{−3/2(1/𝑟−(𝛼−3𝛽)/3𝛼)+1𝛼}𝐶𝑡^{−3/2(1/𝑟−(1−𝛼−𝛽)/(1−𝛼))1−𝛼}≤𝐶𝑡^{−3/2(1/𝑟−1/𝑠)+𝛼}, (2.39) where 1−2𝛼/3−2𝛽=1/𝑠.

Now, in this section, we consider 𝜙(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 𝐼𝐼₃≤𝐶^𝑡₀|𝑥|^𝜎−2_{3/(2−𝜎)2}‖𝐮‖_𝑠1∇×𝜔^𝑖_{𝑡−𝜏𝑠2}𝐮𝑑𝜏≤𝐶_0𝑟^𝑡₀𝜏^{−3/2(1/𝑟−1/𝑠1)}(𝑡−𝜏)^{−1+3/2𝑠2}𝑑𝜏≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+3/2−(2−𝜎)2/2}, (2.40) where 1/𝑠₁+1/𝑠₂=1+1/𝑝 and 1/𝑟−1/𝑠₁<2/3. Also, for ‖𝐼𝐼𝐼‖_𝑝, we obtain ‖𝐼𝐼𝐼‖_𝑝≤^𝑡₀𝐮𝜕_𝑗𝜙∗𝜕_𝑗𝑉^𝑖_𝑝+𝐮Δ𝜙∗𝑉^𝑖_𝑝≤𝑑𝜏_𝑡0|𝑥|^𝜎−1𝐮_𝑠𝜕_𝑗𝑉^𝑖_{𝑝𝑠/(𝑝𝑠+𝑠−𝑝)}+‖𝐮‖_𝑠1∇²𝜙_∞𝜕_𝑘∇×𝜔^𝑖_{𝑡−𝜏𝑠2}𝐮𝑑𝜏≤𝐶_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎−1/2}+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.41) where 1/𝑠₁+1/𝑠₂=1+1/𝑝, 1/𝑟−1/𝑠₁<2/3 and 𝑠₂<3.

Next, we have ‖𝐼𝑉‖_𝑝≤𝑐𝐮_∞^𝑡₀|𝑥|^𝜎−1𝐮_𝑠1𝑉^𝑖_𝑠2𝑑𝜏≤𝐶𝐮_∞𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎},∀𝑡>0, (2.42) where 1/𝑠₁+1/𝑠₂=1+1/𝑝, 𝑠₂<3, and 1/𝑟−1/𝑠₁<2𝜎/3. Also, since ‖|𝑥|𝐰‖_∞<𝐶, we get 𝑉_1𝑝≤𝑐^𝑡₀‖𝐮‖_𝑟1|𝑥|^𝜎−1𝐰_∞𝑉^𝑖_𝑟2𝑑𝜏≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1},∀𝑡>0, (2.43) where 1/𝑟₁+1/𝑟₂=1+1/𝑝, 1/𝑟−1/𝑟₁<2/3, and 𝑟₂<3.

And we obtain 𝑉_2𝑝≤𝑐^𝑡₀|𝑥|^𝜎−1𝐮_𝑟3‖|𝑥|𝐰‖_∞∇𝑉^𝑖_𝑟4𝑑𝜏≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎−1/2},∀𝑡>0, (2.44) where 1/𝑟₃+1/𝑟₄=1+1/𝑝, 1/𝑟−1/𝑟₃<2𝜎/3, and 𝑟₄<3/2.

Next, for any 𝑡>0, we have 𝑉_3𝑝≤^𝑡₀‖𝐮‖_𝑟5‖𝐰‖_𝟑∇²𝜙_∞∇×𝜔^𝑖_𝑟6+|𝑥|^𝜎−1𝐮_𝑟7‖𝐰‖_𝟑𝜕_𝑘∇×𝜔^𝑖_𝑟8𝐮𝑑𝜏≤𝐶_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎−1/2}, (2.45) 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 ‖𝑉‖_𝑝≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1}+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎−1/2},∀𝑡>0. (2.46)

Consider 𝑉𝐼  as follows:𝑉𝐼=−^𝑡_0Ω𝑢_𝑘𝐮⋅𝜕_𝑦𝑘𝜙𝑦𝑉^𝑖𝑦𝜕𝑥−𝑦,𝑡−𝜏+𝜙_𝑦𝑘𝑉^𝑖𝑥−𝑦,𝑡−𝜏+𝜕_𝑦𝑘𝑅^𝑖₁𝑥−𝑦,𝑡−𝜏𝑑𝑦𝑑𝜏≤𝑉𝐼₁+𝑉𝐼₂+𝑉𝐼₃. (2.47)

We have, for any 𝑡>0, 𝑉𝐼_1𝑝≤^𝑡₀|𝑥|^𝜎−1𝐮_𝑠1‖𝐮‖_𝑠2𝑉^𝑖_𝑠3𝑑𝜏≤𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎−1/2}, (2.48) 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 to 𝑉𝐼₁, we get 𝑉𝐼_3𝑝≤^𝑡₀‖𝐮‖_𝑟1‖𝐮‖_𝑟2∇²𝜙_∞∇×𝜔^𝑖_𝑟3+|𝑥|^𝜎−1𝐮_𝑠1‖𝐮‖_𝑠2𝜕_𝑘∇×𝜔^𝑖_𝑠3𝐮𝑑𝜏≤𝐶_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+1/2}+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎},∀𝑡>0, (2.49) 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 ‖𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)+1/2} instead of ‖𝑉^𝑖(𝑡)‖_𝑞≤𝑡^{−3/2(1−1/𝑞)} because of technical reason. Now, we have 𝑉𝐼_2𝑝≤^𝑡₀‖|𝑥|𝐮(𝜏)‖_𝑠1|𝑥|^𝜎−1𝐮(𝜏)_𝑠2∇𝑉^𝑖(𝑡−𝜏)_𝑠3𝐮𝑑𝜏≤𝐶_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎+1/2−3/2𝑟1}, (2.50) 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 𝜙𝐮(𝑡)_𝑝≤‖∇×𝐯(𝑡)‖_𝑝+∇𝑁∗𝐮(𝑡)∇𝜙_𝑝≤‖∇×𝐯(𝑡)‖_𝑝+|𝑥|^𝜎−1𝐮(𝑡)_{3𝑝/(3+𝑝)}≤‖∇×𝐯(𝑡)‖_𝑝+𝐶𝐮_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎−1/2}𝐮,∀𝑡>0≤𝐶_0𝑟𝑡^{−3/2(1/𝑟−1/𝑝)+𝜎},∀𝑡≥1, (2.51) which completes the proof.

Acknowledgments

The author would like to express her appreciation to Professors Hyeong-Ohk Bae and Bum Ja Jin for valuable comments. The author would like to thank the referee for helpful comments. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0023386).