Abstract and Applied Analysis

Volumeย 2011, Article IDย 856896, 15 pages

http://dx.doi.org/10.1155/2011/856896

## 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 for the exterior Navier-Stokes flows with a nonzero constant velocity at infinity. For with nonzero stationary solution , Chen (1993), Kozono and Ogawa (1994), and Borchers and Miyakawa (1995) have studied the temporal stability in spaces for and obtained good stability decay rates. For the spatial direction, we recently obtained some results. For , Heywood (1970, 1972) and Masuda (1975) have studied the temporal stability in space. Shibata (1999) and Enomoto and Shibata (2005) have studied the temporal stability in spaces for . 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 for and obtain stability as Kozono and Ogawa and Borchers and Miyakawa obtained for .

#### 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: where is an exterior domain in with a smooth boundary , and denotes a given constant vector describing the velocity of the fluid at infinity. In this paper, we consider a nonzero constant . The physical model of the exterior Navier-Stokes equations with a nonzero constant can be considered as the motion of water in the sea when a boat is moving with the speed , while the one with zero constant can be considered when a boat is stopped. There are few known results for the case , while, with , many results were obtained for the temporal decay and weighted estimates of solutions of (1.1) (refer [1โ12]).

Now, we set in (1.1) and have

Consider the following linear problem: 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: where , The Helmholtz decomposition of was proved by Fujiwara and Morimoto [15], Miyakawa [16], and Simader and Sohr [17]. Let be a continuous projection from onto .

By applying into (1.3) and setting = , one has where the domain of is given by Then, Enomoto and Shibata [14] proved that generates an analytic semigroup which is called the Oseen semigroup (one can also refer to [16, 18]) and obtained the following properties.

Proposition 1.1. *Let and assume that . Let , then
**
where and ,
**
where and .*

The main purpose of this paper is to discuss the temporal stability of stationary solution of the nonlinear Navier-Stokes equation (1.2). One can note that satisfies the following equations:

For suitable , Shibata [19] proved that, for any given , there exists such that if , then one has where Throughout this paper, we assume that satisfies the assumption in Shibata [19]. Now, we consider the polar coordinate system for , , and . Let be an orthogonal matrix such that = and put = . By a change of variable , See Shibata [19] for the detail. Now, by using the above change of variable, we can see easily that satisfies for small , , where is independent on .

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 , by setting and for , , , in (1.2) and (1.10), we have the following equations in : Here, in fact, the initial data should be , but for our convenience, we denote by for if there is no confusion.

First, Heywood [21, 22] and Masuda [23] have studied the temporal stability in space. Shibata [19] proved that there exists small such that if and , then a unique solution of (1.16) has the following properties: for any , where

After that, Enomoto and Shibata [14] considered the stability for arbitrary by deleting the smallness condition of . But in this case, all constants in their results depend on when . Also, they assumed the existence of stationary solution with for small , and . Then, as a result, they proved (1.16) has a unique solution with as when is small enough in the space .

Also, Bae and Roh [24] improved Enomoto-Shibata's result in some sense. But their result is limited in the space for , while we consider all . Moreover, their result depends on and , while ours only depends on , where and . Also, their optimal decay rate is , while ours is .

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 and a very well-known method. Also, for with , Kozono and Ogawa [12] also used similar method.

Theorem 1.2. *There exists small such that if and , then a unique solution of (1.16) has the following properties:
**
where .*

#### 2. Proof of Main Theorem

First, we consider the following linear problem:

By applying Helmholtz-Leray projection and setting we have And we note that the domain of is

Let be a semigroup generated by the linear operator , then, by Duharmel's Principle, a solution of (2.1) can be written as in the following integral form, where is an analytic semigroup generated by the Oseen operator .

Lemma 2.1. *Let for , then there exists a small such that if and , then a solution represented by (2.5) satisfies with ,
**
and for with ,
*

*Proof. *Before we prove Lemma 2.1 note from (1.15) that we have
for small . In fact, by straight calculations, we can choose any .*Step 1. *Let with and with . We consider the following iteration method to obtain our estimates:
We let and
If , then by Proposition 1.1, for small , we have
where and . If , then we have
where . So, we obtain
which implies
Similarly, we obtain for ,
where . Also, for , we have

Therefore, we get
So if (the constant C is bounded as goes to zero, so we can make by choosing small ), then we have some such that
for all . Hence, by taking the limit, we complete the proof.*Step 2. *Now, we want to prove . For this case, we choose and such that
Then, we have
where and . One can note that and .*Step 3. *Now, we want to prove . For this case, we choose and such that
Similar to Step 2, we have
where and . One can note that and .*Step 4. *At last, we want to prove with . In this case, we can do easily, by interpolation inequality, Steps 1 and 2.

