𝐿^𝑟-𝐿^𝑝 Stability of the Incompressible Flows with Nonzero Far-Field Velocity

Abstract

We consider the stability of stationary solutions 𝐰 for the exterior Navier-Stokes flows with a nonzero constant velocity 𝐮_∞ at infinity. For 𝐮_∞=0 with nonzero stationary solution 𝐰, Chen (1993), Kozono and Ogawa (1994), and Borchers and Miyakawa (1995) have studied the temporal stability in 𝐿^𝑝 spaces for 1<𝑝 and obtained good stability decay rates. For the spatial direction, we recently obtained some results. For 𝐮_∞≠0, Heywood (1970, 1972) and Masuda (1975) have studied the temporal stability in 𝐿² space. Shibata (1999) and Enomoto and Shibata (2005) have studied the temporal stability in 𝐿^𝑝 spaces for 𝑝≥3. Then, Bae and Roh recently improved Enomoto and Shibata's results in some sense. In this paper, we improve Bae and Roh's result in the spaces 𝐿^𝑝 for 𝑝>1 and obtain 𝐿^𝑟-𝐿^𝑝 stability as Kozono and Ogawa and Borchers and Miyakawa obtained for 𝐮_∞=0.

1. Introduction

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=0𝐮|_𝑡=0inΩ×(0,∞),=𝐮₀,𝐮|_𝜕Ω=0,lim_|𝑥|→∞𝐮(𝑥,𝑡)=𝐮_∞,(1.1) where Ω is an exterior domain in 𝑅^𝑛 with a smooth boundary 𝜕Ω, and 𝐮_∞ denotes a given constant vector describing the velocity of the fluid at infinity. In this paper, we consider a nonzero constant 𝐮_∞. The physical model of the exterior Navier-Stokes equations with a nonzero constant 𝐮_∞ can be considered as the motion of water in the sea when a boat is moving with the speed −𝐮_∞, while the one with zero constant 𝐮_∞ can be considered when a boat is stopped. There are few known results for the case 𝐮_∞≠0, while, with 𝐮_∞=0, many results were obtained for the temporal decay and weighted estimates of solutions of (1.1) (refer [1–12]).

Now, we set 𝐮=𝐮_∞+𝐯 in (1.1) and have𝜕𝜕𝑡𝐯−Δ𝐯+𝐮^∞𝐯|_𝑡=0⋅∇𝐯+(𝐯⋅∇)𝐯+∇𝑝₁=𝐟,∇⋅𝐯=0inΩ×(0,∞),=𝐮₀−𝐮_∞,𝐯|_𝜕Ω=−𝐮_∞,lim_|𝑥|→∞𝐯(𝑥,𝑡)=0.(1.2)

Consider the following linear problem:𝜕𝜕𝑡𝐮−Δ𝐮+𝐮^∞𝐮|_𝑡=0⋅∇𝐮+∇𝑝=0,∇⋅𝐮=0inΩ×(0,∞),=𝐮₀,𝐮|_𝜕Ω=0,lim_|𝑥|→∞𝐮(𝑥,𝑡)=0,(1.3) which is referred to as the Oseen equations; see [13].

In order to formulate the problem (1.3), Enomoto and Shibata [14] used the Helmholtz decomposition: 𝐿_𝑝(Ω)^𝑛=𝐽_𝑝(Ω)⊕𝐺_𝑝(Ω),(1.4) where 1<𝑝<∞, 𝐿_𝑝(𝐶Ω)^𝑛=𝑢=𝑢₁,…,𝑢_𝑛∶𝑢_𝑗∈𝐿_𝑝(Ω),𝑗=1,…,𝑛_∞0,𝜎,𝐽_𝑝=𝑢=𝑢₁,…,𝑢_𝑛∈𝐶^∞0(Ω)^𝑛∶∇⋅𝑢=0inΩ,𝐺_𝑝(Ω)=thecompletionof𝐶^∞0,𝜎(Ω),in𝐿_𝑝(Ω)^𝑛,(Ω)=∇𝜋∈𝐿_𝑝(Ω)^𝑛∶𝜋∈𝐿_𝑝,locΩ.(1.5) The Helmholtz decomposition of 𝐿_𝑝(Ω)^𝑛 was proved by Fujiwara and Morimoto [15], Miyakawa [16], and Simader and Sohr [17]. 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_𝑝𝒪_𝐮∞=𝑢∈𝐽_𝑝(Ω)∩𝑊^2𝑝(Ω)^𝑛∶𝑢|_𝜕Ω=0.(1.7) Then, Enomoto and Shibata [14] proved that 𝒪_𝐮∞ generates an analytic semigroup {𝑇(𝑡)}_𝑡≥0 which is called the Oseen semigroup (one can also refer to [16, 18]) 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).

The main purpose of this paper is to discuss the temporal stability of stationary solution 𝐰 of the nonlinear Navier-Stokes equation (1.2). One can note that 𝐰 satisfies the following equations:−Δ𝐰+𝐮_∞𝐰⋅|∇_𝜕Ω𝐰+(𝐰⋅∇)𝐰+∇𝑝₂=𝐟,∇⋅𝐰=0,=−𝐮_∞,lim_|𝑥|→∞𝐰(𝑥)=0.(1.10)

