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), 35-48. 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), 311-324. 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), 409-441.  Google Scholar

[4]

E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom., 10 (2002), 307-339.  Google Scholar

[5]

Y. N. Liu, Gradient flow for the Helfrich functional, Chin. Ann. Math. B, 33 (2012), 931-940. doi: 10.1007/s11401-012-0741-0.  Google Scholar

[6]

J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.  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), 147-178. doi: 10.1007/s00209-010-0720-7.  Google Scholar

[9]

G. Simonett, The Willmore flow near spheres, Differential Integral Equations, 14 (2001), 1005-1014.  Google Scholar

[10]

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

[11]

G. Wheeler, Lifespan Theorem for simple constrained surface diffusion flows, J. Math. Anal. Appl., 375 (2011), 685-698. doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar

[12]

T. Willmore, "Riemannian Geometry," Oxford University Press, New York, 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), 35-48. 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), 311-324. 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), 409-441.  Google Scholar

[4]

E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional, Comm. Anal. Geom., 10 (2002), 307-339.  Google Scholar

[5]

Y. N. Liu, Gradient flow for the Helfrich functional, Chin. Ann. Math. B, 33 (2012), 931-940. doi: 10.1007/s11401-012-0741-0.  Google Scholar

[6]

J. McCoy, The surface area preserving mean curvature flow, Asian J. Math., 7 (2003), 7-30.  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), 147-178. doi: 10.1007/s00209-010-0720-7.  Google Scholar

[9]

G. Simonett, The Willmore flow near spheres, Differential Integral Equations, 14 (2001), 1005-1014.  Google Scholar

[10]

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

[11]

G. Wheeler, Lifespan Theorem for simple constrained surface diffusion flows, J. Math. Anal. Appl., 375 (2011), 685-698. doi: 10.1016/j.jmaa.2010.09.043.  Google Scholar

[12]

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

[1]

Bertram Düring, Daniel Matthes, Josipa Pina Milišić. A gradient flow scheme for nonlinear fourth order equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 935-959. doi: 10.3934/dcdsb.2010.14.935
[2]

Jeremy LeCrone, Yuanzhen Shao, Gieri Simonett. The surface diffusion and the Willmore flow for uniformly regular hypersurfaces. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3503-3524. doi: 10.3934/dcdss.2020242
[3]

Jaime Angulo Pava, Carlos Banquet, Márcia Scialom. Stability for the modified and fourth-order Benjamin-Bona-Mahony equations. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 851-871. doi: 10.3934/dcds.2011.30.851
[4]

Feliz Minhós. Periodic solutions for some fully nonlinear fourth order differential equations. Conference Publications, 2011, 2011 (Special) : 1068-1077. doi: 10.3934/proc.2011.2011.1068
[5]

John B. Greer, Andrea L. Bertozzi. $H^1$ Solutions of a class of fourth order nonlinear equations for image processing. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 349-366. doi: 10.3934/dcds.2004.10.349
[6]

Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677
[7]

Feliz Minhós, João Fialho. On the solvability of some fourth-order equations with functional boundary conditions. Conference Publications, 2009, 2009 (Special) : 564-573. doi: 10.3934/proc.2009.2009.564
[8]

Craig Cowan. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. Communications on Pure & Applied Analysis, 2016, 15 (2) : 519-533. doi: 10.3934/cpaa.2016.15.519
[9]

Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393-402. doi: 10.3934/proc.2003.2003.393
[10]

Mohammed Al Horani, Angelo Favini. First-order inverse evolution equations. Evolution Equations & Control Theory, 2014, 3 (3) : 355-361. doi: 10.3934/eect.2014.3.355
[11]

Alain Haraux, Mitsuharu Ôtani. Analyticity and regularity for a class of second order evolution equations. Evolution Equations & Control Theory, 2013, 2 (1) : 101-117. doi: 10.3934/eect.2013.2.101
[12]

Víctor León, Bruno Scárdua. A geometric-analytic study of linear differential equations of order two. Electronic Research Archive, 2021, 29 (2) : 2101-2127. doi: 10.3934/era.2020107
[13]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure & Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
[14]

Gabriele Bonanno, Beatrice Di Bella. Fourth-order hemivariational inequalities. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 729-739. doi: 10.3934/dcdss.2012.5.729
[15]

Pierre Fabrie, Alain Miranville. Exponential attractors for nonautonomous first-order evolution equations. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 225-240. doi: 10.3934/dcds.1998.4.225
[16]

M. Euler, N. Euler, M. C. Nucci. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4239-4247. doi: 10.3934/dcds.2017181
[17]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[18]

Peng Gao. Carleman estimates for forward and backward stochastic fourth order Schrödinger equations and their applications. Evolution Equations & Control Theory, 2018, 7 (3) : 465-499. doi: 10.3934/eect.2018023
[19]

Frédéric Robert. On the influence of the kernel of the bi-harmonic operator on fourth order equations with exponential growth. Conference Publications, 2007, 2007 (Special) : 875-882. doi: 10.3934/proc.2007.2007.875
[20]

Marcel Braukhoff, Ansgar Jüngel. Entropy-dissipating finite-difference schemes for nonlinear fourth-order parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3335-3355. doi: 10.3934/dcdsb.2020234

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences