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

and Applied Analysis 3 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. Let σ0 > 0 and assume that |u∞| ≤ σ0. Let 1 ≤ r ≤ q ≤ ∞. Then, ‖T t a‖Lq Ω ≤ Cr,q,σ0t 1/r−1/q ‖a‖Lr Ω , t > 0, 1.8 where r, q / 1, 1 and ∞,∞ , ‖∇T t a‖Lq Ω ≤ Cr,q,σ0t 1/r−1/q −1/2‖a‖Lr 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 Ω with ∇ · u0 0, then for any t > 0, ‖|x|u t ‖p ≤ Ct−3/2 1/r−1/p ‖|x|u0‖Lr Ω C|u∞|t−3/2 1/r−1/p ‖u0‖Lr Ω , 1.10 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 , andw satisfies the following equations: −Δw u∞ · ∇ w w · ∇ w ∇p2 f, ∇ ·w 0, w|∂Ω −u∞, lim |x|→∞ x 0. 1.11 For suitable f, Shibata 21 proved that for any given 0 < δ < 1/4 there exists such that if 0 < |u∞| ≤ , then one has ‖w‖L3/ 1 δ1 Ω ‖w‖L3/ 1−δ2 Ω ‖∇w‖L3/ 2 δ1 Ω ‖∇w‖L3/ 2−δ2 Ω ≤ C|u∞|, 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: ∂ ∂t u −Δu u∞ · ∇ u u · ∇ w w · ∇ u u · ∇ u ∇p 0, ∇ · u t, x 0, u x, 0 u0 x for x ∈ Ω, u x, t 0 for x ∈ ∂Ω, lim |x|→∞ u x, t 0. 1.13 4 Abstract and Applied Analysis 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 , EnomotoShibata 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 p, q, r such that if 0 < |u∞| ≤ , and ‖u0‖L3 Ω < , then a unique solution u x, t of 1.13 has ‖u t ‖Lp Ω ≤ C t−3/2 1/r−1/p ‖u0‖r for 1 < r < p ≤ ∞, t > 0, ‖∇u t ‖Lq Ω ≤ C t−3/2 1/r−1/q − 1/2 ‖u0‖r for 1 < r < q ≤ 3, t > 0, 1.14 where u0 ∈ L3 Ω ∩ L 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 p, r such that if 0 < |u∞| ≤ , ‖u0‖L3 Ω < , |x|u0 ∈ L3r/ 3−2r Ω , and ∇ · u0 0, then the solution u x, t of 1.13 satisfies ‖|x|u t ‖Lp Ω ≤ C t−3/2 1/r−1/p ‖u0‖r , ∀t > 0, 1.15 where u0 ∈ L3 Ω ∩ L Ω . Remark 1.4. In Theorem 1.3, the assumption |x|u0 ∈ L3r/ 3−2r Ω is for simple calculations. We also can obtain a similar result where |x|u0 ∈ L 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 p, r such that if 0 < |u∞| ≤ , ‖u0‖L3 Ω < , |x|u0 ∈ L3r/ 3−2r Ω , and ∇ · u0 0, then the solution u x, t of 1.13 satisfies ‖|x|σu t ‖Lp Ω ≤ C t−3/2 1/r−1/p ‖u0‖r , ∀t ≥ 1, 1.16 where u0 ∈ L3 Ω ∩ L Ω . Remark 1.6. For the exterior Navier-Stokes flows with u∞ 0, temporal decay rate with weight function |x| 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| 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 Abstract and Applied Analysis 5 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 .and Applied Analysis 5 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 NavierStokes equations with nonzero far-field velocity. We first consider |x| for a weight function and then |x| 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 φR 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 v x ≡ ∫ R3 N ( x − 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 ∫ Ω N ( x − 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 Γt x e ∇ ∂ ∂xi N ∗ Γt x , 2.4 where Γt x Γ x, t 4πt −3/2e−|x−tu∞| /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 t x ω i x, t N ∗ Γt x e, i 1, 2, 3, where e is the standard unit vector of which the ith term is 1. Then, we have ∇ × ∇ ×ωi −Δωi ∇divωi V . 2.5 6 Abstract and Applied Analysis Hence, we have the identity ∇y × [ φ ( y ∇y ×ωi ( x − y, t − τ ] φ ( y ) V i ( x − y, t − τ Ri1 ( x, y, t − τ, 2.6 where Ri1 ( x, y, t − τ ∇φy × ∇y ×ωi ( x − y, t − τ. 2.7 From straightforward calculations we have that for 1 ≤ s ≤ ∞, ∥ ∥ ∥∂Γt−τ ∥ ∥ ∥ s ≤ c t − τ −3/2 1−1/s − |β|/2 . 2.8 One might note that we may sometimes use ‖V ‖s ≤ ‖Γt‖3s/ 3 s < ct−1 3/2s instead of ‖V ‖s ≤ ‖Γt‖s < ct−3/2 1−1/s because of technical reason. By the definition of V , both inequalities hold for any s ≥ 3/2. We multiply 1.13 by ∇y × φ y ∇y × ω x − y, t − τ and integrate over Ω × 0, t − , and then we have ∫ t− 0 ∫ Ω ( ∂u ∂τ −Δyu ( u∞ · ∇y ) u w · ∇ u u · ∇ w u · ∇ u ) · ∇y × [ φ ( y ∇y ×ωi ( x − y, t − τ ] dy dτ


