Jaiok Roh, " Stability of the Incompressible Flows with Nonzero Far-Field Velocity", Abstract and Applied Analysis, vol. 2011, Article ID 856896, 15 pages, 2011. https://doi.org/10.1155/2011/856896
Stability of the Incompressible Flows with Nonzero Far-Field Velocity
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 .
Now, we set in (1.1) and have
Consider the following linear problem: which is referred to as the Oseen equations; see .
In order to formulate the problem (1.3), Enomoto and Shibata  used the Helmholtz decomposition: where , The Helmholtz decomposition of was proved by Fujiwara and Morimoto , Miyakawa , and Simader and Sohr . 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  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  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 . 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  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  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  have studied the temporal stability in space. Shibata  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  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  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  for and a very well-known method. Also, for with , Kozono and Ogawa  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 1–4.
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 = = .
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
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 1–3, we have
Case 2 (let ). By Step 1–3, 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 1–4, we obtain
Case 2 (let ). By Step 1-Step 3, we have where , , , and .
Therefore, by Step 1–5, we complete the proof of Theorem 1.2.
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