For suitable 𝐟, Shibata [19] proved that, for any given 0<𝛿<1/4, there exists 𝜖 such that if 0<|𝐮_∞|≤𝜖, then one has‖𝐰‖_𝑝,2+||‖𝐰‖||_𝛿+‖‖𝑝₂‖‖_𝑝,1≤||𝐮_∞||^𝛽,(1.11) where ||‖𝐮‖||_𝛿‖𝐮‖_𝑝,𝑚=‖𝜕^𝑚𝐮‖_𝐿^𝑝_(Ω),=sup_𝑥∈Ω(1+|𝑥|)𝑠1+𝑠_𝐮∞(𝑥)^𝛿||𝐮(𝑥)||+sup_𝑥∈Ω(1+|𝑥|)^3/21+𝑠_𝐮∞(𝑥)^1/2+𝛿||∇𝐮(𝑥)_𝐮∞||,(𝑥)=|𝑥|−𝑥^𝑇⋅𝐮^∞||𝐮_∞||𝛿<𝛽<1−𝛿.(1.12) Throughout this paper, we assume that 𝐟 satisfies the assumption in Shibata [19]. Now, we consider the polar coordinate system 𝑦₁=𝑟cos𝜃,𝑦₂=𝑟sin𝜃cos𝜙,𝑦₃=𝑟sin𝜃sin𝜙,(1.13) for 0≤𝜃≤𝜋, 0≤𝜙≤2𝜋, and 0≤𝑟<∞. Let 𝑆 be an orthogonal matrix such that 𝑆𝐮_∞ = |𝐮_∞|(1,0,0)^𝑇 and put 𝑠(𝑦) = |𝑦|−𝑦₁. By a change of variable 𝑦=𝑆𝑥, |𝑥|=||𝑦||=𝑟,𝑠_𝐮∞(𝑥)=𝑠(𝑦)=𝑟(1−cos𝜃).(1.14) See Shibata [19] for the detail. Now, by using the above change of variable, we can see easily that 𝐰 satisfies‖𝐰‖_𝐿^1)3/(1+𝛿_(Ω)+‖𝐰‖_𝐿^{2)3/(1−𝛿}_(Ω)+‖∇𝐰‖_𝐿^1)3/(2+𝛿_(Ω)+‖∇𝐰‖_𝐿^{2)3/(2−𝛿}_(Ω)≤𝐶||𝐮_∞||^1/2,(1.15) for small 𝛿₁, 𝛿₂, where 𝐶 is independent on 𝐮_∞.

One can also refer to [20] for more general cases of the existence and regularity of stationary Navier-Stokes equations.

For the stability of stationary solutions 𝐰, by setting 𝐮=𝐯−𝐰 and 𝑝=𝑝₁−𝑝₂ for 𝐯, 𝑝₁, 𝐰, 𝑝₂ in (1.2) and (1.10), we have the following equations in Ω:𝜕𝜕𝑡𝐮−Δ𝐮+𝐮^∞⋅∇𝐮(𝐮𝑥,+0)(𝐮=⋅∇𝐮∇₀⋅)𝐮𝐰=+0(,𝐰⋅∇)𝐮+(𝐮⋅∇)𝐮+∇𝑝=0,𝐮(𝑥,𝑡)=0(𝑥)for𝑥∈Ω,for𝑥∈𝜕Ω,lim_|𝑥|→∞𝐮(𝑥,𝑡)=0.(1.16) 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.

First, Heywood [21, 22] and Masuda [23] have studied the temporal stability in 𝐿² space. Shibata [19] proved that there exists small 𝜖 such that if 0<|𝐮_∞|≤𝜖 and ‖𝐮₀‖₃≤𝜖, then a unique solution 𝐮(𝑥,𝑡) of (1.16) has the following properties: for any 3<𝑝<∞, lim_𝑡→0+[𝐮]_3,0,𝑡+[𝐮]_{𝑝,𝜇(𝑝),𝑡}+[∇𝐮]_3,1/2,𝑡≤√𝜖,‖‖𝐮(𝑡)−𝐮₀‖‖₃+[𝐮]_{𝑝,𝜇(𝑝),𝑡}+[∇𝐮]_3,1/2,𝑡=0,(1.17) where [𝐳]_{𝑝,𝜌,𝑡}=sup_0<𝑠<𝑡𝑠^𝜌‖𝐳(𝑠,⋅)‖_𝑝,𝜇(𝑝)=12−23𝑝.(1.18)

After that, Enomoto and Shibata [14] considered the stability for arbitrary 𝐮_∞ by deleting the smallness condition of |𝐮_∞|. But in this case, all constants in their results depend on 𝜎₀ when |𝐮_∞|≤𝜎₀. Also, they assumed the existence of stationary solution 𝐰 with ‖𝐰‖_𝐿^1)3/(1+𝛿_(Ω)+‖𝐰‖_𝐿^{2)3/(1−𝛿}_(Ω)+‖∇𝐰‖_𝐿^1)3/(2+𝛿_(Ω)+‖∇𝐰‖_𝐿^{2)3/(2−𝛿}_(Ω)≤𝛼,(1.19) for small 𝛿₁, 𝛿₂ and 𝛼. Then, as a result, they proved (1.16) has a unique solution 𝐮(𝑥,𝑡) with lim‖_𝑡→0+𝐮(‖‖𝑡𝐮)‖(𝑡)−_𝐮(𝑡)𝐮₀‖‖₃+𝑡^1/2‖𝐮(𝑡)‖_𝐿^∞+‖∇𝐮(𝑡)‖_𝐿³=0,=𝑜‖𝑡∇𝐮(𝑡)‖^{−((1/2)−(3/2𝑝))}₃,forany3≤𝑝≤∞,=𝑜𝑡^−1/2(1.20) as 𝑡→∞ when 𝐮₀ is small enough in the space 𝐿³(Ω).

Also, Bae and Roh [24] improved Enomoto-Shibata's result in some sense. But their result is limited in the space 𝐿^𝑝 for 3/2<𝑝, while we consider all 1<𝑝. Moreover, their result depends on 𝑠 and 𝑟, while ours only depends on 𝑟, where 𝐰∈𝐿^𝑠 and 𝐮₀∈𝐿^𝑟. Also, their optimal decay rate is 2/3+𝛿, while ours is 3/2+𝛿.

Now, in the next main Theorem, we settle the temporal stability of stationary solutions for the Navier-Stokes equations with a nonzero constant vector at infinity. The idea of the proof is initiated by Kato [25] for 𝐰=0 and a very well-known method. Also, for 𝐰≠0 with 𝐮_∞=0, Kozono and Ogawa [12] also used similar method.

Theorem 1.2. There exists small 𝜖(𝑝,𝑞,𝑟) such that if0<|𝐮_∞|≤𝜖 and ‖𝐮₀‖_𝐿³_(Ω)<𝜖, then a unique solution 𝐮(𝑥,𝑡) of (1.16) has the following properties: ‖∇‖𝐮𝐮((𝑡𝑡))‖‖_𝐿^𝑝_{(Ω)𝐿𝑞(Ω)}≤𝐶_𝜖𝑡^{−3/2(1/𝑟−1/𝑝)}‖‖𝐮₀‖‖_𝑟for1<𝑟<𝑝≤∞,𝑡>0,≤𝐶_𝜖𝑡^{−3/2(1/𝑟−1/𝑞)−1/2}‖‖𝐮₀‖‖_𝑟for1<𝑟<𝑞≤3,𝑡>0,(1.21) where 𝐮₀∈𝐿³(Ω)∩𝐿^𝑟(Ω).

2. Proof of Main Theorem

First, we consider the following linear problem:𝜕𝜕𝑡𝐮−Δ𝐮+𝐮^∞𝐮|_𝑡=0⋅∇𝐮∇+⋅(𝐰𝐮⋅=∇0),𝐮+(𝐮⋅∇)𝐰+∇𝑝=0,=𝐮₀,𝐮|_𝜕Ω=0,lim_|𝑥|→∞𝐮(𝑥,𝑡)=0.(2.1)

By applying Helmholtz-Leray projection 𝑃 and setting ℒ𝐮==𝒪𝑃_𝐮∞−Δ𝐮+𝐮_∞⋅∇𝐮+(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰𝐮+𝑃[(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰],(2.2) we have 𝐮_𝑡+ℒ𝐮=0,for𝑡>0,𝐮(0)=𝐮₀.(2.3) And we note that the domain of ℒ is 𝒟_𝑝(ℒ)=𝒟_𝑝𝒪_𝐮∞=𝑢∈𝐽_𝑝(Ω)∩𝑊^2𝑝(Ω)^𝑛|𝑢|_𝜕Ω=0.(2.4)

Let 𝑆(𝑡) be a semigroup generated by the linear operator ℒ, then, by Duharmel's Principle, a solution 𝐮(𝑥,𝑡) of (2.1) can be written as in the following integral form,𝐮(𝑥,𝑡)=𝑆(𝑡)𝐮₀=𝑇(𝑡)𝐮₀+^𝑡0𝑇(𝑡−𝜏)𝑃[(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰]𝑑𝜏,(2.5) where 𝑇(𝑡) is an analytic semigroup generated by the Oseen operator 𝒪_𝐮∞.

Lemma 2.1. Let 𝐮₀∈𝐿³(Ω)∩𝐿^𝑟(Ω) for 1<𝑟<3, then there exists a small 𝜖(𝑝,𝑞,𝑟) such that if |𝐮_∞|≤𝜖 and ‖𝐮₀‖_𝐿³_(Ω)<𝜖, then a solution 𝐮(𝑥,𝑡) represented by (2.5) satisfies 1<𝑝≤∞ with 1/𝑟−1/𝑝<2/3, ‖𝐮(𝑡)‖_𝐿^𝑝_(Ω)=‖‖𝑆(𝑡)𝐮₀‖‖_𝐿𝑝(Ω)≤𝐶_𝜖𝑡^{−3/2(1/𝑟−1/𝑝)}‖‖𝐮₀‖‖_𝐿𝑟(Ω),𝑡>0,(2.6) and for 1<𝑞≤3 with 1/𝑟−1/𝑞<1/3, ‖∇𝐮(𝑡)‖_𝐿^𝑞_(Ω)=‖‖∇𝑆(𝑡)𝐮₀‖‖_𝐿𝑞(Ω)≤𝐶_𝜖𝑡^{−3/2(1/𝑟−1/𝑞)−1/2}‖‖𝐮₀‖‖_𝐿𝑟(Ω),𝑡>0.(2.7)

Proof. Before we prove Lemma 2.1 note from (1.15) that we have ‖𝐰‖_𝐿^1)3/(1+𝛿_(Ω)+‖𝐰‖_𝐿^{2)3/(1−𝛿}_(Ω)+‖∇𝐰‖_𝐿^1)3/(2+𝛿_(Ω)+‖∇𝐰‖_𝐿^{2)3/(2−𝛿}_(Ω)≤𝐶||𝐮_∞||^1/2,(2.8) for small 𝛿₁,𝛿₂>0. In fact, by straight calculations, we can choose any 𝛿₁,𝛿₂≤3/16.
Step 1. Let 3<𝑝≤∞ with 1/3≤1/𝑟−1/𝑝<2/3 and 3/2<𝑞≤3 with 1/𝑟−1/𝑞<1/3. We consider the following iteration method to obtain our estimates: 𝐮_𝑘+1(𝑡)=𝑇(𝑡)𝐮₀+^𝑡0𝑇(𝑡−𝜏)𝑃(𝐰⋅∇)𝐮_𝑘+𝐮_𝑘⋅∇𝐰𝑑𝜏.(2.9) We let 1/𝑞−1/𝑝=1/3 and 𝑀^𝑘𝑝=sup_{𝑡∈[0,∞)}𝑡^{𝑛/2(1/𝑟−1/𝑝)}‖‖𝑢^𝑘(𝑡)‖‖_𝑝,𝑁^𝑘𝑞=sup_{𝑡∈(0,∞)}𝑡^{𝑛/2(1/𝑟−1/𝑞)+1/2}‖‖∇𝑢^𝑘(𝑡)‖‖_𝑞.(2.10) If 𝑡≥2, then by Proposition 1.1, for small 𝛿₁,𝛿₂>0, we have ^𝑡0≤‖‖𝑇𝐶(𝑡−𝜏_0𝑡−1)𝑃(𝐰⋅∇)𝐮_𝑘+𝐮_𝑘⋅∇𝐰‖‖_𝑝𝑑𝜏+(𝑡−𝜏)^{−𝑛/2(1/𝑟1−1/𝑝)}‖‖(𝐰⋅∇)𝐮_𝑘‖‖_𝑟1𝑑𝜏+^𝑡𝑡−1(𝑡−𝜏)^{−𝑛/2(1/𝑟2−1/𝑝)}‖‖(𝐰⋅∇)𝐮_𝑘‖‖_𝑟2𝑑𝜏_0𝑡−1≤𝐶||𝐮_∞(𝑡−𝜏)^{−𝑛/2(1/𝑟1−1/𝑝)}‖‖(𝐮_𝑘⋅∇)𝐰‖‖_𝑟1𝑑𝜏+^𝑡𝑡−1(𝑡−𝜏)^{−𝑛/2(1/𝑟2−1/𝑝)}‖‖(𝐮_𝑘⋅∇)𝐰‖‖_𝑟2𝑑𝜏||𝑁^𝑘𝑞^0𝑡−1+𝐶||𝐮_∞(𝑡−𝜏)^{−1−𝛿1/2}𝜏^{−3/2(1/𝑟−1/𝑞)−1/2}𝑑𝜏+^𝑡𝑡−1(𝑡−𝜏)^−1+𝛿2/2𝜏^{−3/2(1/𝑟−1/𝑞)−1/2}𝑑𝜏||𝑀^𝑘𝑝^0𝑡−1≤𝐶||𝐮_∞(𝑡−𝜏)^{−1−𝛿1/2}𝜏^{−3/2(1/𝑟−1/𝑝)}𝑑𝜏+^𝑡𝑡−1(𝑡−𝜏)^−1+𝛿2/2𝜏^{−3/2(1/𝑟−1/𝑝)}𝑑𝜏||𝑀^𝑘𝑝+𝑁^𝑘𝑞𝑡^{−3/2(1/𝑟−1/𝑝)},(2.11) where 1/𝑟₁=1/𝑝+2/3+𝛿₁/3 and 1/𝑟₂=1/𝑝+2/3−𝛿₂/3. If 0<𝑡<2, then we have ^𝑡0≤‖‖𝑇𝐶(𝑡−_𝑡0𝜏)𝑃(𝐰⋅∇)𝐮_𝑘+𝐮_𝑘⋅∇𝐰‖‖_𝑝𝑑𝜏≤𝐶||𝐮_∞(𝑡−𝜏)^{−𝑛/2(1/𝑟3−1/𝑝)}‖‖(𝐰⋅∇)𝐮_𝑘‖‖_𝑟3𝑑𝜏+^𝑡0(𝑡−𝜏)^{−𝑛/2(1/𝑟3−1/𝑝)}‖‖𝐮_𝑘⋅∇𝐰‖‖_𝑟3𝑑𝜏||𝑀^𝑘𝑝+𝑁^𝑘𝑞𝑡^{−3/2(1/𝑟−1/𝑝)},(2.12) where 1/𝑟₃=1/𝑝+2/3−𝛿₂/3. So, we obtain ‖‖𝐮_𝑘+1(𝑡)‖‖_𝑝≤𝐶𝑡^{−𝑛/2(1/𝑟−1/𝑝)}‖‖𝐮₀‖‖_𝑟+𝐶||𝐮_∞||𝑡^{−3/2(1/𝑟−1/𝑝)}𝑀^𝑘𝑝+𝑁^𝑘𝑞,∀𝑡>0,(2.13) which implies 𝑀^𝑝𝑘+1≤𝐶‖‖𝐮₀‖‖_𝑟+𝐶||𝐮_∞||𝑀^𝑘𝑝+𝑁^𝑘𝑞.(2.14) Similarly, we obtain for 𝑡≥2, ‖‖∇≤𝐮𝐶_𝑘+1𝑡(𝑡)‖‖_𝑞≤‖‖∇𝑇_{−𝑛/2(1/𝑟−1/𝑞)−1/2}(𝑡)𝐮₀‖‖_𝑞+^𝑡0‖‖∇𝑇(𝑡−𝜏)𝑃(𝐰⋅∇)𝐮_𝑘+𝐮_𝑘⋅∇𝐰‖‖_𝑞𝑑𝜏‖‖𝐮₀‖‖_𝑟+𝐶||𝐮_∞||𝑁^𝑘𝑞^0𝑡−1+𝐶||𝐮_∞(𝑡−𝜏)^{−1−𝛿1/2}𝜏^{−3/2(1/𝑟−1/𝑞)−1/2}𝑑𝜏||𝑁^𝑘𝑞^𝑡𝑡−1+𝐶||𝐮_∞(𝑡−𝜏)^−1+𝛿2/3𝜏^{−3/2(1/𝑟−1/𝑞)−1/2}𝑑𝜏||𝑀^𝑘𝑝^𝑡0≤𝐶𝑡_{−𝑛/2(1/𝑟−1/𝑞)−1/2}(𝑡−𝜏)^{−𝑛/2(1/𝑟4−1/𝑞)−1/2}𝜏^{−3/2(1/𝑟−1/𝑝)}𝑑𝜏‖‖𝐮₀‖‖_𝑟+𝐶||𝐮_∞||𝑡^{−3/2(1/𝑟−1/𝑞)−1/2}𝑀^𝑘𝑝+𝑁^𝑘𝑞,(2.15) where 1/𝑟₄=2/3+1/𝑝=1/3+1/𝑞. Also, for 0<𝑡<2, we have ^𝑡0‖‖∇≤𝑇𝐶(𝑡||𝐮−_∞𝜏)𝑃(𝐰⋅∇)𝐮_𝑘+𝐮_𝑘⋅∇𝐰‖‖_𝑞𝑑𝜏||𝑀^𝑘𝑝+𝑁^𝑘𝑞𝑡^{−3/2(1/𝑟−1/𝑞)−1/2+𝛿2/2}≤𝐶||𝐮_∞||𝑀^𝑘𝑝+𝑁^𝑘𝑞𝑡^{−3/2(1/𝑟−1/𝑞)−1/2}.(2.16)
Therefore, we get 𝑀^𝑝𝑘+1+𝑁^𝑞𝑘+1≤𝐶‖‖𝐮₀‖‖_𝑟+𝐶||𝐮_∞||𝑀^𝑘𝑝+𝑁^𝑘𝑞.(2.17) So if 𝐶|𝐮_∞|<1 (the constant C is bounded as |𝐮_∞| goes to zero, so we can make 𝐶|𝐮_∞|<1 by choosing small 𝐮_∞), then we have some 𝐾 such that 𝑀^𝑝𝑘+1+𝑁^𝑞𝑘+1<𝐾,(2.18) for all 𝑘. Hence, by taking the limit, we complete the proof.
Step 2. Now, we want to prove 1<𝑟<𝑝≤3. For this case, we choose 3/2<𝑞≤3 and 𝑝₁>3 such that 1𝑟−1𝑞<13,1𝑟−𝑝1₁<23.(2.19) Then, we have ‖𝐮(𝑡)‖≤_𝑝≤𝐶𝑡‖‖𝑇(𝑡)𝐮₀‖‖_{𝑝−3/2(1/𝑟−1/𝑝)}+^𝑡0‖𝑇(𝑡−𝜏)𝑃[(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰]‖_𝑝𝑑𝜏+𝐶_𝑡0‖‖𝐮₀‖‖_𝑟+𝐶^𝑡0(𝑡−𝜏)^{−3/2(1/𝑟1−1/𝑝)}‖𝐰‖₃‖∇𝐮‖_𝑞𝑑𝜏≤𝐶_𝜖(𝑡−𝜏)^{−3/2(1/𝑟2−1/𝑝)}‖𝐮‖_𝑝1‖∇𝐰‖_3/2𝑑𝜏𝑡^{−3/2(1/𝑟−1/𝑝)}‖‖𝐮₀‖‖_𝑟,(2.20) where 1/𝑟₁=1/3+1/𝑞 and 1/𝑟₂=1/𝑝₁+2/3. One can note that 1/𝑟₁−1/𝑝<2/3 and 1/𝑟₂−1/𝑝<2/3.
Step 3. Now, we want to prove 1<𝑟<𝑞≤3/2. For this case, we choose 3/2<𝑞₁≤3 and 𝑝>3 such that 1𝑟−𝑞1₁<13,1𝑟−1𝑝<23.(2.21) Similar to Step 2, we have ‖∇𝑢(𝑡)‖_𝑞≤≤𝐶‖‖∇𝑡𝑇(𝑡)𝐮₀‖‖_𝑞+_{−3/2(1/𝑟−1/𝑞)−1/2}^𝑡0‖∇𝑇(𝑡−𝜏)𝑃[(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰]‖_𝑞𝑑𝜏+𝐶_𝑡0‖‖𝐮₀‖‖_𝑟+𝐶^𝑡0(𝑡−𝜏)^{−3/2(1/𝑟1−1/𝑞)−1/2}‖𝐰‖₃‖∇𝐮‖_𝑞1𝑑𝜏≤𝐶𝑡(𝑡−𝜏)^{−3/2(1/𝑟2−1/𝑞)−1/2}_{−3/2(1/𝑟−1/𝑞)−1/2}‖𝐮‖_𝑝‖∇𝐰‖_3/2𝑑𝜏‖‖𝐮₀‖‖_𝑟,(2.22) where 1/𝑟₁=1/3+1/𝑞₁ and 1/𝑟₂=1/𝑝+2/3. One can note that 1/𝑟₁−1/𝑞<1/3 and 1/𝑟₂−1/𝑞<1/3.
Step 4. At last, we want to prove 3<𝑝<∞ with 1/𝑟−1/𝑝<1/3. In this case, we can do easily, by interpolation inequality, Steps 1 and 2.
Therefore, we complete the proof by Steps 1–4.

