On an asymptotically log-periodic solution to the graphical curve shortening flow equation

Dong-Ho Tsai; Xiao-Liu Wang; Dong-Ho Tsai; Xiao-Liu Wang

doi:10.3934/mine.2022019

Mathematics in Engineering

2022, Volume 4, Issue 3: 1-14. doi: 10.3934/mine.2022019

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On an asymptotically log-periodic solution to the graphical curve shortening flow equation

Dong-Ho Tsai ^{1
,
,},
Xiao-Liu Wang ²

1.
Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan
2.
School of Mathematics, Southeast University, Nanjing 210096, Jiangsu, China

Received: 15 February 2020 Accepted: 26 July 2021 Published: 30 July 2021

With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $ y\left(x, t\right) \ $has the interesting property that it converges to a log-periodic function of the form

$ A\sin \left( \log t\right) +B\cos \left( \log t\right) $

as$ \ t\rightarrow \infty, \ $where $ A, \ B $ are constants. Moreover, for any two numbers $ \alpha < \beta, \ $we are also able to construct a solution satisfying the oscillation limits

$ \liminf\limits_{t\rightarrow \infty}y\left( x,t\right) = \alpha,\ \ \ \limsup\limits _{t\rightarrow \infty}y\left( x,t\right) = \beta,\ \ \ x\in K $

on any compact subset$ \ K\subset \mathbb{R}. $
- curve shortening flow,
- heat equation,
- geometric heat equation,
- log-periodic function,
- prescribing oscillation values
Citation: Dong-Ho Tsai, Xiao-Liu Wang. On an asymptotically log-periodic solution to the graphical curve shortening flow equation[J]. Mathematics in Engineering, 2022, 4(3): 1-14. doi: 10.3934/mine.2022019

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Abstract

With the help of heat equation, we first construct an example of a graphical solution to the curve shortening flow. This solution $ y\left(x, t\right) \ $has the interesting property that it converges to a log-periodic function of the form

$ A\sin \left( \log t\right) +B\cos \left( \log t\right) $

as$ \ t\rightarrow \infty, \ $where $ A, \ B $ are constants. Moreover, for any two numbers $ \alpha < \beta, \ $we are also able to construct a solution satisfying the oscillation limits

$ \liminf\limits_{t\rightarrow \infty}y\left( x,t\right) = \alpha,\ \ \ \limsup\limits _{t\rightarrow \infty}y\left( x,t\right) = \beta,\ \ \ x\in K $

on any compact subset$ \ K\subset \mathbb{R}. $

References

[1]	S. Angenent, On the formation of singularities in the curve shortening flow, J. Diff. Geom., 33 (1991), 601–633.
[2]	P. Broadbridge, P. J. Vassiliouz, The role of symmetry and separation in surface evolution and curve shortening, SIGMA, 7 (2011), 1–19.
[3]	K. S. Chou, X. P. Zhu, The curve shortening problem, Chapman: Hall/CRC, 2001.
[4]	K. S. Chou, X. P. Zhu, Shortening complete plane curves, J. Diff. Geom., 50 (1998), 471–504.
[5]	P. W. Doyle, P. J. Vassiliou, Separation of variables for the 1-dimensional non-linear diffusion equation, Int. J. Non-Linear Mech., 33 (1998), 315–326. doi: 10.1016/S0020-7462(97)00013-9
[6]	M. A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Diff. Geom., 26 (1987), 285–314.
[7]	M. E. Gage, R. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69–96.
[8]	M. E. Gurtin, Thermomechanics of evolving phase boundaries in the plane, New York: The Clarendon Press, 1993.
[9]	M. Nara, M. Taniguchi, The condition on the stability of stationary lines in a curvature flow in the whole plane, J. Diff. Eq., 237 (2007), 61–76. doi: 10.1016/j.jde.2007.02.012
[10]	D. H. Tsai, X. L. Wang, On some simple methods to derive the hairclip and paperclip solutions of the curve shortening flow, Acta Math. Sci., 39 (2019), 1674–1694. doi: 10.1007/s10473-019-0616-5

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