- Research
- Open Access
Boundary value behaviors for solutions of the equilibrium equations with angular velocity
- Jiaofeng Wang^{1},
- Jun Pu^{2},
- Bin Huang^{3}Email author and
- Guojian Shi^{4}
- Received: 31 August 2015
- Accepted: 27 November 2015
- Published: 9 December 2015
Abstract
This work is concerned with a mixed boundary value problems for the slow equilibrium equations with prescribed angular velocity. As an application, we find sufficient conditions for the existence and uniqueness of blow-up solutions under weaker conditions.
Keywords
- axisymmetric
- equilibrium equation
- blow-up solution
1 Introduction
In the study of this model, Auchmuty [1] proved the existence of an equilibrium solution if the angular velocity satisfied certain decay conditions. For a constant angular velocity, Miyamoto [2] has proved that there exists an equilibrium solution if the angular velocity is less than certain constant and that there is no equilibrium for large velocity. Pang et al. [3] talked about the exact numbers of the stationary solutions. For many other interesting results, see references [4–6].
In Section 2, first we prove the existence of a minimizer of the energy functional F in \({\mathcal{A}}_{M}\). Then we give the properties of minimizers; they are stationary solutions of equation (1.1) with finite mass and compact support. The main difficulty in the proof is the loss of compactness due to the unboundedness of \({\mathbb{R}}^{4}\). To prevent the mass from running off to spatial infinity along a minimizing sequence, our variational approach is related to the concentration-compactness principle due to Fang and Li [4]. For many other interesting results, see references [6–8].
2 Minimizer of the energy
In this section, we present some properties of the functional F and prove the existence of a minimizer. It is easy to verify that the function F is invariant under any vertical shift, that is, if \(\rho\in{{\mathcal{A}}_{M}}\), then \(T\rho(x):=\rho(x+ae_{3})\in {\mathcal{A}}_{M}\) and \(F(T\rho)=F(\rho)\) for any \(a\in\mathbb{R}\). Here \(e_{3} =(0,0,1)\). Therefore, if \((\rho_{n})\) is a minimizing sequence of F in \({\mathcal{A}}_{M}\), then \((T\rho_{n})\) is a minimizing sequence of F in \({\mathcal{A}}_{M}\) too. First, we give some estimates.
Lemma 2.1
Proof
The proof can be found in [1]. □
Lemma 2.2
For \(\rho\in L^{1} \cap L^{\frac{4}{3}} ({\mathbb{R}}^{4})\), we have \(\nabla\Phi\in L^{2} ({\mathbb{R}}^{4})\).
Proof
Lemma 2.3
Assume that \(P_{1}\) holds. Then there exists a nonnegative constant C, depending only on \(\frac{1}{|x|}\), M, and \(J(r)\), such that \(F\geq-C\).
Proof
Let \(h_{M}=\inf_{{\mathcal{A}}_{M}} F\). A simple scaling argument shows that \(h_{M} <0\): let \(\overline{\rho}(x)=\varepsilon^{3}\rho(\varepsilon x)\), then \(\int\overline{\rho}=\int\rho\). Since \(\lim_{\rho \rightarrow0}Q(\rho)\rho^{-1}=0\), it is easy to see that for ε small enough, \(\int Q(\overline{\rho})=\int\varepsilon^{-3}Q(\varepsilon^{3} \rho )\rightarrow0\). Therefore, \(h_{M}<0\).
Lemma 2.4
Assume that \(P_{1}\) holds. Then for every \(0<\widetilde{M}\leq M\), we have \(h_{\widetilde{M}}\geq ( \frac{\widetilde{M}}{M})^{\frac{5}{3}} h_{M} \).
Proof
Lemma 2.5
Proof
By Sobolev theorem and Lemma 2.1 we can complete the proof. □
Lemma 2.6
Proof
We are now ready to show the existence of a minimizer of \(h_{M}\), provided that \(P_{1}\) holds.
Theorem 2.1
Remark 2.1
Without admitting the spatial shifts, the assertion of the theorem is false: Given a minimizer \(\rho_{0}\) and a sequence of shift vectors \((a_{n} e_{3} )\in{\mathbb{R}}^{4}\), the functional F is translation invariant, that is, \(F(T\rho)=F(\rho)\). But if \(|a_{n} e_{3} |\rightarrow \infty\), then this minimizing sequence converges weakly to zero, which is not in \({\mathcal{A}}_{M}\).
Proof
Next, we show that the minimizers obtained are steady states of equation (1.1).
Theorem 2.2
Proof
Declarations
Acknowledgements
The authors are very grateful for reviewers’ valuable comments and suggestions in improving this paper.
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.
Authors’ Affiliations
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