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  December 2020 Early access  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 and 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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[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. 

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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. 

[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.

[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.

[36]

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

[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.

[38]

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

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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[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. 

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

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

[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. 

[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.

[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.

[36]

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

[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.

[38]

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

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 and 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 and 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 and 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 and 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]

Yannan Liu, Linfen Cao. Lifespan theorem and gap lemma for the globally constrained Willmore flow. Communications on Pure and Applied Analysis, 2014, 13 (2) : 715-728. doi: 10.3934/cpaa.2014.13.715
[7]

Giovanni De Matteis, Gianni Manno. Lie algebra symmetry analysis of the Helfrich and Willmore surface shape equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 453-481. doi: 10.3934/cpaa.2014.13.453
[8]

Pascal Auscher, Sylvie Monniaux, Pierre Portal. The maximal regularity operator on tent spaces. Communications on Pure and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2016, 36 (12) : 6799-6833. doi: 10.3934/dcds.2016096
[15]

Dinh-Ke Tran, Nhu-Thang Nguyen. On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations. Communications on Pure and Applied Analysis, 2022, 21 (3) : 817-835. doi: 10.3934/cpaa.2021200
[16]

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 and Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
[17]

José A. Carrillo, Dejan Slepčev, Lijiang Wu. Nonlocal-interaction equations on uniformly prox-regular sets. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1209-1247. doi: 10.3934/dcds.2016.36.1209
[18]

Alberto Bressan, Fang Yu. Continuous Riemann solvers for traffic flow at a junction. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4149-4171. doi: 10.3934/dcds.2015.35.4149
[19]

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 and Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156
[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

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (206)
  • HTML views (344)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy