Research Article | Open Access
Jaiok Roh, "Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity", Abstract and Applied Analysis, vol. 2011, Article ID 369571, 15 pages, 2011. https://doi.org/10.1155/2011/369571
Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity
Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in space and by Shibata (1999) and Enomoto-Shibata (2005) in spaces for . However, their results did not include enough information to find the spatial decay. So, Bae-Roh (2010) improved Enomoto-Shibata's results in some sense and estimated the spatial decay even though their results are limited. In this paper, we will prove temporal decay with a weighted function by using decay estimates obtained by Roh (2011). Bae-Roh (2010) proved the temporal rate becomes slower by if a weighted function is for . In this paper, we prove that the temporal decay becomes slower by where if a weighted function is . For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted -norm of solutions. This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008).
When a boat is sailing with a constant velocity , we may think that the water is flowing around the fixed boat with opposite velocity 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 is an exterior domain in with a smooth boundary and denotes a given constant vector describing the velocity of the fluid at infinity. For , 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 . We set in (1.1) and have
In order to formulate the problem (1.3), Enomoto and Shibata  used the Helmholtz decomposition: where , The Helmholtz decomposition of was proved by Fujiwara-Morimoto , Miyakawa , and Simader-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 [17, 19]) and obtained the following properties.
Proposition 1.1. Let and assume that . Let . Then, where and , where and .
And, for the nonstationary Navier-Stokes equations, we discuss the stability of stationary solution of the nonlinear Navier-Stokes equation (1.2), and satisfies the following equations:
For suitable , Shibata  proved that for any given there exists such that if , then one has for small , where is independent of .
By setting and for in (1.2) and (1.11), 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. Heywood [22, 23], Masuda , Shibata , Enomoto-Shibata , Bae-Roh , and Roh  have studied the temporal decay for solutions of (1.13), and we have the followings in .
Proposition 1.2. There exists small such that if , and , then a unique solution of (1.13) has where .
Now, we are in the position to introduce our main theorems which are the weighted stability of stationary solution .
Theorem 1.3. Let and . Then there exists small such that if , , , and , then the solution of (1.13) satisfies where .
Remark 1.4. In Theorem 1.3, the assumption is for simple calculations. We also can obtain a similar result where . For the proof we have to consider delay solution = . Then we can follow the method in Bae and Roh .
Theorem 1.5. Let for and . Then there exists small such that if , , , and , then the solution of (1.13) satisfies where .
Remark 1.6. For the exterior Navier-Stokes flows with , temporal decay rate with weight function becomes slower by ; refer to [1–4, 8, 13]. However, for , we found out from Theorems 1.3 and 1.5 that temporal decay rate with weight function becomes slower by for . In fact, Bae and Roh  concluded that it becomes slower by for . Hence, our decay rate is little faster than the one in Bae and Roh  for .
One of the difficulties for the exterior Navier-Stokes equations is dealing with the boundary of 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 vanishing near the boundary. This astonied method for exterior problem was initiated by He and Xin  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 Navier-Stokes equations with nonzero far-field velocity. We first consider for a weight function and then for . Our method can be applied to the Oseen equations. As a result, we note that we can improve the result of Bae-Jin  by the same method.
2.1. Proof of Theorem 1.3
We define for large , where is a nonnegative cutoff function with , for , and for . When there is no confusion, we use the same notation instead of for convenience.
As in , we set where is the fundamental function of , that is, . By the definition of , we have . Moreover, where
We first estimate and then obtain the estimate of .
Now, we consider the fundamental solutions for the nonstationary Oseen equations, written as where (refer to [15, 28]). In fact, is a translation in the direction of by of the heat kernel , that is, . Set , , where is the standard unit vector of which the th term is 1. Then, we have Hence, we have the identity where
From straightforward calculations we have that for ,
One might note that we may sometimes use instead of because of technical reason. By the definition of , both inequalities hold for any . We multiply (1.13) by and integrate over , and then we have
Applying Young’s convolution and the Calderon-Zygmund inequalities, we obtain if and .
And is bounded by as follows: where , and .
We have where and . Also, we obtain where . Finally, we get where . Hence, for any , we have
Also, we obtain where , and . In the above calculation, we used instead of because of simplicity of calculations.
And we have where and .
Next, for , we have
We get where and . In the above calculation, we used instead of because of simplicity of calculations.
Since (see ), we have where , and . In the above calculation, we used instead of because of simplicity of calculations.
Next, for any , we have where , and .
Hence, we have
Consider as follows:
We have, for any , where , , and . In the above calculation, we used instead of because of technical reason.
Similar to , we get where , , , and ,.
Since for , , from Shibata , we have
Hence, we have
Now, we use the following lemma (refer to ).
Lemma 2.1. Let a function satisfy the inequality, for some , One also assumes that Then, there is so that if , then one has for some independent of .
condition (2.33) satisfies for .
So, by Lemma 2.1, we have
2.2. Proof of Theorem 1.5
By using the results in previous section, for any , we have small such that where .
Now, in this section, we consider , where .
Similar to previous section, for , , and , we obtain the same decay rate with previous section. And for any , we have where and . Also, for , we obtain where , and .
Next, we have where , , and . Also, since , we get where , , and .
And we obtain where , , and .
Next, for any , we have where , , , and .
Hence, we have
Consider as follows:
We have, for any , where , , , and .
Similar to , we get where , , , , , , , and . In the above calculation, we used instead of because of technical reason. Now, we have where , , , and .
So, we obtain which completes the proof.
The author would like to express her appreciation to Professors Hyeong-Ohk Bae and Bum Ja Jin for valuable comments. The author would like to thank the referee for helpful comments. This research was supported by the 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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