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: where Ω is an exterior domain in R 3 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 1.2 Consider the following linear equations of 1.2 : 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: where 1 < p < ∞, The Helmholtz decomposition of L p Ω n was proved by Fujiwara-Morimoto 16 , Miyakawa where the domain of O u ∞ is given by 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.
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|u 0 , u 0 ∈ L r Ω with ∇ • u 0 0, then for any t > 0, 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: For suitable f, Shibata 21 proved that for any given 0 < δ < 1/4 there exists such that if 0 < |u ∞ | ≤ , then one has for small δ 1 , δ 2 , where C is independent of u ∞ .By setting u v − w and p p 1 − p 2 for v, p 1 , w, p 2 in 1.2 and 1.11 , we have the following equations in Ω: Here, in fact, the initial data should be u 0 −u ∞ −w, but for our convenience we denote by u 0 for u 0 − u ∞ − w if there is no confusion.Heywood 14 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 p, r such that if where Remark 1.4.In Theorem 1.3, the assumption |x|u 0 ∈ L 3r/ 3−2r Ω is for simple calculations.
We also can obtain a similar result where |x|u 0 ∈ L r Ω .For the proof we have to consider delay solution u t = u t t 0 .Then we can follow the method in Bae and Roh 4 . where 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 .

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.

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 where where 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 where refer to 15, 28 .In fact, Γ is a translation in the direction of x by tu ∞ of the heat kernel K x, t , where e i is the standard unit vector of which the ith term is 1.Then, we have 2.5 Hence, we have the identity where From straightforward calculations we have that for 1 ≤ s ≤ ∞, One might note that we may sometimes use . 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 − , and then we have ∇p y • ∇ y × φ y ∇ y × ω i x − y, t − τ dy dτ 0.

2.9
We finally get the following integral representation for ∇ × v refer to 2, 3 for the detail : where
Next, for any t > 0, we have
Hence, we have Consider V Ias follows:

2.29
Hence, we have

2.31
Now, we use the following lemma refer to 25 .

2.32
One also assumes that Then, there is ε 0 so that if ε ≤ ε 0 , then one has for some c independent of t. Since

2.37
Hence, by 2.28 , for any t > 0, we have and by taking R → ∞, we complete the proof of Theorem 1.3.

Proof of Theorem 1.5
By using the results in previous section, for any 0 < α < 1, we have small β > 0 such that

2.46
Consider V I as follows: