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

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

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.

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

[14]

J. EscherU. 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. McCoyG. 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. AmannM. 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. DisconziY. Shao and G. Simonett, Some remarks on uniformly regular Riemannian manifolds, Math. Nachr., 289 (2016), 232-242.  doi: 10.1002/mana.201400354.

[14]

J. EscherU. 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. McCoyG. 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.

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