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Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 856896, 15 pages
http://dx.doi.org/10.1155/2011/856896
Research Article

๐ฟ๐‘Ÿ-๐ฟ๐‘ Stability of the Incompressible Flows with Nonzero Far-Field Velocity

Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea

Received 16 June 2011; Accepted 15 September 2011

Academic Editor: Ibrahimย Sadek

Copyright ยฉ 2011 Jaiok Roh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the stability of stationary solutions ๐ฐ for the exterior Navier-Stokes flows with a nonzero constant velocity ๐ฎโˆž at infinity. For ๐ฎโˆž=0 with nonzero stationary solution ๐ฐ, Chen (1993), Kozono and Ogawa (1994), and Borchers and Miyakawa (1995) have studied the temporal stability in ๐ฟ๐‘ spaces for 1<๐‘ and obtained good stability decay rates. For the spatial direction, we recently obtained some results. For ๐ฎโˆžโ‰ 0, Heywood (1970, 1972) and Masuda (1975) have studied the temporal stability in ๐ฟ2 space. Shibata (1999) and Enomoto and Shibata (2005) have studied the temporal stability in ๐ฟ๐‘ spaces for ๐‘โ‰ฅ3. 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 ๐ฟ๐‘ for ๐‘>1 and obtain ๐ฟ๐‘Ÿ-๐ฟ๐‘ stability as Kozono and Ogawa and Borchers and Miyakawa obtained for ๐ฎโˆž=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:๐œ•๐œ•๐‘ก๐ฎโˆ’ฮ”๐ฎ+(๐ฎโ‹…โˆ‡)๐ฎ+โˆ‡๐‘=๐Ÿ,โˆ‡โ‹…๐ฎ=0inฮฉร—(0,โˆž),๐ฎ|๐‘ก=0=๐ฎ0,๐ฎ|๐œ•ฮฉ=0,lim|๐‘ฅ|โ†’โˆž๐ฎ(๐‘ฅ,๐‘ก)=๐ฎโˆž,(1.1) where ฮฉ is an exterior domain in ๐‘…๐‘› with a smooth boundary ๐œ•ฮฉ, and ๐ฎโˆž denotes a given constant vector describing the velocity of the fluid at infinity. In this paper, we consider a nonzero constant ๐ฎโˆž. The physical model of the exterior Navier-Stokes equations with a nonzero constant ๐ฎโˆž can be considered as the motion of water in the sea when a boat is moving with the speed โˆ’๐ฎโˆž, while the one with zero constant ๐ฎโˆž can be considered when a boat is stopped. There are few known results for the case ๐ฎโˆžโ‰ 0, while, with ๐ฎโˆž=0, many results were obtained for the temporal decay and weighted estimates of solutions of (1.1) (refer [1โ€“12]).

Now, we set ๐ฎ=๐ฎโˆž+๐ฏ in (1.1) and have๐œ•๎€ท๐ฎ๐œ•๐‘ก๐ฏโˆ’ฮ”๐ฏ+โˆž๎€ธโ‹…โˆ‡๐ฏ+(๐ฏโ‹…โˆ‡)๐ฏ+โˆ‡๐‘1=๐Ÿ,โˆ‡โ‹…๐ฏ=0inฮฉร—(0,โˆž),๐ฏ|๐‘ก=0=๐ฎ0โˆ’๐ฎโˆž,๐ฏ|๐œ•ฮฉ=โˆ’๐ฎโˆž,lim|๐‘ฅ|โ†’โˆž๐ฏ(๐‘ฅ,๐‘ก)=0.(1.2)

Consider the following linear problem:๐œ•๎€ท๐ฎ๐œ•๐‘ก๐ฎโˆ’ฮ”๐ฎ+โˆž๎€ธโ‹…โˆ‡๐ฎ+โˆ‡๐‘=0,โˆ‡โ‹…๐ฎ=0inฮฉร—(0,โˆž),๐ฎ|๐‘ก=0=๐ฎ0,๐ฎ|๐œ•ฮฉ=0,lim|๐‘ฅ|โ†’โˆž๐ฎ(๐‘ฅ,๐‘ก)=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: ๐ฟ๐‘(ฮฉ)๐‘›=๐ฝ๐‘(ฮฉ)โŠ•๐บ๐‘(ฮฉ),(1.4) where 1<๐‘<โˆž, ๐ฟ๐‘(ฮฉ)๐‘›=๎€ฝ๎€ท๐‘ข๐‘ข=1,โ€ฆ,๐‘ข๐‘›๎€ธโˆถ๐‘ข๐‘—โˆˆ๐ฟ๐‘๎€พ,๐ถ(ฮฉ),๐‘—=1,โ€ฆ,๐‘›โˆž0,๐œŽ=๎€ฝ๎€ท๐‘ข๐‘ข=1,โ€ฆ,๐‘ข๐‘›๎€ธโˆˆ๐ถโˆž0(ฮฉ)๐‘›โˆถโˆ‡โ‹…๐‘ข=0inฮฉ๎€พ,๐ฝ๐‘(ฮฉ)=thecompletionof๐ถโˆž0,๐œŽ(ฮฉ),in๐ฟ๐‘(ฮฉ)๐‘›,๐บ๐‘๎‚†(ฮฉ)=โˆ‡๐œ‹โˆˆ๐ฟ๐‘(ฮฉ)๐‘›โˆถ๐œ‹โˆˆ๐ฟ๐‘,loc๎‚€ฮฉ.๎‚๎‚‡(1.5) The Helmholtz decomposition of ๐ฟ๐‘(ฮฉ)๐‘› was proved by Fujiwara and Morimoto [15], Miyakawa [16], and Simader and Sohr [17]. Let ๐‘ƒ be a continuous projection from ๐ฟ๐‘(ฮฉ)๐‘› onto ๐ฝ๐‘(ฮฉ)๐‘›.

By applying ๐‘ƒ into (1.3) and setting ๐’ช๐ฎโˆž = ๐‘ƒ(โˆ’ฮ”+๐ฎโˆžโ‹…โˆ‡), one has ๐ฎ๐‘ก+๐’ช๐ฎโˆž๐ฎ=0,for๐‘ก>0,๐ฎ(0)=๐ฎ0,(1.6) where the domain of ๐’ช๐ฎโˆž is given by ๐’Ÿ๐‘๎€ท๐’ช๐ฎโˆž๎€ธ=๎€ฝ๐‘ขโˆˆ๐ฝ๐‘(ฮฉ)โˆฉ๐‘Š2๐‘(ฮฉ)๐‘›โˆถ๐‘ข|๐œ•ฮฉ๎€พ.=0(1.7) Then, Enomoto and Shibata [14] proved that ๐’ช๐ฎโˆž generates an analytic semigroup {๐‘‡(๐‘ก)}๐‘กโ‰ฅ0 which is called the Oseen semigroup (one can also refer to [16, 18]) and obtained the following properties.

Proposition 1.1. Let ๐œŽ0>0 and assume that |๐ฎโˆž|โ‰ค๐œŽ0. Let 1โ‰ค๐‘Ÿโ‰ค๐‘žโ‰คโˆž, then โ€–๐‘‡(๐‘ก)๐‘Žโ€–๐ฟ๐‘ž(ฮฉ)โ‰ค๐ถ๐‘Ÿ,๐‘ž,๐œŽ0๐‘กโˆ’(3/2)(1/๐‘Ÿโˆ’1/๐‘ž)โ€–๐‘Žโ€–๐ฟ๐‘Ÿ(ฮฉ),๐‘ก>0,(1.8) where (๐‘Ÿ,๐‘ž)โ‰ (1,1) and (โˆž,โˆž), โ€–โˆ‡๐‘‡(๐‘ก)๐‘Žโ€–๐ฟ๐‘ž(ฮฉ)โ‰ค๐ถ๐‘Ÿ,๐‘ž,๐œŽ0๐‘กโˆ’(3/2)(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–๐‘Žโ€–๐ฟ๐‘Ÿ(ฮฉ),๐‘ก>0,(1.9) where 1โ‰ค๐‘Ÿโ‰ค๐‘žโ‰ค3 and (๐‘Ÿ,๐‘ž)โ‰ (1,1).

The main purpose of this paper is to discuss the temporal stability of stationary solution ๐ฐ of the nonlinear Navier-Stokes equation (1.2). One can note that ๐ฐ satisfies the following equations:๎€ท๐ฎโˆ’ฮ”๐ฐ+โˆž๎€ธโ‹…โˆ‡๐ฐ+(๐ฐโ‹…โˆ‡)๐ฐ+โˆ‡๐‘2=๐Ÿ,โˆ‡โ‹…๐ฐ=0,๐ฐ|๐œ•ฮฉ=โˆ’๐ฎโˆž,lim|๐‘ฅ|โ†’โˆž๐ฐ(๐‘ฅ)=0.(1.10)

For suitable ๐Ÿ, Shibata [19] proved that, for any given 0<๐›ฟ<1/4, there exists ๐œ– such that if 0<|๐ฎโˆž|โ‰ค๐œ–, then one hasโ€–๐ฐโ€–๐‘,2+||||โ€–๐ฐโ€–๐›ฟ+โ€–โ€–๐‘2โ€–โ€–๐‘,1โ‰ค||๐ฎโˆž||๐›ฝ,(1.11) where โ€–๐ฎโ€–๐‘,๐‘š=โ€–๐œ•๐‘š๐ฎโ€–๐ฟ๐‘(ฮฉ),||||โ€–๐ฎโ€–๐›ฟ=sup๐‘ฅโˆˆฮฉ๎€ท(1+|๐‘ฅ|)1+๐‘ ๐ฎโˆž๎€ธ(๐‘ฅ)๐›ฟ||||๐ฎ(๐‘ฅ)+sup๐‘ฅโˆˆฮฉ(1+|๐‘ฅ|)3/2๎€ท1+๐‘ ๐ฎโˆž๎€ธ(๐‘ฅ)1/2+๐›ฟ||||,๐‘ โˆ‡๐ฎ(๐‘ฅ)๐ฎโˆž(๐‘ฅ)=|๐‘ฅ|โˆ’๐‘ฅ๐‘‡โ‹…๐ฎโˆž||๐ฎโˆž||๐›ฟ<๐›ฝ<1โˆ’๐›ฟ.(1.12) Throughout this paper, we assume that ๐Ÿ satisfies the assumption in Shibata [19]. Now, we consider the polar coordinate system ๐‘ฆ1=๐‘Ÿcos๐œƒ,๐‘ฆ2=๐‘Ÿsin๐œƒcos๐œ™,๐‘ฆ3,=๐‘Ÿsin๐œƒsin๐œ™(1.13) for 0โ‰ค๐œƒโ‰ค๐œ‹, 0โ‰ค๐œ™โ‰ค2๐œ‹, and 0โ‰ค๐‘Ÿ<โˆž. Let ๐‘† be an orthogonal matrix such that ๐‘†๐ฎโˆž = |๐ฎโˆž|(1,0,0)๐‘‡ and put ๐‘ (๐‘ฆ) = |๐‘ฆ|โˆ’๐‘ฆ1. By a change of variable ๐‘ฆ=๐‘†๐‘ฅ, ||๐‘ฆ|||๐‘ฅ|==๐‘Ÿ,๐‘ ๐ฎโˆž(๐‘ฅ)=๐‘ (๐‘ฆ)=๐‘Ÿ(1โˆ’cos๐œƒ).(1.14) See Shibata [19] for the detail. Now, by using the above change of variable, we can see easily that ๐ฐ satisfiesโ€–๐ฐโ€–๐ฟ1)3/(1+๐›ฟ(ฮฉ)+โ€–๐ฐโ€–๐ฟ2)3/(1โˆ’๐›ฟ(ฮฉ)+โ€–โˆ‡๐ฐโ€–๐ฟ1)3/(2+๐›ฟ(ฮฉ)+โ€–โˆ‡๐ฐโ€–๐ฟ2)3/(2โˆ’๐›ฟ(ฮฉ)||๐ฎโ‰ค๐ถโˆž||1/2,(1.15) for small ๐›ฟ1, ๐›ฟ2, where ๐ถ is independent on ๐ฎโˆž.

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 ๐ฐ, by setting ๐ฎ=๐ฏโˆ’๐ฐ and ๐‘=๐‘1โˆ’๐‘2 for ๐ฏ, ๐‘1, ๐ฐ, ๐‘2 in (1.2) and (1.10), we have the following equations in ฮฉ:๐œ•๎€ท๐ฎ๐œ•๐‘ก๐ฎโˆ’ฮ”๐ฎ+โˆž๎€ธ๐ฎโ‹…โˆ‡๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ+(๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฎ+โˆ‡๐‘=0,โˆ‡โ‹…๐ฎ=0,(๐‘ฅ,0)=๐ฎ0(๐‘ฅ)for๐‘ฅโˆˆฮฉ,๐ฎ(๐‘ฅ,๐‘ก)=0for๐‘ฅโˆˆ๐œ•ฮฉ,lim|๐‘ฅ|โ†’โˆž๐ฎ(๐‘ฅ,๐‘ก)=0.(1.16) Here, in fact, the initial data should be ๐ฎ0โˆ’๐ฎโˆžโˆ’๐ฐ, but for our convenience, we denote by ๐ฎ0 for ๐ฎ0โˆ’๐ฎโˆžโˆ’๐ฐ if there is no confusion.

First, Heywood [21, 22] and Masuda [23] have studied the temporal stability in ๐ฟ2 space. Shibata [19] proved that there exists small ๐œ– such that if 0<|๐ฎโˆž|โ‰ค๐œ– and โ€–๐ฎ0โ€–3โ‰ค๐œ–, then a unique solution ๐ฎ(๐‘ฅ,๐‘ก) of (1.16) has the following properties: for any 3<๐‘<โˆž, [๐ฎ]3,0,๐‘ก+[๐ฎ]๐‘,๐œ‡(๐‘),๐‘ก+[]โˆ‡๐ฎ3,1/2,๐‘กโ‰คโˆš๐œ–,lim๐‘กโ†’0+๎€บโ€–โ€–๐ฎ(๐‘ก)โˆ’๐ฎ0โ€–โ€–3+[๐ฎ]๐‘,๐œ‡(๐‘),๐‘ก+[]โˆ‡๐ฎ3,1/2,๐‘ก๎€ป=0,(1.17) where [๐ณ]๐‘,๐œŒ,๐‘ก=sup0<๐‘ <๐‘ก๐‘ ๐œŒโ€–โ€–๐ณ(๐‘ ,โ‹…)๐‘1,๐œ‡(๐‘)=2โˆ’3.2๐‘(1.18)

After that, Enomoto and Shibata [14] considered the stability for arbitrary ๐ฎโˆž by deleting the smallness condition of |๐ฎโˆž|. But in this case, all constants in their results depend on ๐œŽ0 when |๐ฎโˆž|โ‰ค๐œŽ0. Also, they assumed the existence of stationary solution ๐ฐ with โ€–๐ฐโ€–๐ฟ1)3/(1+๐›ฟ(ฮฉ)+โ€–๐ฐโ€–๐ฟ2)3/(1โˆ’๐›ฟ(ฮฉ)+โ€–โˆ‡๐ฐโ€–๐ฟ1)3/(2+๐›ฟ(ฮฉ)+โ€–โˆ‡๐ฐโ€–๐ฟ2)3/(2โˆ’๐›ฟ(ฮฉ)โ‰ค๐›ผ,(1.19) for small ๐›ฟ1, ๐›ฟ2 and ๐›ผ. Then, as a result, they proved (1.16) has a unique solution ๐ฎ(๐‘ฅ,๐‘ก) with lim๐‘กโ†’0+๎€ฝโ€–โ€–๐ฎ(๐‘ก)โˆ’๐ฎ0โ€–โ€–3+๐‘ก1/2๎€ทโ€–๐ฎ(๐‘ก)โ€–๐ฟโˆž+โ€–โˆ‡๐ฎ(๐‘ก)โ€–๐ฟ3๎€ธ๎€พ=0,โ€–๐ฎ(๐‘ก)โ€–๐ฎ(๐‘ก)๎€ท๐‘ก=๐‘œโˆ’((1/2)โˆ’(3/2๐‘))๎€ธ,foranyโ€–3โ‰ค๐‘โ‰คโˆž,โ€–โˆ‡๐ฎ(๐‘ก)3๎€ท๐‘ก=๐‘œโˆ’1/2๎€ธ(1.20) as ๐‘กโ†’โˆž when ๐ฎ0 is small enough in the space ๐ฟ3(ฮฉ).

Also, Bae and Roh [24] improved Enomoto-Shibata's result in some sense. But their result is limited in the space ๐ฟ๐‘ for 3/2<๐‘, while we consider all 1<๐‘. Moreover, their result depends on ๐‘  and ๐‘Ÿ, while ours only depends on ๐‘Ÿ, where ๐ฐโˆˆ๐ฟ๐‘  and ๐ฎ0โˆˆ๐ฟ๐‘Ÿ. Also, their optimal decay rate is 2/3+๐›ฟ, while ours is 3/2+๐›ฟ.

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 ๐ฐ=0 and a very well-known method. Also, for ๐ฐโ‰ 0 with ๐ฎโˆž=0, Kozono and Ogawa [12] also used similar method.

Theorem 1.2. There exists small ๐œ–(๐‘,๐‘ž,๐‘Ÿ) such that if0<|๐ฎโˆž|โ‰ค๐œ– and โ€–๐ฎ0โ€–๐ฟ3(ฮฉ)<๐œ–, then a unique solution ๐ฎ(๐‘ฅ,๐‘ก) of (1.16) has the following properties: โ€–๐ฎ(๐‘ก)โ€–๐ฟ๐‘(ฮฉ)โ‰ค๐ถ๐œ–๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿfor1<๐‘Ÿ<๐‘โ‰คโˆž,๐‘ก>0,โ€–โˆ‡๐ฎ(๐‘ก)โ€–๐ฟ๐‘ž(ฮฉ)โ‰ค๐ถ๐œ–๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿfor1<๐‘Ÿ<๐‘žโ‰ค3,๐‘ก>0,(1.21) where ๐ฎ0โˆˆ๐ฟ3(ฮฉ)โˆฉ๐ฟ๐‘Ÿ(ฮฉ).

2. Proof of Main Theorem

First, we consider the following linear problem:๐œ•๎€ท๐ฎ๐œ•๐‘ก๐ฎโˆ’ฮ”๐ฎ+โˆž๎€ธโ‹…โˆ‡๐ฎ+(๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ+โˆ‡๐‘=0,โˆ‡โ‹…๐ฎ=0,๐ฎ|๐‘ก=0=๐ฎ0,๐ฎ|๐œ•ฮฉ=0,lim|๐‘ฅ|โ†’โˆž๐ฎ(๐‘ฅ,๐‘ก)=0.(2.1)

By applying Helmholtz-Leray projection ๐‘ƒ and setting ๎€บ๎€ท๐ฎโ„’๐ฎ=๐‘ƒโˆ’ฮ”๐ฎ+โˆž๎€ธ๎€ปโ‹…โˆ‡๐ฎ+(๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ=๐’ช๐ฎโˆž[],๐ฎ+๐‘ƒ(๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ(2.2) we have ๐ฎ๐‘ก+โ„’๐ฎ=0,for๐‘ก>0,๐ฎ(0)=๐ฎ0.(2.3) And we note that the domain of โ„’ is ๐’Ÿ๐‘(โ„’)=๐’Ÿ๐‘๎€ท๐’ช๐ฎโˆž๎€ธ=๎€ฝ๐‘ขโˆˆ๐ฝ๐‘(ฮฉ)โˆฉ๐‘Š2๐‘(ฮฉ)๐‘›|๐‘ข|๐œ•ฮฉ๎€พ.=0(2.4)

Let ๐‘†(๐‘ก) be a semigroup generated by the linear operator โ„’, then, by Duharmel's Principle, a solution ๐ฎ(๐‘ฅ,๐‘ก) of (2.1) can be written as in the following integral form,๐ฎ(๐‘ฅ,๐‘ก)=๐‘†(๐‘ก)๐ฎ0=๐‘‡(๐‘ก)๐ฎ0+๎€œ๐‘ก0[(]๐‘‡(๐‘กโˆ’๐œ)๐‘ƒ๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ๐‘‘๐œ,(2.5) where ๐‘‡(๐‘ก) is an analytic semigroup generated by the Oseen operator ๐’ช๐ฎโˆž.

Lemma 2.1. Let ๐ฎ0โˆˆ๐ฟ3(ฮฉ)โˆฉ๐ฟ๐‘Ÿ(ฮฉ) for 1<๐‘Ÿ<3, then there exists a small ๐œ–(๐‘,๐‘ž,๐‘Ÿ) such that if |๐ฎโˆž|โ‰ค๐œ– and โ€–๐ฎ0โ€–๐ฟ3(ฮฉ)<๐œ–, then a solution ๐ฎ(๐‘ฅ,๐‘ก) represented by (2.5) satisfies 1<๐‘โ‰คโˆž with 1/๐‘Ÿโˆ’1/๐‘<2/3, โ€–๐ฎ(๐‘ก)โ€–๐ฟ๐‘(ฮฉ)=โ€–โ€–๐‘†(๐‘ก)๐ฎ0โ€–โ€–๐ฟ๐‘(ฮฉ)โ‰ค๐ถ๐œ–๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐ฟ๐‘Ÿ(ฮฉ),๐‘ก>0,(2.6) and for 1<๐‘žโ‰ค3 with 1/๐‘Ÿโˆ’1/๐‘ž<1/3, โ€–โˆ‡๐ฎ(๐‘ก)โ€–๐ฟ๐‘ž(ฮฉ)=โ€–โ€–โˆ‡๐‘†(๐‘ก)๐ฎ0โ€–โ€–๐ฟ๐‘ž(ฮฉ)โ‰ค๐ถ๐œ–๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐ฟ๐‘Ÿ(ฮฉ),๐‘ก>0.(2.7)

Proof. Before we prove Lemma 2.1 note from (1.15) that we have โ€–๐ฐโ€–๐ฟ1)3/(1+๐›ฟ(ฮฉ)+โ€–๐ฐโ€–๐ฟ2)3/(1โˆ’๐›ฟ(ฮฉ)+โ€–โˆ‡๐ฐโ€–๐ฟ1)3/(2+๐›ฟ(ฮฉ)+โ€–โˆ‡๐ฐโ€–๐ฟ2)3/(2โˆ’๐›ฟ(ฮฉ)||๐ฎโ‰ค๐ถโˆž||1/2,(2.8) for small ๐›ฟ1,๐›ฟ2>0. In fact, by straight calculations, we can choose any ๐›ฟ1,๐›ฟ2โ‰ค3/16.
Step 1. Let 3<๐‘โ‰คโˆž with 1/3โ‰ค1/๐‘Ÿโˆ’1/๐‘<2/3 and 3/2<๐‘žโ‰ค3 with 1/๐‘Ÿโˆ’1/๐‘ž<1/3. We consider the following iteration method to obtain our estimates: ๐ฎ๐‘˜+1(๐‘ก)=๐‘‡(๐‘ก)๐ฎ0+๎€œ๐‘ก0๎€บ(๐‘‡(๐‘กโˆ’๐œ)๐‘ƒ๐ฐโ‹…โˆ‡)๐ฎ๐‘˜+๎€ท๐ฎ๐‘˜๎€ธ๐ฐ๎€ปโ‹…โˆ‡๐‘‘๐œ.(2.9) We let 1/๐‘žโˆ’1/๐‘=1/3 and ๐‘€๐‘˜๐‘=sup๐‘กโˆˆ[0,โˆž)๐‘ก๐‘›/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐‘ข๐‘˜โ€–โ€–(๐‘ก)๐‘,๐‘๐‘˜๐‘ž=sup๐‘กโˆˆ(0,โˆž)๐‘ก๐‘›/2(1/๐‘Ÿโˆ’1/๐‘ž)+1/2โ€–โ€–โˆ‡๐‘ข๐‘˜โ€–โ€–(๐‘ก)๐‘ž.(2.10) If ๐‘กโ‰ฅ2, then by Proposition 1.1, for small ๐›ฟ1,๐›ฟ2>0, we have ๎€œ๐‘ก0โ€–โ€–๎€บ(๐‘‡(๐‘กโˆ’๐œ)๐‘ƒ๐ฐโ‹…โˆ‡)๐ฎ๐‘˜+๎€ท๐ฎ๐‘˜๎€ธ๐ฐ๎€ปโ€–โ€–โ‹…โˆ‡๐‘๎‚ธ๎€œ๐‘‘๐œโ‰ค๐ถ0๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ1โˆ’1/๐‘)โ€–โ€–(๐ฐโ‹…โˆ‡)๐ฎ๐‘˜โ€–โ€–๐‘Ÿ1๎€œ๐‘‘๐œ+๐‘ก๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ2โˆ’1/๐‘)โ€–โ€–(๐ฐโ‹…โˆ‡)๐ฎ๐‘˜โ€–โ€–๐‘Ÿ2+๎€œ๐‘‘๐œ0๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ1โˆ’1/๐‘)โ€–โ€–(๐ฎ๐‘˜โ€–โ€–โ‹…โˆ‡)๐ฐ๐‘Ÿ1๎€œ๐‘‘๐œ+๐‘ก๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ2โˆ’1/๐‘)โ€–โ€–(๐ฎ๐‘˜โ€–โ€–โ‹…โˆ‡)๐ฐ๐‘Ÿ2๎‚น||๐ฎ๐‘‘๐œโ‰ค๐ถโˆž||๐‘๐‘˜๐‘ž๎‚ธ๎€œ0๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’1โˆ’๐›ฟ1/2๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2๎€œ๐‘‘๐œ+๐‘ก๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’1+๐›ฟ2/2๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2๎‚น||๐ฎ๐‘‘๐œ+๐ถโˆž||๐‘€๐‘˜๐‘๎‚ธ๎€œ0๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’1โˆ’๐›ฟ1/2๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)๎€œ๐‘‘๐œ+๐‘ก๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’1+๐›ฟ2/2๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)๎‚น||๐ฎ๐‘‘๐œโ‰ค๐ถโˆž||๎€ท๐‘€๐‘˜๐‘+๐‘๐‘˜๐‘ž๎€ธ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘),(2.11) where 1/๐‘Ÿ1=1/๐‘+2/3+๐›ฟ1/3 and 1/๐‘Ÿ2=1/๐‘+2/3โˆ’๐›ฟ2/3. If 0<๐‘ก<2, then we have ๎€œ๐‘ก0โ€–โ€–๎€บ(๐‘‡(๐‘กโˆ’๐œ)๐‘ƒ๐ฐโ‹…โˆ‡)๐ฎ๐‘˜+๎€ท๐ฎ๐‘˜๎€ธ๐ฐ๎€ปโ€–โ€–โ‹…โˆ‡๐‘๎‚ธ๎€œ๐‘‘๐œโ‰ค๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ3โˆ’1/๐‘)โ€–โ€–(๐ฐโ‹…โˆ‡)๐ฎ๐‘˜โ€–โ€–๐‘Ÿ3๎€œ๐‘‘๐œ+๐‘ก0(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ3โˆ’1/๐‘)โ€–โ€–๎€ท๐ฎ๐‘˜๎€ธ๐ฐโ€–โ€–โ‹…โˆ‡๐‘Ÿ3๎‚น||๐ฎ๐‘‘๐œโ‰ค๐ถโˆž||๎€ท๐‘€๐‘˜๐‘+๐‘๐‘˜๐‘ž๎€ธ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘),(2.12) where 1/๐‘Ÿ3=1/๐‘+2/3โˆ’๐›ฟ2/3. So, we obtain โ€–โ€–๐ฎ๐‘˜+1โ€–โ€–(๐‘ก)๐‘โ‰ค๐ถ๐‘กโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ||๐ฎ+๐ถโˆž||๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)๎€บ๐‘€๐‘˜๐‘+๐‘๐‘˜๐‘ž๎€ป,โˆ€๐‘ก>0,(2.13) which implies ๐‘€๐‘๐‘˜+1โ€–โ€–๐ฎโ‰ค๐ถ0โ€–โ€–๐‘Ÿ||๐ฎ+๐ถโˆž||๎€ท๐‘€๐‘˜๐‘+๐‘๐‘˜๐‘ž๎€ธ.(2.14) Similarly, we obtain for ๐‘กโ‰ฅ2, โ€–โ€–โˆ‡๐ฎ๐‘˜+1(โ€–โ€–๐‘ก)๐‘žโ‰คโ€–โ€–โˆ‡๐‘‡(๐‘ก)๐ฎ0โ€–โ€–๐‘ž+๎€œ๐‘ก0โ€–โ€–๎€บ(โˆ‡๐‘‡(๐‘กโˆ’๐œ)๐‘ƒ๐ฐโ‹…โˆ‡)๐ฎ๐‘˜+๎€ท๐ฎ๐‘˜๎€ธ๐ฐ๎€ปโ€–โ€–โ‹…โˆ‡๐‘ž๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ||๐ฎ+๐ถโˆž||๐‘๐‘˜๐‘ž๎€œ0๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’1โˆ’๐›ฟ1/2๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2||๐ฎ๐‘‘๐œ+๐ถโˆž||๐‘๐‘˜๐‘ž๎€œ๐‘ก๐‘กโˆ’1(๐‘กโˆ’๐œ)โˆ’1+๐›ฟ2/3๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2||๐ฎ๐‘‘๐œ+๐ถโˆž||๐‘€๐‘˜๐‘๎€œ๐‘ก0(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ4โˆ’1/๐‘ž)โˆ’1/2๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ||๐ฎ+๐ถโˆž||๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2๎€บ๐‘€๐‘˜๐‘+๐‘๐‘˜๐‘ž๎€ป,(2.15) where 1/๐‘Ÿ4=2/3+1/๐‘=1/3+1/๐‘ž. Also, for 0<๐‘ก<2, we have ๎€œ๐‘ก0โ€–โ€–๎€บ(โˆ‡๐‘‡(๐‘กโˆ’๐œ)๐‘ƒ๐ฐโ‹…โˆ‡)๐ฎ๐‘˜+๎€ท๐ฎ๐‘˜๎€ธ๐ฐ๎€ปโ€–โ€–โ‹…โˆ‡๐‘ž||๐ฎ๐‘‘๐œโ‰ค๐ถโˆž||๎€ท๐‘€๐‘˜๐‘+๐‘๐‘˜๐‘ž๎€ธ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2+๐›ฟ2/2||๐ฎโ‰ค๐ถโˆž||๎€ท๐‘€๐‘˜๐‘+๐‘๐‘˜๐‘ž๎€ธ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2.(2.16)
Therefore, we get ๐‘€๐‘๐‘˜+1+๐‘๐‘ž๐‘˜+1โ€–โ€–๐ฎโ‰ค๐ถ0โ€–โ€–๐‘Ÿ||๐ฎ+๐ถโˆž||๎€ท๐‘€๐‘˜๐‘+๐‘๐‘˜๐‘ž๎€ธ.(2.17) So if ๐ถ|๐ฎโˆž|<1 (the constant C is bounded as |๐ฎโˆž| goes to zero, so we can make ๐ถ|๐ฎโˆž|<1 by choosing small ๐ฎโˆž), then we have some ๐พ such that ๐‘€๐‘๐‘˜+1+๐‘๐‘ž๐‘˜+1<๐พ,(2.18) for all ๐‘˜. Hence, by taking the limit, we complete the proof.

Step 2. Now, we want to prove 1<๐‘Ÿ<๐‘โ‰ค3. For this case, we choose 3/2<๐‘žโ‰ค3 and ๐‘1>3 such that 1๐‘Ÿโˆ’1๐‘ž<13,1๐‘Ÿโˆ’1๐‘1<23.(2.19) Then, we have (โ€–๐ฎ๐‘ก)โ€–๐‘โ‰คโ€–โ€–๐‘‡(๐‘ก)๐ฎ0โ€–โ€–๐‘+๎€œ๐‘ก0([(]โ€–โ€–๐‘‡๐‘กโˆ’๐œ)๐‘ƒ๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ๐‘๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ๎€œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘Ÿ1โˆ’1/๐‘)โ€–๐ฐโ€–3โ€–โˆ‡๐ฎโ€–๐‘ž๎€œ๐‘‘๐œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘Ÿ2โˆ’1/๐‘)โ€–๐ฎโ€–๐‘1โ€–โˆ‡๐ฐโ€–3/2๐‘‘๐œโ‰ค๐ถ๐œ–๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ,(2.20) where 1/๐‘Ÿ1=1/3+1/๐‘ž and 1/๐‘Ÿ2=1/๐‘1+2/3. One can note that 1/๐‘Ÿ1โˆ’1/๐‘<2/3 and 1/๐‘Ÿ2โˆ’1/๐‘<2/3.
Step 3. Now, we want to prove 1<๐‘Ÿ<๐‘žโ‰ค3/2. For this case, we choose 3/2<๐‘ž1โ‰ค3 and ๐‘>3 such that 1๐‘Ÿโˆ’1๐‘ž1<13,1๐‘Ÿโˆ’1๐‘<23.(2.21) Similar to Step 2, we have (โ€–โˆ‡๐‘ข๐‘ก)โ€–๐‘žโ‰คโ€–โ€–โˆ‡๐‘‡(๐‘ก)๐ฎ0โ€–โ€–๐‘ž+๎€œ๐‘ก0([(]โ€–โ€–โˆ‡๐‘‡๐‘กโˆ’๐œ)๐‘ƒ๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ๐‘ž๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ๎€œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘Ÿ1โˆ’1/๐‘ž)โˆ’1/2โ€–๐ฐโ€–3โ€–โˆ‡๐ฎโ€–๐‘ž1๎€œ๐‘‘๐œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘Ÿ2โˆ’1/๐‘ž)โˆ’1/2โ€–๐ฎโ€–๐‘โ€–โˆ‡๐ฐโ€–3/2๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ,(2.22) where 1/๐‘Ÿ1=1/3+1/๐‘ž1 and 1/๐‘Ÿ2=1/๐‘+2/3. One can note that 1/๐‘Ÿ1โˆ’1/๐‘ž<1/3 and 1/๐‘Ÿ2โˆ’1/๐‘ž<1/3.
Step 4. At last, we want to prove 3<๐‘<โˆž with 1/๐‘Ÿโˆ’1/๐‘<1/3. In this case, we can do easily, by interpolation inequality, Steps 1 and 2.
Therefore, we complete the proof by Steps 1โ€“4.

Now, by applying the Helmholtz-Leray projection ๐‘ƒ into (1.16), we can obtain๐ฎ๐‘ก[]+โ„’๐ฎ+๐‘ƒ(๐ฎโ‹…โˆ‡)๐ฎ=0,for๐‘ก>0,๐ฎ(0)=๐ฎ0,(2.23) where ๎€บ๎€ท๐ฎโ„’๐ฎ=๐‘ƒโˆ’ฮ”๐ฎ+โˆž๎€ธ๎€ปโ‹…โˆ‡๐ฎ+(๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ=๐’ช๐ฎโˆž[],๐’Ÿ๐ฎ+๐‘ƒ(๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ๐‘(โ„’)=๐’Ÿ๐‘๎€ท๐’ช๐ฎโˆž๎€ธ=๎€ฝ๐‘ขโˆˆ๐ฝ๐‘(ฮฉ)โˆฉ๐‘Š2๐‘(ฮฉ)๐‘›|๐‘ข|๐œ•ฮฉ๎€พ.=0(2.24)

One can note from of [14, Lemma 2.6] that for 1<๐‘<โˆž and ๐ฎโˆˆ๐’Ÿ๐‘(โ„’)=๐’Ÿ๐‘(๐’ช๐ฎโˆž), โ€–๐ฎโ€–๐‘Š2,๐‘(ฮฉ)โ‰ค๐ถ๐‘๎‚€โ€–โ€–๐’ช๐ฎโˆž๐ฎโ€–โ€–๐‘+โ€–๐ฎโ€–๐‘๎‚.(2.25) Also, from (1.11), we have โ€–โ€–(๐ฐโ‹…โˆ‡)๐ฎ+(๐ฎโ‹…โˆ‡)๐ฐ๐‘โ‰ค๎€ทโ€–๐ฐโ€–โˆž+โ€–โˆ‡๐ฐโ€–โˆž๎€ธโ€–๐ฎโ€–๐‘Š2,๐‘(ฮฉ)โ‰ค||๐ฎโˆž||โ€–๐ฎโ€–๐‘Š2,๐‘(ฮฉ)โ‰ค๐ถ๐‘||๐ฎโˆž||๎‚€โ€–โ€–๐’ช๐ฎโˆž๐ฎโ€–โ€–๐‘+โ€–๐ฎโ€–๐‘๎‚.(2.26) Since the linear operator ๐’ช๐ฎโˆž generates an analytic semigroup ๐‘‡(๐‘ก) (refer to [14, 19]), we obtain an analytic semigroup ๐‘†(๐‘ก) generated by the linear operator โ„’ if |๐ฎโˆž| 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 1/๐‘Ÿโˆ’1/๐‘<2/3 for โ€–๐ฎ(๐‘ก)โ€–๐ฟ๐‘(ฮฉ) and 1/๐‘Ÿโˆ’1/๐‘<1/3 for โ€–โˆ‡๐ฎ(๐‘ก)โ€–๐ฟ๐‘(ฮฉ), because we have ๐ฎ(๐‘ฅ,๐‘ก) = ๐‘†(๐‘ก)๐ฎ0 = ๐‘†(๐‘ก/2)๐‘†(๐‘ก/2)๐ฎ0.

Now, we are in the position to prove Theorem 1.2. For the proof, we consider a solution ๐ฎ(๐‘ฅ,๐‘ก) (1.16) as the limit of the following usual iteration method:๐ฎ๐‘˜+1(๐‘ก)=๐‘†(๐‘ก)๐ฎ0โˆ’๎€œ๐‘ก0๐ฎ๐‘†(๐‘กโˆ’๐œ)๐‘ƒ๎€บ๎€ท๐‘˜๎€ธ๐ฎโ‹…โˆ‡๐‘˜๎€ป๐‘‘๐œ.(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 ๐‘>3, we have โ€–โˆ‡๐ฎ(๐‘ก)โ€–3<๐ถ๐‘กโˆ’1/2,โ€–๐ฎ(๐‘ก)โ€–๐‘<๐ถ๐‘กโˆ’1/2+3/2๐‘,โˆ€๐‘ก>0.(2.28)
Let ๐‘€๐‘˜๐‘=sup๐‘กโˆˆ[0,โˆž)๐‘ก1/2โˆ’3/2๐‘โ€–โ€–๐‘ข๐‘˜โ€–โ€–(๐‘ก)๐‘,for๐‘๐‘>3,๐‘˜3=sup๐‘กโˆˆ(0,โˆž)๐‘ก1/2โ€–โ€–โˆ‡๐‘ข๐‘˜โ€–โ€–(๐‘ก)3.(2.29) By Lemma 2.1 and (2.27), we obtain โ€–โ€–๐ฎ๐‘˜+1(โ€–โ€–๐‘ก)๐‘โ‰ค๐ถ๐‘กโˆ’1/2+3/2๐‘โ€–โ€–๐ฎ0โ€–โ€–3๎€œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’1/2โ€–โ€–๐ฎ๐‘˜(โ€–โ€–๐‘ก)๐‘โ€–โ€–โˆ‡๐ฎ๐‘˜(โ€–โ€–๐‘ก)3๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’1/2+3/2๐‘โ€–โ€–๐ฎ0โ€–โ€–3+๐ถ๐‘€๐‘˜๐‘๐‘๐‘˜3๎€œ๐‘ก0(๐‘กโˆ’๐œ)โˆ’1/2๐œโˆ’1/2+3/2๐‘๐œโˆ’1/2๐‘‘๐œโ‰ค๐‘กโˆ’1/2+3/2๐‘๎€บ๐ถโ€–โ€–๐ฎ0โ€–โ€–3+๐ถ๐‘€๐‘˜๐‘๐‘๐‘˜3๎€ป,(2.30) which implies ๐‘€๐‘๐‘˜+1โ€–โ€–๐ฎโ‰ค๐ถ0โ€–โ€–3+๐ถ๐‘€๐‘˜๐‘๐‘๐‘˜3.(2.31) Similarly, we have โ€–โ€–โˆ‡๐ฎ๐‘˜+1(โ€–โ€–๐‘ก)3โ‰ค๐ถ๐‘กโˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–3๎€œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2๐‘โˆ’1/2โ€–โ€–๐ฎ๐‘˜(โ€–โ€–๐‘ก)๐‘โ€–โ€–โˆ‡๐ฎ๐‘˜(โ€–โ€–๐‘ก)3๐‘‘๐œโ‰ค๐‘กโˆ’1/2๎€บ๐ถโ€–โ€–๐ฎ0โ€–โ€–3+๐ถ๐‘€๐‘˜๐‘๐‘๐‘˜3๎€ป,(2.32) which implies ๐‘3๐‘˜+1โ€–โ€–๐ฎโ‰ค๐ถ0โ€–โ€–3+๐ถ๐‘€๐‘˜๐‘๐‘๐‘˜3.(2.33)
Hence, we have ๐‘€๐‘๐‘˜+1+๐‘3๐‘˜+1โ€–โ€–๐ฎ<๐ถ0โ€–โ€–3๎€ท๐‘€+๐ถ๐‘˜๐‘+๐‘๐‘˜3๎€ธ2.(2.34)
Now, we have a sequence of the form ๐‘ฅ๐‘˜+1โ‰ค๐›ผ+๐›ฝ๐‘ฅ2๐‘˜,(2.35) and we know that such sequence satisfies ๐‘ฅ๐‘˜โ‰ค๐‘€โ‰ก1โˆ’(1โˆ’4๐›ผ๐›ฝ)1/2<12๐›ฝ,2๐›ฝif1๐›ผ<.4๐›ฝ(2.36)
Therefore, by recurrence estimates, smallness of โ€–๐ฎ0โ€–3 implies ๐‘€๐‘๐‘˜+1+๐‘3๐‘˜+1<๐พ,(2.37) for some constant ๐พ. Finally, we obtain โ€–โˆ‡๐ฎ(๐‘ก)โ€–3<๐ถ๐‘กโˆ’1/2,โ€–๐ฎ(๐‘ก)โ€–๐‘<๐ถ๐‘กโˆ’1/2+3/2๐‘,โˆ€๐‘ก>0.(2.38)

Step 2. We prove that if 3/2<๐‘ with 1/๐‘Ÿโˆ’1/๐‘<1/3 and ๐ฎ0โˆˆ๐ฟ๐‘Ÿ(ฮฉ)โˆฉ๐ฟ3(ฮฉ), then we have โ€–๐ฎ(๐‘ก)โ€–๐‘โ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘),โˆ€๐‘ก>0.(2.39) Let ๐‘€๐‘=sup๐‘กโˆˆ(0,โˆž)๐‘ก3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–๐‘ข(๐‘ก)โ€–๐‘.(2.40)
From estimates of Step 1, one can note that we have โ€–โˆ‡๐ฎ(๐‘ก)โ€–3โ‰ค๐ถ๐‘กโˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–3,โˆ€๐‘ก>0.(2.41)
So, we have (โ€–๐ฎ๐‘ก)โ€–๐‘โ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ๎€œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ8โˆ’1/๐‘)(โ€–๐ฎ๐‘ก)โ€–๐‘(โ€–โˆ‡๐ฎ๐‘ก)โ€–3๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–3๎€œ๐‘ก0(๐‘กโˆ’๐œ)โˆ’1/2๐œโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘)๐œโˆ’1/2๐‘‘๐œโ‰ค๐‘กโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘)๎€บ๐ถโ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–3๐‘€๐‘๎€ป,(2.42) which implies ๐‘€๐‘โ€–โ€–๐ฎ<๐ถ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–3๐‘€๐‘,(2.43) where 1/๐‘Ÿ8=1/3+1/๐‘.
Hence, we complete the proof with ๐ถโ€–๐ฎ0โ€–3<1.

Step 3. We prove that if 3/2<๐‘žโ‰ค3 with 1/๐‘Ÿโˆ’1/๐‘ž<1/3 and ๐ฎ0โˆˆ๐ฟ๐‘Ÿ(ฮฉ)โˆฉ๐ฟ3(ฮฉ), then we have โ€–โˆ‡๐ฎ(๐‘ก)โ€–๐‘žโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2,โˆ€๐‘ก>0.(2.44)
Let ๐‘๐‘ž=sup๐‘กโˆˆ(0,โˆž)๐‘ก๐‘›/2(1/๐‘Ÿโˆ’1/๐‘ž)+1/2โ€–โˆ‡๐‘ข(๐‘ก)โ€–๐‘ž.(2.45) We choose some ๐‘1โ‰ˆ3 with ๐‘1>3 such that โ€–โˆ‡๐ฎโ€–๐‘žโ‰ค๐ถ๐‘กโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ๎€œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’๐‘›/2(1/๐‘Ÿ7โˆ’1/๐‘ž)โˆ’1/2โ€–๐ฎโ€–๐‘1โ€–โˆ‡๐ฎโ€–๐‘ž๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–3๐‘๐‘ž๎€œ๐‘ก0(๐‘กโˆ’๐œ)โˆ’1/2โˆ’3/2๐‘๐œโˆ’1/2+3/2๐‘1๐œโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2๐‘‘๐œโ‰ค๐‘กโˆ’๐‘›/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2๎€บ๐ถโ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–3๐‘๐‘ž๎€ป.(2.46) So we complete the proof with ๐ถโ€–๐ฎ0โ€–3<1.

Step 4. We prove that if 1<๐‘Ÿ<๐‘<โˆž, 1<๐‘Ÿ<3, and ๐ฎ0โˆˆ๐ฟ๐‘Ÿ(ฮฉ)โˆฉ๐ฟ3(ฮฉ), then we have โ€–๐ฎ(๐‘ก)โ€–๐‘โ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘),โˆ€๐‘ก>0.(2.47)
Case 1 (let ๐‘>3/2). Since we proved for 1/๐‘Ÿโˆ’1/๐‘<1/3 in Step 2, we can assume that 1/3โ‰ค1/๐‘Ÿโˆ’1/๐‘. One notes that we can rewrite solutions ๐ฎ(๐‘ก) in the form ๎‚€๐‘ก๐ฎ(๐‘ก)=๐‘†2๎‚๐ฎ๎‚€๐‘ก2๎‚โˆ’๎€œ๐‘ก๐‘ก/2[(]๐‘†(๐‘กโˆ’๐œ)๐‘ƒ๐ฎโ‹…โˆ‡)๐ฎ๐‘‘๐œ.(2.48) For any ๐‘Ÿ>1, we choose ๐‘™>3/2 such that 1/๐‘Ÿโˆ’1/๐‘™<1/3 and 1/๐‘™โˆ’1/๐‘<2/3. Also, for any 1<๐‘Ÿ<๐‘โ‰คโˆž with 1<๐‘Ÿ<3, we choose ๐‘ 1>3 and 3/2<๐‘ 2<3 such that 1๐‘Ÿโˆ’1๐‘ 2<13,1๐‘ 1+1๐‘ 2โˆ’1๐‘<23.(2.49) Then, by Steps 1โ€“3, we have (โ€–๐ฎ๐‘ก)โ€–๐‘โ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘™โˆ’1/๐‘)โ€–โ€–โ€–๐ฎ๎‚€๐‘ก2๎‚โ€–โ€–โ€–๐‘™๎€œ+๐ถ๐‘ก๐‘ก/2(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘ โˆ’1/๐‘)โ€–(๐ฎโ‹…โˆ‡)๐ฎโ€–๐‘ ๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–๐‘Ÿ๎€œ๐‘ก๐‘ก/2(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘ โˆ’1/๐‘)๐œโˆ’1/2โˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ 2)๐œโˆ’1/2+3/2๐‘ 1๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ,โˆ€๐‘ก>0.(2.50)
Case 2 (let 1<๐‘โ‰ค3/2). By Step 1โ€“3, we have (โ€–๐ฎ๐‘ก)โ€–๐‘โ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ๎€œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘ โˆ’1/๐‘)โ€–(๐ฎโ‹…โˆ‡)๐ฎโ€–๐‘ ๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–๐‘Ÿ๎€œ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘ โˆ’1/๐‘)๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ 1)๐œโˆ’1/2๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘)โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ,โˆ€๐‘ก>0,(2.51) where ๐‘ 1>3/2, 1/๐‘Ÿโˆ’1/๐‘ 1<1/3, 1/๐‘ =1/๐‘ 1+1/3.

Step 5. We prove that if 1<๐‘Ÿ<๐‘žโ‰ค3 and ๐ฎ0โˆˆ๐ฟ๐‘Ÿ(ฮฉ)โˆฉ๐ฟ3(ฮฉ), then โ€–โˆ‡๐ฎ(๐‘ก)โ€–๐‘žโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2.(2.52)
Case 1 (let 3/2<๐‘žโ‰ค3). Since we proved 1/๐‘Ÿโˆ’1/๐‘ž<1/3 in Step 3, we can assume that 1/3โ‰ค1/๐‘Ÿโˆ’1/๐‘ž. Now, we choose ๐‘™>3/2 such that 1/๐‘Ÿโˆ’1/๐‘™<1/3 and 1/๐‘™โˆ’1/๐‘ž<1/3. We also can have ๐‘ 1>3 and 3/2<๐‘ 2<3 with 1๐‘ =1๐‘ 1+1๐‘ 2,1๐‘Ÿโˆ’1๐‘ 2<13,1๐‘ โˆ’1๐‘ž<13.(2.53) So, by Step 1โ€“4, we obtain (โ€–โˆ‡๐ฎ๐‘ก)โ€–๐‘žโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘™โˆ’1/๐‘ž)โˆ’1/2โ€–โ€–โ€–๐ฎ๎‚€๐‘ก2๎‚โ€–โ€–โ€–๐‘™๎€œ+๐ถ๐‘ก๐‘ก/2(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘ โˆ’1/๐‘ž)โˆ’1/2(โ€–๐ฎ๐‘ก)โ€–๐‘ 1(โ€–โˆ‡๐ฎ๐‘ก)โ€–๐‘ 2๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–๐‘Ÿร—๎€œ๐‘ก๐‘ก/2(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘ โˆ’1/๐‘ž)โˆ’1/2๐œโˆ’1/2+3/2๐‘ 1๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ 2)โˆ’1/2๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ.(2.54)
Case 2 (let 1<๐‘žโ‰ค3/2). By Step 1-Step 3, we have (โ€–โˆ‡๐ฎ๐‘ก)โ€–๐‘žโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ๎€œ+๐ถ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘ โˆ’1/๐‘ž)โˆ’1/2โ€–(๐ฎโ‹…โˆ‡)๐ฎโ€–๐‘ ๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿโ€–โ€–๐ฎ+๐ถ0โ€–โ€–๐‘Ÿ๎€œ๐‘ก0(๐‘กโˆ’๐œ)โˆ’3/2(1/๐‘ โˆ’1/๐‘ž)โˆ’1/2๐œโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ 1)๐œโˆ’1/2๐‘‘๐œโ‰ค๐ถ๐‘กโˆ’3/2(1/๐‘Ÿโˆ’1/๐‘ž)โˆ’1/2โ€–โ€–๐ฎ0โ€–โ€–๐‘Ÿ,โˆ€๐‘ก>0,(2.55) where ๐‘ 1>3/2, 1/๐‘Ÿโˆ’1/๐‘ 1<1/3, 1/๐‘ =1/๐‘ 1+1/3, and 1/๐‘ โˆ’1/๐‘ž<1/3.
Therefore, by Step 1โ€“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] โ€œWeighted ๐ฟ๐‘-Estimates for Stokes Flow in ๐‘…๐‘›+, with Applications to the Non-Stationary Navier-Stokes Flowโ€ 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).

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