American Institute of Mathematical Sciences

Advanced
December  2020, 13(12): 3503-3524. doi: 10.3934/dcdss.2020242

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

Jeremy LeCrone 1, , Yuanzhen Shao 2, and  Gieri Simonett 3,,

1. 

Department of Mathematics & Computer Science, University of Richmond, Richmond, VA 23173, USA
2. 

Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA
3. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

* Corresponding author: Gieri Simonett

Received  January 2019 Published  January 2020

Fund Project: This work was supported by a grant from the Simons Foundation (#426729, Gieri Simonett)

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are $ C^{1+\alpha} $–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are $ C^{1+\alpha} $–close to a sphere, and we prove that these solutions become spherical as time goes to infinity.

Keywords: Surface diffusion flow, Willmore flow, uniformly regular manifolds, geometric evolution equations, continuous maximal regularity, critical spaces, stability of spheres.
Mathematics Subject Classification: Primary: 35K55, 53C44; Secondary: 54C35, 35B65, 35B35.
Citation: 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
References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems: Volume Ⅰ., Abstract Linear Theory. Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[2]

H. Amann, Elliptic operators with infinite-dimensional state spaces, J. Evol. Equ., 1 (2001), 143-188.  doi: 10.1007/PL00001367.  Google Scholar

[3]

H. Amann, Function spaces on singular manifolds, Math. Nachr., 286 (2013), 436-475.  doi: 10.1002/mana.201100157.  Google Scholar

[4]

H. Amann, Anisotropic function spaces on singular manifolds., arXiv: 1204.0606. Google Scholar

[5]

H. Amann, Uniformly regular and singular Riemannian manifolds, In: Elliptic and Parabolic Equations, 1–43, Springer Proc. Math. Stat., 119, Springer, Cham, 2015. doi: 10.1007/978-3-319-12547-3_1.  Google Scholar

[6]

H. Amann, Cauchy problems for parabolic equations in Sobolev-Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds, J. Evol. Equ., 17 (2017), 51-100.  doi: 10.1007/s00028-016-0347-1.  Google Scholar

[7]

H. Amann, M. Hieber and G. Simonett, Bounded $H_\infty$-calculus for elliptic operators., Differential Integral Equations, 7 (1994), 613-653.   Google Scholar

[8]

T. Asai, Quasilinear parabolic equation and its applications to fourth order equations with rough initial data, J. Math. Sci. Univ. Tokyo., 19 (2012), 507-532.   Google Scholar

[9]

Y. Bernard, G. Wheeler and V.-M. Wheeler, Concentration-compactness and finite-time singularities for Chen's flow, arXiv: 1706.01707. Google Scholar

[10]

S. Blatt, A singular example for the Willmore flow, Analysis (Munich), 29 (2009), 407-430.   Google Scholar

[11]

Ph. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations, J. Evol. Equ., 1 (2001), 39-67.  doi: 10.1007/PL00001364.  Google Scholar

[12]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and Problems of Elliptic and Parabolic Type, Mem. Amer. Math. Soc., 166 (2003), ⅷ+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[13]

M. Disconzi, Y. Shao and G. Simonett, Some remarks on uniformly regular Riemannian manifolds, Math. Nachr., 289 (2016), 232-242.  doi: 10.1002/mana.201400354.  Google Scholar

[14]

J. Escher, U. F. Mayer and G. Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29 (1998), 1419-1433.  doi: 10.1137/S0036141097320675.  Google Scholar

[15]

J. Escher and P. B. Mucha, The surface diffusion flow on rough phase spaces, Discrete Contin. Dyn. Syst., 26 (2010), 431-453.  doi: 10.3934/dcds.2010.26.431.  Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[17]

H. Koch and T. Lamm, Geometric flows with rough initial data, Asian J. Math., 16 (2012), 209-235.  doi: 10.4310/AJM.2012.v16.n2.a3.  Google Scholar

[18]

E. Kuwert and R. Schätzle, The Willmore flow with small initial energy,, J. Differential Geom., 57 (2001), 409-441.  doi: 10.4310/jdg/1090348128.  Google Scholar

[19]

E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional,, Comm. Anal. Geom., 10 (2002), 307-339.  doi: 10.4310/CAG.2002.v10.n2.a4.  Google Scholar

[20]

E. Kuwert and R. Schätzle, Removability of point singularities of Willmore surfaces,, Ann. of Math. (2), 160 (2004), 315-357.  doi: 10.4007/annals.2004.160.315.  Google Scholar

[21]

J. LeCrone and G. Simonett, On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow, SIAM J. Math. Anal., 45 (2013), 2834-2869.   Google Scholar

[22]

J. LeCrone and G. Simonett, On the flow of non-axisymmetric perturbations of cylinders via surface diffusion, J. Differential Equations, 260 (2016), 5510-5531.  doi: 10.1016/j.jde.2015.12.008.  Google Scholar

[23]

J. LeCrone and G. Simonett, On quasilinear parabolic equations and continuous maximal regularity, Evolution Equations & Control Theory, 9 (2020), 61-86.  doi: 10.3934/eect.2020017.  Google Scholar

[24]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[25]

U. F. Mayer and G. Simonett, A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow, Interfaces Free Bound, 4 (2002), 89-109.  doi: 10.4171/IFB/54.  Google Scholar

[26]

U. F. Mayer and G. Simonett, Self-intersections for Willmore flow., Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics (Levico Terme, 2000), 341–348, Progr. Nonlinear Differential Equations Appl., 55, Birkhäuser, Basel, 2003.  Google Scholar

[27]

J. McCoy and G. Wheeler, Finite time singularities for the locally constrained Willmore flow of surfaces, Comm. Anal. Geom., 24 (2016), 843-886.  doi: 10.4310/CAG.2016.v24.n4.a7.  Google Scholar

[28]

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

[29]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\mathbb{R}^n$, Discrete Contin. Dyn. Syst., 33 (2013), 5407-5428.  doi: 10.3934/dcds.2013.33.5407.  Google Scholar

[30]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics. Birkhäuser Verlag, 2016. doi: 10.1007/978-3-319-27698-4.  Google Scholar

[31]

J. Prüss and M. Wilke, Addendum to the paper "On quasilinear parabolic evolution equations in weighted $L_p$–spaces Ⅱ", J. Evol. Equ., 17 (2017), 1381-1388.  doi: 10.1007/s00028-017-0382-6.  Google Scholar

[32]

H. Samelson, Orientability of hypersurfaces in $\mathbb{R}^n$, Proc. Amer. Math. Soc., 22 (1969), 301-302.  doi: 10.2307/2036976.  Google Scholar

[33]

Y. Shao, Real analytic solutions to the Willmore flow, Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, Electron. J. Diff. Eqns., Conf., 20 (2013), 151-164.   Google Scholar

[34]

Y. Shao, A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 45-85.  doi: 10.1007/s00030-014-0275-0.  Google Scholar

[35]

Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds, J. Evol. Equ., 1 (2014), 211-248.  doi: 10.1007/s00028-014-0218-6.  Google Scholar

[36]

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

[37]

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

[38]

G. Wheeler, Surface diffusion flow near spheres, Calc. Var. Partial Differential Equations, 44 (2012), 131-151.  doi: 10.1007/s00526-011-0429-4.  Google Scholar

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems: Volume Ⅰ., Abstract Linear Theory. Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.  Google Scholar

[2]

H. Amann, Elliptic operators with infinite-dimensional state spaces, J. Evol. Equ., 1 (2001), 143-188.  doi: 10.1007/PL00001367.  Google Scholar

[3]

H. Amann, Function spaces on singular manifolds, Math. Nachr., 286 (2013), 436-475.  doi: 10.1002/mana.201100157.  Google Scholar

[4]

H. Amann, Anisotropic function spaces on singular manifolds., arXiv: 1204.0606. Google Scholar

[5]

H. Amann, Uniformly regular and singular Riemannian manifolds, In: Elliptic and Parabolic Equations, 1–43, Springer Proc. Math. Stat., 119, Springer, Cham, 2015. doi: 10.1007/978-3-319-12547-3_1.  Google Scholar

[6]

H. Amann, Cauchy problems for parabolic equations in Sobolev-Slobodeckii and Hölder spaces on uniformly regular Riemannian manifolds, J. Evol. Equ., 17 (2017), 51-100.  doi: 10.1007/s00028-016-0347-1.  Google Scholar

[7]

H. Amann, M. Hieber and G. Simonett, Bounded $H_\infty$-calculus for elliptic operators., Differential Integral Equations, 7 (1994), 613-653.   Google Scholar

[8]

T. Asai, Quasilinear parabolic equation and its applications to fourth order equations with rough initial data, J. Math. Sci. Univ. Tokyo., 19 (2012), 507-532.   Google Scholar

[9]

Y. Bernard, G. Wheeler and V.-M. Wheeler, Concentration-compactness and finite-time singularities for Chen's flow, arXiv: 1706.01707. Google Scholar

[10]

S. Blatt, A singular example for the Willmore flow, Analysis (Munich), 29 (2009), 407-430.   Google Scholar

[11]

Ph. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations, J. Evol. Equ., 1 (2001), 39-67.  doi: 10.1007/PL00001364.  Google Scholar

[12]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and Problems of Elliptic and Parabolic Type, Mem. Amer. Math. Soc., 166 (2003), ⅷ+114 pp. doi: 10.1090/memo/0788.  Google Scholar

[13]

M. Disconzi, Y. Shao and G. Simonett, Some remarks on uniformly regular Riemannian manifolds, Math. Nachr., 289 (2016), 232-242.  doi: 10.1002/mana.201400354.  Google Scholar

[14]

J. Escher, U. F. Mayer and G. Simonett, The surface diffusion flow for immersed hypersurfaces, SIAM J. Math. Anal., 29 (1998), 1419-1433.  doi: 10.1137/S0036141097320675.  Google Scholar

[15]

J. Escher and P. B. Mucha, The surface diffusion flow on rough phase spaces, Discrete Contin. Dyn. Syst., 26 (2010), 431-453.  doi: 10.3934/dcds.2010.26.431.  Google Scholar

[16]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.  Google Scholar

[17]

H. Koch and T. Lamm, Geometric flows with rough initial data, Asian J. Math., 16 (2012), 209-235.  doi: 10.4310/AJM.2012.v16.n2.a3.  Google Scholar

[18]

E. Kuwert and R. Schätzle, The Willmore flow with small initial energy,, J. Differential Geom., 57 (2001), 409-441.  doi: 10.4310/jdg/1090348128.  Google Scholar

[19]

E. Kuwert and R. Schätzle, Gradient flow for the Willmore functional,, Comm. Anal. Geom., 10 (2002), 307-339.  doi: 10.4310/CAG.2002.v10.n2.a4.  Google Scholar

[20]

E. Kuwert and R. Schätzle, Removability of point singularities of Willmore surfaces,, Ann. of Math. (2), 160 (2004), 315-357.  doi: 10.4007/annals.2004.160.315.  Google Scholar

[21]

J. LeCrone and G. Simonett, On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow, SIAM J. Math. Anal., 45 (2013), 2834-2869.   Google Scholar

[22]

J. LeCrone and G. Simonett, On the flow of non-axisymmetric perturbations of cylinders via surface diffusion, J. Differential Equations, 260 (2016), 5510-5531.  doi: 10.1016/j.jde.2015.12.008.  Google Scholar

[23]

J. LeCrone and G. Simonett, On quasilinear parabolic equations and continuous maximal regularity, Evolution Equations & Control Theory, 9 (2020), 61-86.  doi: 10.3934/eect.2020017.  Google Scholar

[24]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel, 1995.  Google Scholar

[25]

U. F. Mayer and G. Simonett, A numerical scheme for axisymmetric solutions of curvature-driven free boundary problems, with applications to the Willmore flow, Interfaces Free Bound, 4 (2002), 89-109.  doi: 10.4171/IFB/54.  Google Scholar

[26]

U. F. Mayer and G. Simonett, Self-intersections for Willmore flow., Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics (Levico Terme, 2000), 341–348, Progr. Nonlinear Differential Equations Appl., 55, Birkhäuser, Basel, 2003.  Google Scholar

[27]

J. McCoy and G. Wheeler, Finite time singularities for the locally constrained Willmore flow of surfaces, Comm. Anal. Geom., 24 (2016), 843-886.  doi: 10.4310/CAG.2016.v24.n4.a7.  Google Scholar

