TY - JOUR

T1 - Curvature effect on asymptotic profiles of spiral curves

AU - Tsai, Je Chiang

AU - Zhang, Zhengyang

N1 - Funding Information:
The authors are grateful to the anonymous referees for their useful suggestions and comments which improve the exposition of the paper. The authors would like to thank Prof. Vladimir Zykov for pointing out that the limiting profile of a spiral curve given in Theorem 1.1 (i) is the involute of the inner circle. JCT is supported by MOST, Taiwan (MOST 107-2115-M-007-011-MY2 ) and NCTS, Taiwan . ZZ is supported by National Natural Science Foundation of China (Grant Numbers No. 11811530272 and No. 11471044 ), and MOST, Taiwan (MOST 107-2811-M-007-1036 and 108-2811-M-007-536 ).
Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/12

Y1 - 2020/12

N2 - We study the shape of spiral curves in an annulus which is governed by curvature flow equations with a driving force term. We establish that as the model parameter μ (which is the coefficient of the curvature) approaches ∞, the profile of the spiral curve tends to a line segment, while as μ approaches 0+, the limiting profile of the spiral curve is the involute of the inner circle of the annulus and the associated limiting rotational speed is the ratio of a constant c, which is the propagation speed of the planar wave, to the inner radius of the annulus. Hence the model parameter μ can be viewed as a twisted parameter. Finally, the spiral curve under consideration is shown to be with sign-changing curvature and exponentially stable.

AB - We study the shape of spiral curves in an annulus which is governed by curvature flow equations with a driving force term. We establish that as the model parameter μ (which is the coefficient of the curvature) approaches ∞, the profile of the spiral curve tends to a line segment, while as μ approaches 0+, the limiting profile of the spiral curve is the involute of the inner circle of the annulus and the associated limiting rotational speed is the ratio of a constant c, which is the propagation speed of the planar wave, to the inner radius of the annulus. Hence the model parameter μ can be viewed as a twisted parameter. Finally, the spiral curve under consideration is shown to be with sign-changing curvature and exponentially stable.

KW - Asymptotic profiles

KW - Curvature flow equation

KW - Involute of circles

KW - Steadily rotating spiral curve

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U2 - 10.1016/j.physd.2020.132657

DO - 10.1016/j.physd.2020.132657

M3 - Article

AN - SCOPUS:85088861669

SN - 0167-2789

VL - 413

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

M1 - 132657

ER -