Therefore, we complete the proof by Steps 1โ4.

Now, by applying the Helmholtz-Leray projection into (1.16), we can obtain where

One can note from of [14, Lemma 2.6] that for and , Also, from (1.11), we have Since the linear operator generates an analytic semigroup (refer to [14, 19]), we obtain an analytic semigroup generated by the linear operator if 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 for and for , because we have = = .

Now, we are in the position to prove Theorem 1.2. For the proof, we consider a solution (1.16) as the limit of the following usual iteration method:

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 , we have

Let
By Lemma 2.1 and (2.27), we obtain
which implies
Similarly, we have
which implies

Hence, we have

Now, we have a sequence of the form
and we know that such sequence satisfies

Therefore, by recurrence estimates, smallness of implies
for some constant . Finally, we obtain

*Step 2. *We prove that if with and , then we have
Let

From estimates of Step 1, one can note that we have

So, we have
which implies
where .

Hence, we complete the proof with .

*Step 3. *We prove that if with and , then we have

Let
We choose some with such that
So we complete the proof with .

*Step 4. *We prove that if , , and , then we have
*Case 1 (let ). *Since we proved for in Step 2, we can assume that . One notes that we can rewrite solutions in the form
For any , we choose such that and . Also, for any with , we choose and such that
Then, by Steps 1โ3, we have
*Case 2 (let ). *By Step 1โ3, we have
where , , .

*Step 5. *We prove that if and , then
*Case 1 (let ). *Since we proved in Step 3, we can assume that . Now, we choose such that and . We also can have and with
So, by Step 1โ4, we obtain
*Case 2 (let ). *By Step 1-Step 3, we have
where , , , and .

Therefore, by Step 1โ5, 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] โWeighted -Estimates for Stokes Flow in , 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).

#### References

- H.-O. Bae and B. J. Jin, โAsymptotic behavior for the Navier-Stokes equations in 2D exterior domains,โ
*Journal of Functional Analysis*, vol. 240, no. 2, pp. 508โ529, 2006. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - H.-O. Bae and B. J. Jin, โTemporal and spatial decay rates of Navier-Stokes solutions in exterior domains,โ
*Bulletin of the Korean Mathematical Society*, vol. 44, no. 3, pp. 547โ567, 2007. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - H.-O. Bae and J. Roh, โWeighted estimates for the incompressible fluid in exterior domains,โ
*Journal of Mathematical Analysis and Applications*, vol. 355, no. 2, pp. 846โ854, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - H.-O. Bae and J. Roh, โOptimal weighted estimates of the flows in exterior domains,โ
*Nonlinear Analysis*, vol. 73, no. 5, pp. 1350โ1363, 2010. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - W. Borchers and T. Miyakawa, โOn stability of exterior stationary Navier-Stokes flows,โ
*Acta Mathematica*, vol. 174, no. 2, pp. 311โ382, 1995. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - Z. M. Chen, โSolutions of the stationary and nonstationary Navier-Stokes equations in exterior domains,โ
*Pacific Journal of Mathematics*, vol. 159, no. 2, pp. 227โ240, 1993. View at Google Scholar ยท View at Zentralblatt MATH - C. He, โWeighted estimates for nonstationary Navier-Stokes equations,โ
*Journal of Differential Equations*, vol. 148, no. 2, pp. 422โ444, 1998. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - C. He and T. Miyakawa, โOn weighted-norm estimates for nonstationary incompressible Navier-Stokes flows in a 3D exterior domain,โ
*Journal of Differential Equations*, vol. 246, no. 6, pp. 2355โ2386, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - H. Iwashita, โ${L}_{q}\text{-}{L}_{p}$ estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in ${L}_{q}$ spaces,โ
*Mathematische Annalen*, vol. 285, no. 2, pp. 265โ288, 1989. View at Publisher ยท View at Google Scholar - H. Kozono and T. Ogawa, โSome ${L}^{p}$ estimate for the exterior Stokes flow and an application to the nonstationary Navier-Stokes equations,โ
*Indiana University Mathematics Journal*, vol. 41, no. 3, pp. 789โ808, 1992. View at Publisher ยท View at Google Scholar - H. Kozono and T. Ogawa, โTwo-dimensional Navier-Stokes flow in unbounded domains,โ
*Mathematische Annalen*, vol. 297, no. 1, pp. 1โ31, 1993. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - H. Kozono and T. Ogawa, โOn stability of Navier-Stokes flows in exterior domains,โ
*Archive for Rational Mechanics and Analysis*, vol. 128, no. 1, pp. 1โ31, 1994. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - C. W. Oseen,
*Neuere Methoden und Ergebnisse in der Hydrodynamik*, Akademische, Leipzig, Germany, 1927. - Y. Enomoto and Y. Shibata, โOn the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation,โ
*Journal of Mathematical Fluid Mechanics*, vol. 7, no. 3, pp. 339โ367, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - D. Fujiwara and H. Morimoto, โAn ${L}_{r}$-theorem of the Helmholtz decomposition of vector fields,โ
*Journal of the Faculty of Science, the University of Tokyo: Section I*, vol. IX, pp. 59โ102, 1961. View at Google Scholar - T. Miyakawa, โOn nonstationary solutions of the Navier-Stokes equations in an exterior domain,โ
*Hiroshima Mathematical Journal*, vol. 12, no. 1, pp. 115โ140, 1982. View at Google Scholar ยท View at Zentralblatt MATH - C. G. Simader and H. Sohr, โThe Helmholtz decomposition in ${L}^{q}$ and related topics,โ in
*Mathematical Problems Related to the Navier-Stokes Equations*, G. P. Galdi, Ed., vol. 11 of*Advances in Mathematics for Applied Science*, pp. 1โ35, World Scientific Publishing, River Edge, NJ, USA, 1992. View at Google Scholar - Y. Enomoto and Y. Shibata, โLocal energy decay of solutions to the Oseen equation in the exterior domains,โ
*Indiana University Mathematics Journal*, vol. 53, no. 5, pp. 1291โ1330, 2004. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - Y. Shibata, โOn an exterior initial-boundary value problem for Navier-Stokes equations,โ
*Quarterly of Applied Mathematics*, vol. 57, no. 1, pp. 117โ155, 1999. View at Google Scholar ยท View at Zentralblatt MATH - G. P. Galdi,
*An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II: Nonlinear Steady Problem*, vol. 39 of*Springer Tracts in Natural Philosophy*, Springer, New York, NY, USA, 1994. - J. G. Heywood, โOn stationary solutions of the Navier-Stokes equations as limits of nonstationary solutions,โ
*Archive for Rational Mechanics and Analysis*, vol. 37, pp. 48โ60, 1970. View at Google Scholar - J. G. Heywood, โThe exterior nonstationary problem for the Navier-Stokes equations,โ
*Acta Mathematica*, vol. 129, no. 1-2, pp. 11โ34, 1972. View at Google Scholar ยท View at Zentralblatt MATH - K. Masuda, โOn the stability of incompressible viscous fluid motions past objects,โ
*Journal of the Mathematical Society of Japan*, vol. 27, pp. 294โ327, 1975. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH - H.-O. Bae and J. Roh, โStability for the 3D Navier-Stokes Equations with nonzero far field velocity on exterior domains,โ
*Journal of Mathematical Fluid Mechanics*. In press. View at Publisher ยท View at Google Scholar - T. Kato, โStrong ${L}^{p}$-solutions of the Navier-Stokes equation in ${\mathbb{R}}^{n}$, with applications to weak solutions,โ
*Mathematische Zeitschrift*, vol. 187, no. 4, pp. 471โ480, 1984. View at Publisher ยท View at Google Scholar - T. Kato,
*Perturbation Theory for Linear Operators*, Springer, New York, NY, USA, 1980. - C. He and L. Wang, โWeighted ${L}^{p}$-estimates for Stokes flow in ${R}^{n}{}_{+}$ with applications to the non-stationary Navier-Stokes flow,โ
*Science China. Mathematics*, vol. 54, no. 3, pp. 573โ586, 2011. View at Publisher ยท View at Google Scholar