Now, by applying the Helmholtz-Leray projection 𝑃 into (1.16), we can obtain𝐮_𝑡+ℒ𝐮+𝑃[(𝐮⋅∇)𝐮]=0,for𝑡>0,𝐮(0)=𝐮₀,(2.23) where ℒ𝐮==𝒪𝑃_𝐮∞−Δ𝐮+𝐮_∞⋅∇𝐮+(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰𝒟𝐮+𝑃[(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰_𝑝],(ℒ)=𝒟_𝑝𝒪_𝐮∞=𝑢∈𝐽_𝑝(Ω)∩𝑊^2𝑝(Ω)^𝑛|𝑢|_𝜕Ω=0.(2.24)

One can note from of [14, Lemma 2.6] that for 1<𝑝<∞ and 𝐮∈𝒟_𝑝(ℒ)=𝒟_𝑝(𝒪_𝐮∞), ‖𝐮‖_𝑊^2,𝑝_(Ω)≤𝐶_𝑝‖‖𝒪^𝐮_∞𝐮‖‖_𝑝+‖𝐮‖_𝑝.(2.25) Also, from (1.11), we have ‖(𝐰⋅∇)𝐮+(𝐮⋅∇)𝐰‖_𝑝≤‖𝐰‖_∞+‖∇𝐰‖_∞‖𝐮‖_𝑊^2,𝑝_(Ω)≤||𝐮_∞||‖𝐮‖_{𝑊2,𝑝(Ω)}≤𝐶_𝑝||𝐮_∞||‖‖𝒪_𝐮∞𝐮‖‖_𝑝+‖𝐮‖_𝑝.(2.26) Since the linear operator 𝒪_𝐮∞ generates an analytic semigroup 𝑇(𝑡) (refer to [14, 19]), we obtain an analytic semigroup 𝑆(𝑡) generated by the linear operator ℒ if |𝐮_∞| is small enough. The proof is from perturbation theory of analytic semigroup (refer to [26, Theorem 2.4, page 499]).

Remark 2.2. In Lemma 2.1, by the property of a semigroup, we can remove the conditions 1/𝑟−1/𝑝<2/3 for ‖𝐮(𝑡)‖_𝐿^𝑝_(Ω) and 1/𝑟−1/𝑝<1/3 for ‖∇𝐮(𝑡)‖_𝐿^𝑝_(Ω), because we have 𝐮(𝑥,𝑡) = 𝑆(𝑡)𝐮₀ = 𝑆(𝑡/2)𝑆(𝑡/2)𝐮₀.

Now, we are in the position to prove Theorem 1.2. For the proof, we consider a solution 𝐮(𝑥,𝑡) (1.16) as the limit of the following usual iteration method:𝐮_𝑘+1(𝑡)=𝑆(𝑡)𝐮₀−^𝑡0𝑆(𝑡−𝜏)𝑃𝐮_𝑘⋅∇𝐮_𝑘𝑑𝜏.(2.27)

Here, we will prove by a similar method with the proof of Lemma 2.1. One can note that we will prove without Remark 2.2.

Step 1. We prove that, for any 𝑝>3, we have ‖∇𝐮(𝑡)‖₃<𝐶𝑡^−1/2,‖𝐮(𝑡)‖_𝑝<𝐶𝑡^{−1/2+3/2𝑝},∀𝑡>0.(2.28)
Let 𝑀^𝑘𝑝=𝑁sup_{𝑡∈[0,∞)}𝑡^{1/2−3/2𝑝}‖‖𝑢^𝑘(𝑡)‖‖_𝑝,for𝑝>3,_𝑘3=sup_{𝑡∈(0,∞)}𝑡^1/2‖‖∇𝑢^𝑘(𝑡)‖‖₃.(2.29) By Lemma 2.1 and (2.27), we obtain ‖‖𝐮_𝑘+1(𝑡)‖‖_𝑝≤≤𝐶𝐶𝑡𝑡^{−1/2+3/2𝑝}_{−1/2+3/2𝑝}‖‖𝐮₀‖‖₃+𝐶^𝑡0(𝑡−𝜏)^−1/2‖‖𝐮_𝑘(𝑡)‖‖_𝑝‖‖∇𝐮_𝑘(𝑡)‖‖₃𝑑𝜏‖‖𝐮₀‖‖₃+𝐶𝑀^𝑘𝑝𝑁^𝑘3^𝑡0≤𝑡_{−1/2+3/2𝑝}(𝑡−𝜏)^−1/2𝜏^{−1/2+3/2𝑝}𝜏^−1/2𝑑𝜏𝐶‖‖𝐮₀‖‖₃+𝐶𝑀^𝑘𝑝𝑁^𝑘3,(2.30) which implies 𝑀^𝑝𝑘+1≤𝐶‖‖𝐮₀‖‖₃+𝐶𝑀^𝑘𝑝𝑁^𝑘3.(2.31) Similarly, we have ‖‖∇𝐮_𝑘+1(𝑡)‖‖₃≤𝐶𝑡^−1/2‖‖𝐮₀‖‖₃+𝐶^𝑡0(𝑡−𝜏)^{−3/2𝑝−1/2}‖‖𝐮_𝑘(𝑡)‖‖_𝑝‖‖∇𝐮_𝑘(𝑡)‖‖₃𝑑𝜏≤𝑡^−1/2𝐶‖‖𝐮₀‖‖₃+𝐶𝑀^𝑘𝑝𝑁^𝑘3,(2.32) which implies 𝑁^3𝑘+1≤𝐶‖‖𝐮₀‖‖₃+𝐶𝑀^𝑘𝑝𝑁^𝑘3.(2.33)
Hence, we have 𝑀^𝑝𝑘+1+𝑁^3𝑘+1<𝐶‖‖𝐮₀‖‖₃+𝐶𝑀^𝑘𝑝+𝑁^𝑘3².(2.34)
Now, we have a sequence of the form 𝑥_𝑘+1≤𝛼+𝛽𝑥^2𝑘,(2.35) and we know that such sequence satisfies 𝑥_𝑘≤𝑀≡1−(1−4𝛼𝛽)^1/22𝛽<21𝛽,if𝛼<14𝛽.(2.36)
Therefore, by recurrence estimates, smallness of ‖𝐮₀‖₃ implies 𝑀^𝑝𝑘+1+𝑁^3𝑘+1<𝐾,(2.37) for some constant 𝐾. Finally, we obtain ‖∇𝐮(𝑡)‖₃<𝐶𝑡^−1/2,‖𝐮(𝑡)‖_𝑝<𝐶𝑡^{−1/2+3/2𝑝},∀𝑡>0.(2.38)

