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Existence of solutions of a system of 3D axisymmetric inviscid stagnation flows
Boundary Value Problems volume 2012, Article number: 153 (2012)
A system of two integral equations is presented to describe the system of 3D axisymmetric inviscid stagnation flows related to Navier-Stokes equations and existence of its solutions is studied. Utilizing it, we construct analytically the similarity solutions of the 3D system. A nonexistence result is obtained. Previous study was only supported by numerical results.
The following system of two differential equations arising in the boundary layer problems in fluid mechanics
with boundary conditions
has been used to describe the system of 3D axisymmetric inviscid stagnation flow [1, 2], which consists of three partial differential equations [2, 3], where λ is a parameter related to the external flow components.
A solution of (1.1)-(1.3) is called a similarity solution and can be used to express the solutions of the 3D system. Regarding the study of (1.1)-(1.3), Howarth  presented a numerical study for the case which can be applied to the stagnation region of an ellipsoid. Davey  investigated numerically the stagnation region near a saddle point (). The two-dimensional cases, or and , and the special cases of the Falkner-Skan equation were solved by Hiemenz  and by Homann , respectively. Regarding the Falkner-Skan problems, further analytical study can be found in [6–10]. Also, one may refer to recent review of similarity solutions of the Navier-Stokes equations .
However, up to now, there has been very little analytical study on the existence of solutions of (1.1)-(1.3).
The main aim of this paper is to study the existence of solutions of (1.1)-(1.3) analytically for the case of . The method is to present a system of two integral equations and study the existence of its solutions and then use it to construct the solutions of (1.1)-(1.3). Also, a nonexistence result is obtained.
2 A system of two integral equations related to (1.1)-(1.3)
In this section, we present a system of two integral equations to describe a system of (1.1)-(1.3) under suitable conditions, which will be utilized in Section 4.
Lemma 2.1 Ifis a solution of (1.1)-(1.3), then.
Proof Since , we have
Notice that , , , and for .
If , we know and then is decreasing on , which implies that exists. Hence, by (2.1).
If , we have by (1.2). By (2.1), there exists such that and then there exists such that . Obviously, by . We prove that is decreasing on .
In fact, if there exist with such that . Let such that , then and .
Differentiating (1.2) with η, we have
a contradiction. Hence, is decreasing on and then .
This completes the proof. □
Theorem 2.1 Ifis a solution of (1.1)-(1.2), then
has a solution, wheredenotes the Green function forwithanddefined by
Proof Assume that . Let for be the inverse function to . It follows that is strictly increasing on and with , . Let for , by Lemma 2.1, . This implies that for and x is continuous on . By Lemma 2.1, we see that x is continuous from the left at 1. Hence, we have and , i.e., .
Using the chain rule to , we obtain and by the inverse function theorem, we have
This, together with , implies
Integrating the last equality from 0 to t implies
Then . By , we know that y is continuous from the left at 1 and then .
Notice that , , we have .
Differentiating with t, we have
From this, we have for and .
Differentiating with t and utilizing , we have
Substituting g, , , and f into (1.2) implies
Integrating (2.5) from t to 1, we have
By , then
Substituting f, , , and g into (1.1) implies
By , we have
where is defined by (2.4). Hence, is a solution of (2.2)-(2.3) in Q. □
3 Positive solutions of the system (2.2)-(2.3)
In this section, we will use the fixed point theorem to study the existence of positive solutions of the system (2.2)-(2.3).
It is easy to verify
We define some functions
By computation, , , there exists such that for and .
In order to study the existence of solutions of (2.2)-(2.3) in Q for , we denote the norm of the Banach space by
Let and be a natural number, we define
where , ,
where is defined by (2.4).
Let , we define an operator F as follows:
It is easy to verify that , θ are continuous operators from into and , , we know the following proposition holds:
Lemma 3.1is a continuous and compact operator fromto.
Lemma 3.2 Let and such that
Then the following assertions hold:
and , where is a total variation of y on .
If , then is increasing on and then for .
Proof We shall use the basic fact: let and () be local minimum (maximum), then (≤0).
If there exists such that , by , we know that there exists such that . Differentiating (3.3) with t twice, we have(3.4)
By and (3.4), we have
a contradiction. Hence, for .
If there exists such that , let , by and , we may assume such that . This implies , i.e., , and . By (3.4) and , we know
a contradiction. Hence, (i) holds.
Let such that and . If , we prove that is increasing on and decreasing on .
Since and , then . Let . If there exist with such that , let such that , then by (i). From , and (3.4), we know
If there exist with such that , let such that , then by (i). Analogously, we know easily
a contradiction. Hence,
and , i.e., (ii) holds.
Let . By (i) and , we know and then is increasing on and then for . Hence, (iii) holds. □
Let E be a Banach space, D be a bounded open set of E and, is compact. Iffor anyand, then F has a fixed point in.
Lemma 3.4 Let, then F has a fixed pointin, i.e., there existssuch that
where . We prove for and with .
In fact, if there exist and μ with and such that , by Lemma 3.2(i) and (iii), we have .
Since and for , this, together with for and , implies
And then , i.e., .
By (3.3), we have
Noticing that and for , we obtain for . This, together with (3.7) and Lemma 3.2(ii), implies
i.e., . Hence,
By Lemmas 3.1 and 3.3, F has a fixed point in . □
Lemma 3.5 Letbe in Lemma 3.4, then
is bounded on .
is bounded on for any .
Proof By Lemma 3.3(i), we know . By (3.5), we have
For , we know for , i.e., is decreasing in , by , for . By for and (3.5), we have(3.9)
And then for . Obviously, for . This, together with the decrease in , implies
Let , μ defined by
where defined in (3.1).
It is easy to verify for . And then
The last two inequalities imply that is bounded on .
we know that is bounded on for any . □
Lemma 3.6 Letbe in Lemma 3.4, then
and is increasing in .
is bounded and equicontinuous in for any .
Lemma 3.2(i) and (iii) imply the desired results.
For , let such that . Since on , by Lemma 3.2(ii), , we obtain .
Differentiating (3.6) with t twice, we have . Integrating this equality from 0 to , we have
Noticing that and for and Lemma 3.2(ii), we know
i.e., is bounded on . Let (where ), we know
This implies that is equicontinuous on . □
Theorem 3.1 There exists such that
Proof Let be in Lemma 3.4, by Lemma 3.5(ii) and (iii), we know that is bounded and equicontinuous on for any . Letting (), utilizing the diagonal principle and the Arzela-Ascoli theorem, we know that there exists a subsequence of and such that converges to for . Without loss of generality, we assume that is itself of .
By Lemma 3.6, we know that is bounded and equicontinuous on for any and then is bounded and equicontinuous on . Let (), the diagonal principle and the Arzela-Ascoli theorem imply that there exist y and in and two subsequences and with such that converges to for with and converges to for each . For the sake of convenience, we assume that and are itself of . By , we obtain and then for .
converges to and converges to for , by the Lebesgue dominated theorem (the dominated function , , we have that satisfies (3.11) and .
Fix and choose such that , then
Noticing that and converges to for , by the Lebesgue dominated theorem (the dominated function on ), we have
Differentiating the last equality twice, we know
By (i), we know and and then . This, together with (2.4), implies that satisfies (3.12). Clearly, . □
Theorem 3.2 For, the system (2.2)-(2.3) has at least a solutionin Q.
Proof Let in Theorem 3.1. It is clear that we only prove . If , by (3.10), we obtain for and then . Next, we prove for for .
Let such that , then
From this and , we obtain and .
Let and , we prove
By and , there exists such that . Since for , i.e., is concave down on , then for and for . Hence, .
By (3.11), we have
Integrating the last inequality from to 1 and utilizing , we have
Hence, (3.13) holds.
By , and , we have and , then
This, together with (3.13), implies
Since for , we have
And then for .
Finally, we prove for .
In fact, if there exists such that , by , there exists such that for and .
and , we obtain
By (3.11), we have
i.e., , . Integrating this inequality from to δ, we have
and then , a contradiction.
This completes the proof. □
4 Existence of solutions of (1.1)-(1.3)
In this section, we use positive solutions obtained in Theorem 3.2 to construct the solutions of (1.1)-(1.3) in Γ.
Theorem 4.1 For, the system (1.1)-(1.3) has at least a solution.
Proof Let , by Theorem 3.2, the system (2.2)-(2.3) has at least a solution in Q. By and (2.2), we know
Let , and then
we have .
Then is strictly increasing on and
Let be the inverse function to , we define the function
From (4.1), we have
Differentiating (4.2) with respect to η, we have
Then for .
Differentiating (4.3) with respect to η, we have
Differentiating (2.2) with respect to t, we have
By setting and utilizing and (4.3), we have
By (4.3), (4.4), (4.5) and (4.6), we have
By (4.1), we have . Differentiating with respect to η, we have
Differentiating (2.3) with t twice and combining (4.5) and (4.6), we obtain
This completes the proof. □
Remark 4.1 For , by Theorem 1 , (1.1)-(1.3) has no solution such that with for , is a constant.
Utilizing the system (2.2)-(2.3), we know easily that (1.1)-(1.3) has no solution in Γ for .
In fact, if (1.1)-(1.3) has a solution for some , by Theorem 2.1, then (1.1)-(1.3) has a solution in . Noticing that
This research uses integrals of equations to investigate the existence of solutions of the 3D axisymmetric inviscid stagnation flows related to Navier-Stokes equations and supplies a gap of analytical study in this field.
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The authors wish to thank the anonymous referees for their valuable comments. This research was supported by the National Natural Science Foundation of China (Grant No. 11171046) and the Scientific Research Foundation of the Education Department of Sichuan Province, China.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.