- Research
- Open Access
Hyperbolic curve flows in the plane
- Zhe Zhou^{1},
- Chuan-Xi Wu^{1} and
- Jing Mao^{1}Email author
https://doi.org/10.1186/s13660-019-2005-y
© The Author(s) 2019
- Received: 11 August 2018
- Accepted: 18 February 2019
- Published: 28 February 2019
Abstract
In this paper, we investigate the evolution of a strictly convex closed planar curve driven by a hyperbolic normal flow. The asymptotical behavior of the evolving curves has also been shown if the velocity of the initial curve is nonnegative.
Keywords
- Hyperbolic partial differential equations
- Curve flows
- Short-time existence
- Convergence
MSC
- 58J45
- 58J47
1 Introduction and our main conclusions
Theorem 1.1
(Local existence and uniqueness)
Suppose that \(F_{0}\) is a smooth strictly convex closed curve in \(\mathbb{R}^{2}\). Then there exist a positive constant \(T>0\) and a family of smooth strictly convex closed planar curves \(F(u,t)\) satisfying (1.1).
If furthermore the normal velocity of the initial curve \(F(\cdot ,0)\) is nonnegative, we can also describe the asymptotical behavior for the hyperbolic flow (1.1).
Theorem 1.2
- (i)
the solution \(F(\cdot ,t)\) converges to a point, that is to say, the curvature of the limit curve becomes unbounded;
- (ii)
the curvature k of the evolving curve is discontinuous so that the solution \(F(\cdot ,t)\) converges to a piecewise smooth curve, which implies that shocks and propagating discontinuities may be generated within the hyperbolic flow (1.1).
Remark 1.3
If \(\alpha =1\), then the hyperbolic flow (1.1) degenerates into the one considered in [6], and correspondingly, our Theorem 1.1 would become [6, Theorem 1.2]. Therefore, our paper here is an interesting extension of [6]. Besides, as in [7], one can also add a term \(c(t)F(u,t)\) to the RHS of the evolution equation in (1.1), which is actually a forcing term in the direction of the position vector, and then using the methods in [7] and the paper here, the evolution and the asymptotical behavior of the new hyperbolic planar flow can be expected without any big difficulty. The research of curve flows and related topics is important and has many interesting applications in other scientific branches (see, e.g., [1–4, 8, 9, 11, 12]).
2 Proof of Theorem 1.1
In this section, we will reparameterize the evolving curves so that the hyperbolic partial differential equation (PDE for short) can be derived for the support function defined by (2.5) below, which leads to the short-time existence and the uniqueness of the solution to the flow (1.1).
Definition 2.1
A curve \(F:\mathbb{S}^{1}\times [0,T)\rightarrow \mathbb{R}^{2}\) evolves normally if and only if its tangential velocity vanishes.
It is easy to know that the flow (1.1) is a normal flow.
Lemma 2.2
The curve flow (1.1) is a normal flow.
Proof
By the definition of υ, (2.1), (2.2), and (2.3), we can obtain the following.
Lemma 2.3
The derivative of υ with respect to t is \(\frac{\partial \upsilon }{\partial t}=-k\sigma \upsilon \).
Proof
Here we would like to get the short-time existence of the flow (2.8) by the linearization method. First, we have the following conclusion.
Lemma 2.4
Next, we want to consider the linearization of (2.8) around \(S_{0}\).
Lemma 2.5
Proof
By Lemma 2.5, we have the following.
Lemma 2.6
Suppose that \(F_{0}\) is a smooth strictly convex closed curve and \(k_{0}>0\) is the curvature of the curve \(F_{0}\). Then there exist some \(T>0\) and a family of strictly convex closed curves \(F(\cdot ,t)\) such that (2.8) has a unique solution \(S\in C^{\infty }(\mathbb{S}^{1} \times [0,T))\) with S the support function of \(F(\cdot ,t)\).
Proof
3 An interesting example
Example 3.1
Case (I). Assume that \(r_{1}\leq 0\). Since \(r_{tt}=-\frac{1}{r^{ \alpha }}<0\), which implies the acceleration and the initial velocity are in the same direction, it is easy to know that \(r(t)\) decreases and then there must exist a finite time \(t_{0}>0\) such that \(r(t_{0})=0\). That is to say, the initial circle \(F(u,0)\) contracts to a single point as \(t\rightarrow t_{0}\). Especially, by [5, Lemma 3.1], if \(\alpha =1\) and \(r_{1}=0\), then \(t_{0}=\sqrt{\frac{\pi }{2}}r_{0}\).
4 Some propositions of the hyperbolic flow
Using the above facts, one can easily get the following maximum principle for the trip adjacent to the θ-axis (see, e.g., [6, 10]).
Lemma 4.1
Lemma 4.1 can be used to get the following principle.
Proposition 4.2
(Containment principle)
Let \(F_{1}\) and \(F_{2}:\mathbb{S}^{1}\times [0,T) \rightarrow \mathbb{R}^{2}\) be two convex solutions of (2.8). Suppose that \(F_{2}(u,0)\) lies in the domain enclosed by \(F_{1}(u,0)\), and \(f_{2}(u)\geq f_{1}(u)\geq 0\). Then \(F_{2}(u,t)\) is contained in the domain enclosed by \(F_{1}(u,t)\) for all \(t\in [0,T)\).
Proof
Proposition 4.3
(Preserving convexity)
Proof
We also need the following properties of the evolving curves \(F(\cdot ,t)\) and the Blaschke selection theorem.
Lemma 4.4
Proof
Lemma 4.5
Proof
Theorem 4.6
(Blaschke selection theorem)
Let \(\{K_{j}\}\) be a sequence of convex sets which are contained in a bounded set. Then there exist a subsequence \(K_{jk}\) and a convex set K such that \(\{K_{jk}\}\) converges to K in the Hausdorff metric.
5 Proof of Theorem 1.2
By reasonably using Example 3.1 and the containment principle, we can get the convergence of the hyperbolic flow (1.1).
Proof of Theorem 1.2
Let \([0,T_{\max })\) be the maximal time interval of the hyperbolic flow (1.1). We divide the proof into the following several steps.
Step 1. Preserving convexity
By Proposition 4.3, the evolving curves \(F(\cdot ,t)\) remain strictly convex on \([0,T_{\max })\) and their curvatures have a uniformly positive lower bound \(\max_{\theta \in \mathbb{S} ^{1}}k_{0}(\theta )\) on \(\mathbb{S}^{1}\times [0,T_{\max })\).
Step 2. Short-time existence
Enclose the initial curve F by a large enough round circle \(\gamma _{0}\), and then let this circle evolve under the hyperbolic flow (1.1) with the initial velocity \(\min_{u\in \mathbb{S}^{1}}f(u)\) to get a solution \(\gamma (\cdot ,t)\). By Example 3.1, we know that the solution \(\gamma (\cdot ,t)\) exists only at a finite time interval \([0,T^{\ast })\), with \(T^{\ast }<\infty \), and \(\gamma (\cdot ,t)\) shrinks into a point as \(t\rightarrow T^{\ast }\). By Proposition 4.2, we know that \(F(\cdot ,t)\) is always enclosed by \(\gamma (\cdot ,t)\) for all \(t\in [0,T^{\ast })\). Therefore, \(F(\cdot ,t)\) must become singular at some time \(T_{\max } \leq T^{\ast }\).
Step 3. Hausdorff convergence
From Example 3.1, since \(f(u)\) is nonnegative, the round circle \(\gamma (\cdot ,t)\) constructed in Step 2 is shrinking. Since, for any time \(t\in [0,T_{\max })\), \(F(\cdot ,t)\) is enclosed by \(\gamma (\cdot ,t)\) and \(\gamma (\cdot ,t)\) is shrinking, every convex set \(K_{F(\cdot ,t)}\) enclosed by \(F(\cdot ,t)\) must be contained in an open bounded disk enclosed by \(\gamma (\cdot ,0)=\gamma _{0}\). By Theorem 4.6, we know that \(F(\cdot ,t)\) converges to a (maybe degenerate and non-smooth) weakly convex curve \(F(\cdot ,T_{\max })\) in the Hausdorff metric.
Step 4. Asymptotical behavior
Case I. \(T_{0}\leq T_{\max }\). On the one hand, there exists a unique solution of the evolution equation (1.1) on the interval \([0,T_{0})\). On the other hand, \(\ell (t)\rightarrow 0\) as \(t\rightarrow T_{0}\), which implies that the curvature k goes to infinity when \(t\rightarrow T_{0}\), and then \(F(\cdot ,t)\) will blow up at time \(T_{0}\). Hence, by the definition of \(T_{\max }\), we have \(T_{0}=T_{\max }\). That is to say, \(F(\cdot ,t)\) converges to a point as \(t\rightarrow T_{\max }\).
- (1)
\(\|F(u,T_{\max })\|=\sup_{u\in \mathbb{S}^{1}}|F(u,T _{\max })|=\infty \). However, as shown in Step 2, we know that \(F(\cdot ,t)\) is contained by the initial curve \(F_{0}\), and then \(\|F(u,T_{\max })\|\) must be bounded, which is a contradiction. So, this case is impossible.
- (2)\(\|F_{u}(u,T_{\max })\|=\infty \), then the length of the limit curve satisfieswhich is contradict with the fact \(\ell (T_{\max })<\ell (0)< \infty \). So, this case is also impossible.$$\begin{aligned} \ell (T_{\max }) =&\lim_{t\rightarrow T_{\max }} \int _{F(u,t)}\,ds \\ =&\lim_{t\rightarrow T_{\max }} \int _{F(u,t)} \bigl\vert F_{u}(u,t) \bigr\vert \,du \\ =& \int _{F(u,t)}\lim_{t\rightarrow T_{\max }} \bigl\vert F_{u}(u,t) \bigr\vert \,du \\ =&\infty , \end{aligned}$$
- (3)
The curvature k is discontinuous. We cannot exclude this case, and then this phenomenon will occur if the above shocks are impossible.
Declarations
Acknowledgements
The authors would like to thank the referees for their careful reading and interesting comments such that the paper appears as its present version. The corresponding author, Prof. J. Mao, wants to thank the Department of Mathematics, Instituto Superior Técnico, University of Lisbon for its hospitality during his visit from September 2018 to September 2019.
Funding
This research was supported in part by the NSF of China (Grant No. 11401131), China Scholarship Council, the Fok Ying-Tung Education Foundation (China), and Hubei Key Laboratory of Applied Mathematics (Hubei University).
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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