Step 2. We prove that if 3/2<𝑝 with 1/𝑟−1/𝑝<1/3 and 𝐮₀∈𝐿^𝑟(Ω)∩𝐿³(Ω), then we have ‖𝐮(𝑡)‖_𝑝≤𝐶𝑡^{−3/2(1/𝑟−1/𝑝)},∀𝑡>0.(2.39) Let 𝑀_𝑝=sup_{𝑡∈(0,∞)}𝑡^{3/2(1/𝑟−1/𝑝)}‖𝑢(𝑡)‖_𝑝.(2.40)
From estimates of Step 1, one can note that we have ‖∇𝐮(𝑡)‖₃≤𝐶𝑡^−1/2‖‖𝐮₀‖‖₃,∀𝑡>0.(2.41)
So, we have ‖𝐮(𝑡)‖_𝑝≤≤𝐶𝐶𝑡𝑡^{−3/2(1/𝑟−1/𝑝)}_{−𝑛/2(1/𝑟−1/𝑝)}‖‖𝐮₀‖‖_𝑟+𝐶^𝑡0(𝑡−𝜏)^{−𝑛/2(1/𝑟8−1/𝑝)}‖𝐮(𝑡)‖_𝑝‖∇𝐮(𝑡)‖₃𝑑𝜏‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖₃^𝑡0≤𝑡_{−𝑛/2(1/𝑟−1/𝑝)}(𝑡−𝜏)^−1/2𝜏^{−𝑛/2(1/𝑟−1/𝑝)}𝜏^−1/2𝑑𝜏𝐶‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖₃𝑀_𝑝,(2.42) which implies 𝑀_𝑝<𝐶‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖₃𝑀_𝑝,(2.43) where 1/𝑟₈=1/3+1/𝑝.
Hence, we complete the proof with 𝐶‖𝐮₀‖₃<1.

Step 3. We prove that if 3/2<𝑞≤3 with 1/𝑟−1/𝑞<1/3 and 𝐮₀∈𝐿^𝑟(Ω)∩𝐿³(Ω), then we have ‖∇𝐮(𝑡)‖_𝑞≤𝐶𝑡^{−3/2(1/𝑟−1/𝑞)−1/2},∀𝑡>0.(2.44)
Let 𝑁_𝑞=sup_{𝑡∈(0,∞)}𝑡^{𝑛/2(1/𝑟−1/𝑞)+1/2}‖∇𝑢(𝑡)‖_𝑞.(2.45) We choose some 𝑝₁≈3 with 𝑝₁>3 such that ‖∇𝐮‖_𝑞≤≤𝐶𝐶𝑡𝑡^{−𝑛/2(1/𝑟−1/𝑞)−1/2}_{−𝑛/2(1/𝑟−1/𝑞)−1/2}‖‖𝐮₀‖‖_𝑟+𝐶^𝑡0(𝑡−𝜏)^{−𝑛/2(1/𝑟7−1/𝑞)−1/2}‖𝐮‖_𝑝1‖∇𝐮‖_𝑞𝑑𝜏‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖₃𝑁_𝑞^𝑡0≤𝑡_{−𝑛/2(1/𝑟−1/𝑞)−1/2}(𝑡−𝜏)^{−1/2−3/2𝑝}𝜏^{−1/2+3/2𝑝1}𝜏^{−𝑛/2(1/𝑟−1/𝑞)−1/2}𝑑𝜏𝐶‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖₃𝑁_𝑞.(2.46) So we complete the proof with 𝐶‖𝐮₀‖₃<1.

Step 4. We prove that if 1<𝑟<𝑝<∞, 1<𝑟<3, and 𝐮₀∈𝐿^𝑟(Ω)∩𝐿³(Ω), then we have ‖𝐮(𝑡)‖_𝑝≤𝐶𝑡^{−3/2(1/𝑟−1/𝑝)},∀𝑡>0.(2.47)
Case 1 (let 𝑝>3/2). Since we proved for 1/𝑟−1/𝑝<1/3 in Step 2, we can assume that 1/3≤1/𝑟−1/𝑝. One notes that we can rewrite solutions 𝐮(𝑡) in the form 𝐮(𝑡)=𝑆𝑡2𝐮2𝑡−^𝑡𝑡/2𝑆(𝑡−𝜏)𝑃[(𝐮⋅∇)𝐮]𝑑𝜏.(2.48) For any 𝑟>1, we choose 𝑙>3/2 such that 1/𝑟−1/𝑙<1/3 and 1/𝑙−1/𝑝<2/3. Also, for any 1<𝑟<𝑝≤∞ with 1<𝑟<3, we choose 𝑠₁>3 and 3/2<𝑠₂<3 such that 1𝑟−𝑠1₂<13,𝑠1₁+1𝑠₂−1𝑝<23.(2.49) Then, by Steps 1–3, we have ‖𝐮(𝑡)‖_𝑝≤𝐶𝑡^{−3/2(1/𝑙−1/𝑝)}‖‖‖𝐮𝑡2‖‖‖_𝑙≤𝐶𝑡_{−3/2(1/𝑟−1/𝑝)}+𝐶^𝑡𝑡/2(𝑡−𝜏)^{−3/2(1/𝑠−1/𝑝)}‖(𝐮⋅∇)𝐮‖_𝑠𝑑𝜏‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖_𝑟^𝑡𝑡/2≤𝐶𝑡_{−3/2(1/𝑟−1/𝑝)}(𝑡−𝜏)^{−3/2(1/𝑠−1/𝑝)}𝜏^{−1/2−3/2(1/𝑟−1/𝑠2)}𝜏^{−1/2+3/2𝑠1}𝑑𝜏‖‖𝐮₀‖‖_𝑟,∀𝑡>0.(2.50)
Case 2 (let 1<𝑝≤3/2). By Step 1–3, we have ‖𝐮(𝑡)‖_𝑝≤≤𝐶𝐶𝑡𝑡^{−3/2(1/𝑟−1/𝑝)}_{−3/2(1/𝑟−1/𝑝)}‖‖𝐮₀‖‖_𝑟+𝐶^𝑡0(𝑡−𝜏)^{−3/2(1/𝑠−1/𝑝)}‖(𝐮⋅∇)𝐮‖_𝑠𝑑𝜏‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖_𝑟^𝑡0≤𝐶𝑡_{−3/2(1/𝑟−1/𝑝)}(𝑡−𝜏)^{−3/2(1/𝑠−1/𝑝)}𝜏^{−3/2(1/𝑟−1/𝑠1)}𝜏^−1/2𝑑𝜏‖‖𝐮₀‖‖_𝑟,∀𝑡>0,(2.51) where 𝑠₁>3/2, 1/𝑟−1/𝑠₁<1/3, 1/𝑠=1/𝑠₁+1/3.

Step 5. We prove that if 1<𝑟<𝑞≤3 and 𝐮₀∈𝐿^𝑟(Ω)∩𝐿³(Ω), then ‖∇𝐮(𝑡)‖_𝑞≤𝐶𝑡^{−3/2(1/𝑟−1/𝑞)−1/2}.(2.52)
Case 1 (let 3/2<𝑞≤3). Since we proved 1/𝑟−1/𝑞<1/3 in Step 3, we can assume that 1/3≤1/𝑟−1/𝑞. Now, we choose 𝑙>3/2 such that 1/𝑟−1/𝑙<1/3 and 1/𝑙−1/𝑞<1/3. We also can have 𝑠₁>3 and 3/2<𝑠₂<3 with 1𝑠=𝑠1₁+1𝑠₂,1𝑟−𝑠1₂<13,1𝑠−1𝑞<13.(2.53) So, by Step 1–4, we obtain ‖∇𝐮(𝑡)‖_𝑞≤𝐶𝑡^{−3/2(1/𝑙−1/𝑞)−1/2}‖‖‖𝐮𝑡2‖‖‖_𝑙≤𝐶𝑡_{−3/2(1/𝑟−1/𝑞)−1/2}+𝐶^𝑡𝑡/2(𝑡−𝜏)^{−3/2(1/𝑠−1/𝑞)−1/2}‖𝐮(𝑡)‖_𝑠1‖∇𝐮(𝑡)‖_𝑠2𝑑𝜏‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖_𝑟≤×𝐶𝑡^𝑡𝑡/2(𝑡−𝜏)^{−3/2(1/𝑠−1/𝑞)−1/2}_{−3/2(1/𝑟−1/𝑞)−1/2}𝜏^{−1/2+3/2𝑠1}𝜏^{−3/2(1/𝑟−1/𝑠2)−1/2}𝑑𝜏‖‖𝐮₀‖‖_𝑟.(2.54)
Case 2 (let 1<𝑞≤3/2). By Step 1-Step 3, we have ‖∇𝐮(𝑡)‖_𝑞≤≤𝐶𝐶𝑡𝑡^{−3/2(1/𝑟−1/𝑞)−1/2}_{−3/2(1/𝑟−1/𝑞)−1/2}‖‖𝐮₀‖‖_𝑟+𝐶^𝑡0(𝑡−𝜏)^{−3/2(1/𝑠−1/𝑞)−1/2}‖(𝐮⋅∇)𝐮‖_𝑠𝑑𝜏‖‖𝐮₀‖‖_𝑟+𝐶‖‖𝐮₀‖‖_𝑟^𝑡0≤𝐶𝑡_{−3/2(1/𝑟−1/𝑞)−1/2}(𝑡−𝜏)^{−3/2(1/𝑠−1/𝑞)−1/2}𝜏^{−3/2(1/𝑟−1/𝑠1)}𝜏^−1/2𝑑𝜏‖‖𝐮₀‖‖_𝑟,∀𝑡>0,(2.55) where 𝑠₁>3/2, 1/𝑟−1/𝑠₁<1/3, 1/𝑠=1/𝑠₁+1/3, and 1/𝑠−1/𝑞<1/3.
Therefore, by Step 1–5, we complete the proof of Theorem 1.2.

Acknowledgment

The author would like to thank Professor Hyeong-Ohk Bae for valuable comments. The author also thanks deeply Professors Cheng He and Lizhen Wang for the preprint [27] “Weighted 𝐿^𝑝-Estimates for Stokes Flow in 𝑅^𝑛+, with Applications to the Non-Stationary Navier-Stokes Flow” which motivates the idea of the proof. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0023386).