[28]

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

[29]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\mathbb{R}^n$, Discrete Contin. Dyn. Syst., 33 (2013), 5407-5428.  doi: 10.3934/dcds.2013.33.5407.  Google Scholar

[30]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Monographs in Mathematics. Birkhäuser Verlag, 2016. doi: 10.1007/978-3-319-27698-4.  Google Scholar

[31]

J. Prüss and M. Wilke, Addendum to the paper "On quasilinear parabolic evolution equations in weighted $L_p$–spaces Ⅱ", J. Evol. Equ., 17 (2017), 1381-1388.  doi: 10.1007/s00028-017-0382-6.  Google Scholar

[32]

H. Samelson, Orientability of hypersurfaces in $\mathbb{R}^n$, Proc. Amer. Math. Soc., 22 (1969), 301-302.  doi: 10.2307/2036976.  Google Scholar

[33]

Y. Shao, Real analytic solutions to the Willmore flow, Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, Electron. J. Diff. Eqns., Conf., 20 (2013), 151-164.   Google Scholar

[34]

Y. Shao, A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 45-85.  doi: 10.1007/s00030-014-0275-0.  Google Scholar

[35]

Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds, J. Evol. Equ., 1 (2014), 211-248.  doi: 10.1007/s00028-014-0218-6.  Google Scholar

[36]

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

[37]

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

[38]

G. Wheeler, Surface diffusion flow near spheres, Calc. Var. Partial Differential Equations, 44 (2012), 131-151.  doi: 10.1007/s00526-011-0429-4.  Google Scholar

Figure 1.   
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Joachim Escher, Piotr B. Mucha. The surface diffusion flow on rough phase spaces. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 431-453. doi: 10.3934/dcds.2010.26.431
[2]

Yuanzhen Shao. Continuous maximal regularity on singular manifolds and its applications. Evolution Equations & Control Theory, 2016, 5 (2) : 303-335. doi: 10.3934/eect.2016006
[3]

Tôn Việt Tạ. Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4507-4542. doi: 10.3934/dcds.2017193
[4]

Jeremy LeCrone, Gieri Simonett. On quasilinear parabolic equations and continuous maximal regularity. Evolution Equations & Control Theory, 2020, 9 (1) : 61-86. doi: 10.3934/eect.2020017
[5]

Jeremy LeCrone, Gieri Simonett. Continuous maximal regularity and analytic semigroups. Conference Publications, 2011, 2011 (Special) : 963-970. doi: 10.3934/proc.2011.2011.963
[6]

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
[7]

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
[8]

Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2213-2219. doi: 10.3934/cpaa.2012.11.2213
[9]

Thomas H. Otway. Compressible flow on manifolds. Conference Publications, 2001, 2001 (Special) : 289-294. doi: 10.3934/proc.2001.2001.289
[10]

Gunduz Caginalp, Mark DeSantis. Multi-group asset flow equations and stability. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 109-150. doi: 10.3934/dcdsb.2011.16.109
[11]

Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228
[12]

Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe, Atsushi Yagi. Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 973-987. doi: 10.3934/dcds.2008.22.973
[13]

Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51.

[14]

Matthias Erbar, Max Fathi, Vaios Laschos, André Schlichting. Gradient flow structure for McKean-Vlasov equations on discrete spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6799-6833. doi: 10.3934/dcds.2016096
[15]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
[16]

José A. Carrillo, Dejan Slepčev, Lijiang Wu. Nonlocal-interaction equations on uniformly prox-regular sets. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1209-1247. doi: 10.3934/dcds.2016.36.1209
[17]

Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149
[18]

Haibo Cui, Qunyi Bie, Zheng-An Yao. Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156
[19]

Said Boulite, S. Hadd, L. Maniar. Critical spectrum and stability for population equations with diffusion in unbounded domains. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 265-276. doi: 10.3934/dcdsb.2005.5.265
[20]

Iordanka N. Panayotova, Pai Song, John P. McHugh. Spatial stability of horizontally sheared flow. Conference Publications, 2013, 2013 (special) : 611-618. doi: 10.3934/proc.2013.2013.611

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (92)
  • HTML views (343)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences