Instability of the Rayleigh problem with piecewise smooth steady states
- Jiangang Qi^{1},
- Bing Xie^{1}Email author and
- Shaozhu Chen^{1}
Received: 20 February 2016
Accepted: 10 May 2016
Published: 16 May 2016
Abstract
The present paper investigates the instability of the Rayleigh equation with piecewise smooth steady states. An existence result of a number of unstable modes for the Rayleigh problem is obtained.
Keywords
MSC
1 Introduction
So for shear flows, the instability problem is reduced to study the Rayleigh eigenvalue problems (1.2) and (1.3). This problem has a long history, going back to scientists such as Rayleigh and Kelvin in the 19th century (see [3–5]).
The flow is linearly unstable if some nontrivial solutions to (1.2) and (1.3) exist, with the imaginary part of c satisfying \(c_{i}:=\operatorname{Im}c>0\). A classical result of Rayleigh [1] is the necessary condition for instability that the basic velocity profile should have an inflection point at some points \(y=y_{s}\), that is, \(U''(y_{s})=0\). This condition was later improved by Fjörtoft [6]. However, it is far more difficult to obtain effective sufficient conditions for instability.
In 1935, Tollmien [7] obtained an unstable solution to (1.2) by formally perturbing around a neutral mode (i.e., c is a real number) for symmetric flows. In 1964, Rosenbluth and Simon [8] gave a necessary and sufficient integral condition for the monotone flows. Recently, some instability criteria for the special flows \(U(y)=\cos my\) in [9] and \(U(y)=\sin my\) in [10] have been obtained. These results were much improved and extended by Lin to more general odd symmetric flows in [11] and other classes of shear flows in [12].
Proposition 1.1
(cf. Theorem 1.2 in [13])
The present paper mainly focuses on the instability of the Rayleigh equation with piecewise smooth and point-symmetric velocity profiles \(U(y)\), satisfying the requirement that \(U'(y)\) exists continuously on \([-1,1]\) and \(U''(y)\) exists on \([-1,1]\) except for a finite number of points. More general cases with piecewise smooth velocity profile were studied in [13].
Proposition 1.2
(cf. Theorem 1.4 in [13])
Theorem 1.3
Note that Proposition 1.1 studies the case where \(K\in L^{1}[-1,1]\) and in Theorem 1.3 the function K is allowed not to be in \(L^{1}[-1,1]\) (\(K\in L_{\mathrm{loc}} ([-1,0)\cup(0,1] )\)). Besides, Proposition 1.2 only gives the existence of an unstable mode in one interval of wave numbers, while Theorem 1.3 gives the existence of many unstable modes in a number of intervals of wave number if \(H(0)\) has more than one negative eigenvalue. At the end of the present paper, we give an illustrative example in which \(H_{1}(0)\oplus H_{2}(0)\) has more than one negative eigenvalue.
Since \(U''(y)\) is allowed to be discontinuous at finite points in \([-1,1]\), this will result in different definitions of solutions of the Rayleigh equation and different Sturm-Liouville problems. For this reason, we will put much of attention on the properties of such solutions and corresponding eigenvalue problems. See Lemmas 2.1, 2.2, 2.4, and 2.5.
The main tools in the proof of Theorem 1.3 are the perturbation theory of operators in Hilbert spaces and the spectral theory of \(\mathcal{P}\mathcal{T}\)-symmetric differential operators.
Following this section, Section 2 presents some preliminary knowledge about the properties of solutions to the Rayleigh equation with piecewise smooth velocity profiles, the spectral properties of singular Sturm-Liouville problems with one singular end point and regular Sturm-Liouville problems with \(\mathcal{P}\mathcal{T}\)-symmetric potentials. The proof of the main result is given in Section 3 and the illustrative example is also given at the end of that section.
2 Preliminary knowledge
In this section, we first introduce properties of solutions of the Rayleigh equation with piecewise smooth velocity profiles and spectral properties of regular and singular Sturm-Liouville problems, especially the regular one with \(\mathcal{P}\mathcal{T}\)-symmetric potentials. We note that the assumptions of all lemmas in Section 2 and Section 3 are the same as in Theorem 1.3 and so are omitted for the sake of brevity.
2.1 Solutions of the Rayleigh equations
Since \(U''(y)\) may be discontinuous at some junctions inside the interval, we must redefine the solution of the governing equation (1.2) and the eigenfunction to the Rayleigh eigenvalue problem in a weaker sense than the classical one.
Lemmas 3.1 and 3.2 in [13] show that such a solution is also uniquely determined by an initial value condition.
Lemma 2.1
(cf. Lemmas 3.1 and 3.2 in [13])
Note, Lemma 2.1 implies that the corresponding eigen-subspace of every eigenvalue of (2.1) and (1.3) has exactly one dimension.
Lemma 2.2
Proof
Since \(\varphi_{1}\), \(\varphi_{2}\) are classical solutions of (2.1), \(W[\varphi_{1},\varphi_{2}](y)\equiv C_{\pm j}\) on \(I_{\pm j}\), \(0\le j\le N\). On the other hand, \(B\varphi_{1}\) and \(B\varphi_{2}\) are both continuous on \([-1,1]\) and \(U_{0}W[\varphi_{1},\varphi_{2}](y)=\varphi_{1}B\varphi_{2}-\varphi _{2}B\varphi_{1}\) is continuous by the continuity of \(\varphi_{k}\) on \([-1,1]\), \(k=1,2\), and hence \(U(y)-c\ne0\) implies that \(W[\varphi_{1},\varphi_{2}]\) is continuous on \([-1,1]\). As a result, \(C_{\pm j}\equiv C\) for \(1\le j\le N\). This proves Lemma 2.2. □
By Lemma 2.2 we can define the linearly dependent(resp. independent) solutions. Let \(\varphi_{1}\), \(\varphi_{2}\) be solutions of (2.1). If \(W[\varphi_{1},\varphi_{2}](y_{0})=0\) for some \(y_{0}\in[-1,1]\), then we say that \(\varphi_{1}\), \(\varphi_{2}\) are linearly dependent. Otherwise we say they are linearly independent.
Lemma 2.3
Proof
2.2 Spectral properties of the corresponding differential operators
Since (2.4) is regular, the operator \(H(c_{i})\) has only countable discrete eigenvalues. The properties of eigenvalues were studied in [13] and it has been proved that the real parts of eigenvalues of (2.4) are bounded from below. The eigenvalues can be arranged in the dictionary order according to their real parts and imaginable parts. Then we write the countably many eigenvalues of \(H(c_{i})\) as \(\{\lambda_{n}(c_{i})\}_{n\ge1}\) in such an order and conclude from Theorem 2.2 of [13] the following.
Lemma 2.4
(cf. Theorem 2.2 of [13])
Lemma 2.5
The operator \(H(0)\) is self-adjoint and \(\sigma_{0}\) contains only discrete, real and algebraically simple eigenvalues.
Proof
2.3 Convergence properties of solutions of (2.4) as \(c_{i}\to0+\)
Lemma 2.6
Proof
Lemma 2.7
Let \(\varphi(y;\lambda ,c_{i})\) be the solution of (2.1) such that \(\varphi(1;\lambda ,c_{i})=0\) and \(\varphi'(1;\lambda ,c_{i})=1/U_{0}(1)\) for \(c_{i}\ge0\). Then \(\varphi(\cdot; \lambda ,c_{i})\to\varphi(\cdot; \lambda ,0)\) as \(c_{i}\to0+\) in \(L^{2}[0,1]\).
Proof
Lemma 2.8
Under the assumptions in Theorem 1.3, \(|W(c_{i})|\to\infty\) as \(c_{i}\to0+\).
Proof
Now we can outline the proof of Theorem 1.3 given in the next section. Recall the definition of \(\lambda _{k}(c_{i})\), the eigenvalue of (2.4). Since there exists an unstable mode for a wave number α if and only if \(\lambda _{k}(c_{i})=0\) for \(k\ge1\), we only need to prove that \(H(c_{i})\) has at least one zero eigenvalue for some \(c_{i}>0\) under the assumptions in Theorem 1.3.
Consider the eigenvalues \(\lambda (c_{i})\) of \(H(c_{i})\) as functions of the variable \(c_{i}\in[0,\infty)\). Applying the Kato-Rellich theorem (see e.g. [3], Theorem XII.8, p.15, IV, or [16], pp.437-439) as well as the resolvent convergence of \(H(c_{i})\) in Lemma 3.1, we prove that \(\lambda (c_{i})\) is a continuous function in the sense of (3.1). The main step is to prove the continuity of \(\lambda (c_{i})\) at \(c_{i}=0\). If this is done and \(\lambda _{k}(c_{i})\neq0\) for all \(k\ge1\) and \(c_{i}>0\), then we can introduce an auxiliary function \(D(c_{i})\) defined in (3.21) and prove that \(D(c_{i})\) is continuous on \((0,\infty]\), \(D(c_{i})>0\) for sufficiently large \(c_{i}>0\), and \(D(c_{i})<0\) for all sufficiently small \(c_{i}>0\) by using several technical lemmas in Section 3. Therefore, the desired contradiction would appear.
3 The proof of Theorem 1.3
The first lemma gives the continuity of \(\lambda (c_{i})\) on \((0,\infty)\).
Lemma 3.1
\(\lambda (c_{i})\) is continuous on \((0,\infty)\).
Proof
With a similar argument to the above one can prove the following.
Lemma 3.2
Proof
The method in the proof of Lemmas 3.1 and 3.2 cannot be applied straightforwardly to the case \(c^{0}_{i}=0\) since \(K(y,0)=K(y)\) is not integrable in \(L^{1}[-1,1]\). Then we first present necessary notations before proving the continuity of \(\lambda (c_{i})\) at \(c_{i}=0\). Let \(H(0)\) be defined as in (2.8) with spectrum \(\sigma_{0}\). Denote by \(\lambda (0)\) the eigenvalue of \(H(0)\). Then we can prove the following.
Lemma 3.3
\(\lambda (c_{i})\) is continuous at \(c_{i}=0\).
Proof
For the convenience of applications of Lemmas 3.1-3.3 in the proof of Theorem 1.3, we give a detailed description of the continuity of \(\lambda (c_{i})\) by summing up the conclusions in Lemmas 3.1-3.3.
Lemma 3.4
The eigenvalue \(\lambda(c_{i})\) of \(H(c_{i})\) is continuous on \([0,\infty]\) in the sense that, for a fixed eigenvalue \(\lambda(c^{0}_{i})\) of \(H(c^{0}_{i})\) with the algebraically multiplicity m and a neighborhood \(O(\lambda(c^{0}_{i}))\) of \(\lambda(c^{0}_{i})\) such that \(O(\lambda(c^{0}_{i}))\cap\sigma(H(c^{0}_{i}))=\{\lambda(c^{0}_{i})\}\), there exists a neighborhood \(O(c^{0}_{i})\) of \(c^{0}_{i}\) such that, for \(c_{i}\in O(c^{0}_{i})\), \(O(\lambda(c^{0}_{i}))\) contains m eigenvalues \(\lambda(c_{i})\) of \(H(c_{i})\).
In fact, Lemma 3.3 gives the resolvent convergence of \(H(c_{i})\) as \(c_{i}\to0+\). Using this fact we can prove the following.
Lemma 3.5
If \(c^{n}_{i}\to0\) and the limit λ̂ of \(\lambda (c^{n}_{i})\) is finite as \(n\to\infty\), then λ̂ is an eigenvalue of \(H(0)\).
Proof
Let \(\varphi_{n}\in L^{2}(-1,1)\) be the corresponding normalized eigenvector corresponding to the eigenvalue \(\lambda _{n}\) of \(H(c^{n}_{i})-z_{0}\), or \(\varphi_{n}=\lambda _{n}G_{n}\varphi_{n}\).
The following result gives a lower bound of the real eigenvalues of \(H(c_{i})\).
Lemma 3.6
There exists \(M>0\) such that for any real eigenvalue \(\lambda (c_{i})\) of \(H(c_{i})\) for \(c_{i}>0\) we have \(\lambda (c_{i})\ge-M\).
Proof
Proof of Theorem 1.3
Note that \(1-e^{-\lambda _{k}(\infty)}>0\) for \(k\ge1\) since \(\lambda _{k}(\infty)= (\frac{k\pi}{2} )^{2}+\alpha ^{2}>0\) by the definition of \(H(\infty)\) in Lemma 3.2. Then \(D(\infty)>0\). If \(D(c_{i})<0\) for some \(c_{i}>0\) and \(D(c_{i})\) is continuous on \((0,\infty)\), then we will have the desired contradiction.
Now we prove the continuity of \(D(c_{i})\) on \((0, \infty)\) provided that \(D(c_{i})\neq0\). Let \(c_{i}^{0}\in(0,\infty)\) and \(\varepsilon>0\) be given. Suppose that \(\mu_{1},\ldots,\mu_{m}\) are the first m distinct eigenvalues of \(H(c^{0}_{i})\) and that each \(\mu_{j}\) has a multiplicity \(n_{j}\) and \(n_{1}+\cdots+n_{m}=M_{m}\). Then \(\{M_{m}\}\) is increasing and the continuity of \(D(c_{i})\) is equivalent to the continuity of the partial product \(D_{m}(c_{i}):=\prod_{k=1}^{M_{m}}(1-e^{-\lambda_{k}(c_{i})})^{n_{k}(c_{i})}\) for every m because of the uniform convergence of the infinite product.
Example 3.7
Declarations
Acknowledgements
This research was partially supported by the NNSF of China (Grants 11271229 and 11471191), the NSF of Shandong (Grants ZR2015AM019), the BSRP of Shandong University (Grants 2015ZQXM2010 and 2015ZQXM2012), the PIP of Shandong (Grant 201301010) and the PSF of China (Grants 125367 and 2015M580583).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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