American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Non-autonomous Honesty theory in abstract state spaces with applications to linear kinetic equations
  • CPAA Home
  • This Issue
  • Next Article
    Formal equivalence between normal forms of reversible and hamiltonian dynamical systems
March  2014, 13(2): 715-728. doi: 10.3934/cpaa.2014.13.715

Lifespan theorem and gap lemma for the globally constrained Willmore flow

Yannan Liu 1, and  Linfen Cao 2,

1. 

Department of Mathematics, Beijing Technology and Business University, Beijing 100048, China
2. 

College of Mathematics and Information Science, Henan Normal University, Henan, 453007

Received  March 2013 Revised  July 2013 Published  October 2013

We study a fourth-order flow, which can be seen as a globally constrained Willmore flow. We obtain a lower bound on the lifespan of the smooth solution, which depends on the concentration of curvature for the initial surface and the constrained term. We also give a gap lemma for this flow, which is an important lemma in the study of the blowup analysis.
Keywords: Geometric evolution equations, fourth order, Willmore flow..
Mathematics Subject Classification: Primary: 35J60, 35K45, 52K44,53A0.
Citation: Yannan Liu, Linfen Cao. Lifespan theorem and gap lemma for the globally constrained Willmore flow. Communications on Pure & Applied Analysis, 2014, 13 (2) : 715-728. doi: 10.3934/cpaa.2014.13.715
References:
[1]

G. Huisken, The volume preserving mean curvature flow,, J. Reine Angew. Math., 382 (1987), 35.  doi: 10.1515/crll.1987.382.35.  Google Scholar

[2]

H. Y. Jian and Y. N. Liu, Long-time existence of mean curvature flow with external force fields,, Pacific J. Math., 234 (2008), 311.  doi: 10.2140/pjm.2008.234.311.  Google Scholar

[3]

E. Kuwert and R. Schätzle, The Willmore flow with small initial energy,, J. Differential Geom., 57 (2001), 409.   Google Scholar

[4]

E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional,, Comm. Anal. Geom., 10 (2002), 307.   Google Scholar

[5]

Y. N. Liu, Gradient flow for the Helfrich functional,, Chin. Ann. Math. B, 33 (2012), 931.  doi: 10.1007/s11401-012-0741-0.  Google Scholar

[6]

J. McCoy, The surface area preserving mean curvature flow,, Asian J. Math., 7 (2003), 7.   Google Scholar

[7]

J. McCoy and G. Wheeler, Finite time singularities for the locally constrained willmore flow of surfaces, preprint,, \arXiv{1201.4541}., ().   Google Scholar

[8]

J.McCoy, G. Wheeler and G. Williams, Lifespan theorem for constrained surface diffusion flows,, Math. Z., 269 (2011), 147.  doi: 10.1007/s00209-010-0720-7.  Google Scholar

[9]

G. Simonett, The Willmore flow near spheres,, Differential Integral Equations, 14 (2001), 1005.   Google Scholar

[10]

G. Wheeler, "Fourth Order Geometric Evolution Equations,", Ph.D thesis, (2010).  doi: 10.1017/s0004972710001863.  Google Scholar

[11]

G. Wheeler, Lifespan Theorem for simple constrained surface diffusion flows,, J. Math. Anal. Appl., 375 (2011), 685.  doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar

[12]

T. Willmore, "Riemannian Geometry,", Oxford University Press, (1993).  doi: 10.2307/3612154.  Google Scholar

show all references

References:
[1]

G. Huisken, The volume preserving mean curvature flow,, J. Reine Angew. Math., 382 (1987), 35.  doi: 10.1515/crll.1987.382.35.  Google Scholar

[2]

H. Y. Jian and Y. N. Liu, Long-time existence of mean curvature flow with external force fields,, Pacific J. Math., 234 (2008), 311.  doi: 10.2140/pjm.2008.234.311.  Google Scholar

[3]

E. Kuwert and R. Schätzle, The Willmore flow with small initial energy,, J. Differential Geom., 57 (2001), 409.   Google Scholar

[4]

E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional,, Comm. Anal. Geom., 10 (2002), 307.   Google Scholar

[5]

Y. N. Liu, Gradient flow for the Helfrich functional,, Chin. Ann. Math. B, 33 (2012), 931.  doi: 10.1007/s11401-012-0741-0.  Google Scholar

[6]

J. McCoy, The surface area preserving mean curvature flow,, Asian J. Math., 7 (2003), 7.   Google Scholar

[7]

J. McCoy and G. Wheeler, Finite time singularities for the locally constrained willmore flow of surfaces, preprint,, \arXiv{1201.4541}., ().   Google Scholar

[8]

J.McCoy, G. Wheeler and G. Williams, Lifespan theorem for constrained surface diffusion flows,, Math. Z., 269 (2011), 147.  doi: 10.1007/s00209-010-0720-7.  Google Scholar

[9]

G. Simonett, The Willmore flow near spheres,, Differential Integral Equations, 14 (2001), 1005.   Google Scholar

[10]

G. Wheeler, "Fourth Order Geometric Evolution Equations,", Ph.D thesis, (2010).  doi: 10.1017/s0004972710001863.  Google Scholar

[11]

G. Wheeler, Lifespan Theorem for simple constrained surface diffusion flows,, J. Math. Anal. Appl., 375 (2011), 685.  doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar

[12]

T. Willmore, "Riemannian Geometry,", Oxford University Press, (1993).  doi: 10.2307/3612154.  Google Scholar

[1]

Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104
[2]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[3]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252
[4]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383
[5]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377
[6]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246
[7]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
[8]

Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265
[9]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117
[10]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305
[11]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[12]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032
[13]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, 2021, 20 (1) : 405-425. doi: 10.3934/cpaa.2020274
[14]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045
[15]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278
[16]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115
[17]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[18]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[19]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[20]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences