
-
Previous Article
Large data solutions for semilinear higher order equations
- DCDS-S Home
- This Issue
-
Next Article
A quantitative Hopf-type maximum principle for subsolutions of elliptic PDEs
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 |
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.
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. 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. |
[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. Google Scholar |
[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. 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. |
[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. Google Scholar |
[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. |
[1] |
Lekbir Afraites, Chorouk Masnaoui, Mourad Nachaoui. Shape optimization method for an inverse geometric source problem and stability at critical shape. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021006 |
[2] |
Ling-Bing He, Li Xu. On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021005 |
[3] |
Gui-Qiang Chen, Beixiang Fang. Stability of transonic shock-fronts in three-dimensional conical steady potential flow past a perturbed cone. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 85-114. doi: 10.3934/dcds.2009.23.85 |
[4] |
Do Lan. Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021002 |
[5] |
Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234 |
[6] |
Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 |
[7] |
João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321 |
[8] |
Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341 |
[9] |
Caterina Balzotti, Simone Göttlich. A two-dimensional multi-class traffic flow model. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2020034 |
[10] |
Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028 |
[11] |
Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020390 |
[12] |
Joan Carles Tatjer, Arturo Vieiro. Dynamics of the QR-flow for upper Hessenberg real matrices. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1359-1403. doi: 10.3934/dcdsb.2020166 |
[13] |
Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389 |
[14] |
Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021020 |
[15] |
Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561 |
[16] |
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 |
[17] |
Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020336 |
[18] |
Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021008 |
[19] |
Peter Frolkovič, Viera Kleinová. A new numerical method for level set motion in normal direction used in optical flow estimation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 851-863. doi: 10.3934/dcdss.2020347 |
[20] |
Kohei Nakamura. An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1093-1102. doi: 10.3934/dcdss.2020385 |
2019 Impact Factor: 1.233
Tools
Metrics
Other articles
by authors
[Back to Top]