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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 240387, 9 pages
The Inviscid Limit Behavior for Smooth Solutions of the Boussinesq System
College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang 314001, China
Received 13 January 2013; Accepted 30 March 2013
Academic Editor: Jesús I. Díaz
Copyright © 2013 Junlei Zhu. 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.
The inviscid limit problem for the smooth solutions of the Boussinesq system is studied in this paper. We prove the convergence result of this system as the diffusion and the viscosity coefficients vanish with the initial data belonging to . Moreover, the convergence rate is given if we allow more regularity on the initial data.
1. Introduction and the Main Results
The two-dimensional Boussinesq system for the homogeneous incompressible fluids with diffusion and viscosity is given by where the space variable is in , is the velocity, denotes the scalar pressure and the scalar temperature, , and and denote, respectively, the molecular diffusion and the viscosity. Such Boussinesq systems are simple models widely used in the modeling of oceanic and atmospheric motions, and these models also appear in many other physical problems; see [1, 2] for more discussions. It is also interesting to consider the system (1) without diffusion and viscosity namely (for the sake of convenience for our limit argument, we use different notation), Moreover, it is known that the two-dimensional viscous (resp., inviscid) Boussinesq equations are closely related to the three-dimensional axisymmetric Navier-Stokes equations (resp., Euler equations) with swirl. Therefore, the Boussinesq systems, especially in two-dimensional case, have been widely studied by many researchers, and we refer, for instance, to [3–9] and the references therein.
It is well known that the system (1) has a unique global in time regularity solution. Moreover, Hou and Li in  obtained the global existence of smooth solution for (1) even with the zero diffusivity case (i.e., and ). Meanwhile, Chae in  also proved global regularity for the 2D Boussinesq system (1) both with the zero diffusivity case ( and ) and the zero viscosity case ( and ). However, for the case and , it is only known that smooth solution exists locally in time (see, e.g., ), and it is not known whether such smooth solutions can develop singularities in finite time. In fact, as well as the famous blow-up problem for the Navier-Stokes equations or Euler equations, the regularity or singularity question for the locally smooth solution of the system (2) appears also as an outstanding open problem in the mathematical fluid mechanics; see .
In this paper, we are interested in studying the limit behavior of the smooth solution for (1) as ; that is, we study the vanishing viscosity limit of solutions of (1). This type limit problem appears not only in the community of applied mathematics, but also in physical reality. A good example of this problem is the vanishing viscosity limit of solutions of the Navier-Stokes equations which appears as a singular limit especially in bounded domains due to the boundary layers effect, and we refer to [11–17] and the references therein. Most of the previous convergence results require some loss of derivatives; namely, if the initial data lies in the space , usually one can obtain the convergence results in with . In this literature, Masmoudi in  obtained the inviscid limit results for the Navier-Stokes equations without loss of derivatives. Inspired by , in this paper, we obtain the convergence of the solution with the initial data belonging to the same space.
Now we state our main results of the paper.
Theorem 1. Let , , and satisfying , , and as . Assume that is the classical solution of (1) with initial data , and is the classical solution of the inviscid system (2) with initial data ; here; is the maximal existence time of the solution . Then, for any and for any , one has(1)(convergence rate in the norm) (2) (convergence rate in the norm with ) (3) (convergence rate in the norm with ) (4) (convergence in the norm) where the constants in (4)–(6) are dependent of and the norm of the initial data , but independent of the parameters and .
Of course, we can generalize the previous results to arbitrary spatial dimension case with replaced by . The important part of Theorem 1 is the convergence result (7). This result tells us that the convergence can be maintained by the solution at its arbitrary existence time. We emphasize that is not assumed to be small; indeed, the standard energy estimate yields that the classical solution blows up at time if and only if as . Note that the rate of convergence depends on how we regularize our initial data; see (75) in the next section. Moreover, if one allows more regularity on the initial data , then we can obtain the following convergence rate.
Theorem 2. Suppose that the same assumptions as Theorem 1 hold. Moreover, one assumes that with . Then, for any , there hold where the constants and depend only on , , and .
Finally, we end this section by setting some notations which will be used throughout the paper. For , denotes the norm in the Lebesgue space . We set the operator , and for , we denote by the nonhomogeneous Sobolev spaces with the norm defined as If , for brevity, we write instead of . Obviously, . In some places, we use the notation to mean that this space consists of vector-valued functions with each component of belonging to . If there is no confusion, the spaces and will be simply denoted by . For , we denote by the usual inner product of and ; namely, For any Banach space , the space consists of all strongly measurable functions equipped with the norm for , and And the space denotes the set of continuous functions with In this paper, the letter is a generic constant and its value may change at each appearance. Moreover, every is independent of the parameters and .
2. Proof of Theorem 1
Lemma 3. Assume that and . If , the Schwartz class, then with such that
Lemma 4. Let . Then,(1)for any with , there hold (2)for any , with , there holds
Proof. Using the divergence free condition and the communicator estimate (15), one sees where and satisfy . Since and , we can always choose such that and . In the case , we choose and . Hence, the estimates (18) and (19) follow immediately. For the estimate (20), the case is treated by the Hölder inequality, and for , we use (16); then, With the condition, we can choose and satisfying , , and , and thus (20) follows.
Lemma 5. With the same hypotheses as Theorem 1, then there exist and such that for sufficiently small , , there holds with and both depending only on the norm of and not depending on and .
Proof. From the first equation of (1), we have Multiplying this equation by and integrating the result, while noting that where we have used the estimate (18) in the previous inequality, then we obtain which gives Using the same argument to the second equation of (1), we can obtain Then, we have Hence, it concludes from the estimates (27) and (29) that Solving this ODE gives Since (3) holds, we may assume for small and that here, depends only on and . Hence, we finally arrive at The estimate (23) follows from the previous inequality provided that we select such that (e.g., we can choose ). The proof of Lemma 5 is complete.
Remark 7. For fixed , without loss of generality, we may assume that the time determined by Lemma 5 satisfies . Indeed, as will be seen in the proof of Theorem 1, no matter how small the is, we can always use bootstrap argument to extend the interval into our desired time interval .
In the following, we define and . Considering the equations, for and , we see satisfying For the sake of convenience, we often omit the superscripts and in the succeeding arguments; hence, means , stands for , and so on.
Proof of Theorem 1. We split the proof into several steps.
Step 1. We first show that (4) holds on . By using (18) and (20) with , we can obtain the energy for as where we have used the uniform estimates (23) and (33) in the last step. Hence, we get Similarly, the energy estimate for can be written as which gives Therefore, one has Then, the Gronwall inequality yields that for all , where with depending on and .
Step 2. We show that (5) holds on . Applying the estimates (18) and (20) with , then we can easily obtain the energy for as Since , we have . By the Gagliardo-Nirenberg interpolation inequality and Young's inequality, we have Inserting this inequality into (41) and using Young's inequality again, we thus get With similar argument as abovementioned, we can also obtain the energy for as The previous two estimates give and by the Gronwall inequality we get which implies that the estimate (5) holds on with .
Step 3. To prove (7), we need to regularize the initial data. Define by where the constant is selected so that . Let , and define the mollification of by . By this definition, one can see ; moreover, if , we have in as and For the proof of these properties, see Lemma 3.5 in . Now let be the solution of the inviscid system (2) with initial data ; namely, solves the equations So, the energy for and can be written as With the same discussion as Lemma 5, we know that there exist and both only depending on the norm of such that Moreover, taking the energy for and and using (19), then for , Using (48), we deduce from the Previous energy estimate that where depends only on , , and . Without loss of generality, in the following, we may assume , where that is determined by Lemma 5.
Step 4. Set and ; then, from (2) and (49), we know that satisfies Using (18) with , we have By the Cauchy-Schwarz inequality and (16), one has With these two estimates, it is easy to obtain the following energy estimate for : where we have used (54) and the uniform estimate (51) in the last step. Similarly, we can obtain the following energy estimate for : Now, we have to estimate . From (55), we take the energy for and obtain (using (18) and (20)) Then, the Gronwall inequality and (48) yield for all , which in turn by Sobolev embedding theorem gives since . Inserting this estimate into (58) and (59), we can see where we have used the relation in the last step since the value of at each appearance may be different. Hence, the Gronwall inequality gives
Step 5. Let , , and recall that here and ; so, one can deduce from (2) and (55) that solves Using the same reasonings that lead to (58), we have the energy estimate for as which yields Similarly, we can obtain the energy for as Now, we should estimate . Note that ; so, By (40), (62), and Sobolev embedding theorem, we have which gives Inserting this estimate into (67) and (68), one has where and . By the Gronwall inequality, one obtains Recall that therefore, it follows from (64) and (73) that for all , where we use in the last step and Note that (3) gives as , and the property of the operator yields as . Now, we choose satisfying the following properties: (1), (2), (3). Hence, combining the previous convergence results, it is easy to obtain from (75) that
Step 6. By now, we have proved that (4), (5), and (7) hold on the time interval . Now our aim is to show that these three results still hold on . Define ; now choose and as the new initial data, and one can see that the limit relation (3) still holds in the time . Moreover, from Lemma 3, we know that , , , and depend only on the norm of the initial data . Then, we repeat the previous argument and find a positive sequence such that (4), (5), and (7) hold on . We assert that . Indeed, if , and then the blow-up criterion implies that we can still extend to some bigger interval, so we can continue this procedure as long as , and by the blow up criterion again, we get our assertion. Since , after finite times iteration, we obtain the convergence results (4), (5), and (7).
Finally, since (7) holds, we have ; then, the convergence result (6) follows from (5) and the following interpolation inequality: Therefore, we finish the proof of Theorem 1.
3. The Convergence Rate with Some Loss of Derivatives
Proof of Theorem 2. Since (7) holds, without loss of generality, we may assume that
for small and . By the extra regularity of the initial data , using the same reasonings that lead to (54) and the same extension method as Step 6 in the proof of Theorem 1, then we obtain
Now the proof is divided into two cases.
Case 1 (). In this case, the estimate (82) implies Using the estimates and (84), we deduce from the first inequality of (58) that Similarly, one can infer from the first inequality of (59) that The previous two inequalities together with (48) give On the other hand, from (81) and the first inequality of (66), we can get where we have used (83) in the last step. Simultaneously, the energy estimate for is estimated as Then Gronwall inequality yields that Combining (87) and (90), we can arrive at for some , from which we know that (8) holds by choosing .
Case 2 (). In this case, we have And also one obtains from (48) and the first inequality of (62) Applying the previous two estimates into (58) and (59), we get which yields that On the other hand, the estimates (71) and (93) imply Then, we insert this inequality and (83) and (92) into (66) and obtain In the same way, the estimate (68) is replaced by Hence, by the Gronwall inequality, we have for some . This inequality, together with (95), gives for some . So, (9) follows from (100) provided that we choose satisfying .
The author would like to thank the referee's valuable suggestions which improved the presentation considerably. This work is supported by the National Science Foundation (Grant no. 11271334), Zhejiang Provincial Natural Science Foundation of China (Grant no. LY12A01019) and the Science Research Foundation of Jiaxing University.
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