May  2020, 19(5): 2655-2677. doi: 10.3934/cpaa.2020116

Existence of weak solution for mean curvature flow with transport term and forcing term

Department of Mathematics/Hakubi Center, Kyoto University, Kitashirakawa-Oiwakecho Sakyo Kyoto 606-8502, Japan

Received  February 2019 Revised  October 2019 Published  March 2020

Fund Project: This work was supported by JSPS KAKENHI Grant Numbers JP16K17622, JP18H03670, and JSPS Leading Initiative for Excellent Young Researchers (LEADER) operated by Funds for the Development of Human Resources in Science and Technology

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove the global existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.

Citation: Keisuke Takasao. Existence of weak solution for mean curvature flow with transport term and forcing term. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2655-2677. doi: 10.3934/cpaa.2020116
References:
[1]

W. Allard, On the first variation of a varifold, Ann. Math., 95 (1972), 417-491.  doi: 10.2307/1970868.

[2]

S. M. Allen and J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metal., 27 (1979), 1085-1095. 

[3]

F. J. AlmgrenJ. E. Taylor and L. H. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020.

[4]

L. BertiniP. Buttà and A. Pisante, Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound, Arch. Ration. Mech. Anal., 224 (2017), 659-707.  doi: 10.1007/s00205-017-1086-3.

[5] K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Princeton University Press, Princeton, N.J., 1978. 
[6]

L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff. Equ., 90 (1991), 211-237.  doi: 10.1016/0022-0396(91)90147-2.

[7]

W. K. BurtonN. Cabrera and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Philos. Trans. R. Soc. London. Ser. A., 243 (1951), 299-358.  doi: 10.1098/rsta.1951.0006.

[8]

Y. G. ChenY. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differ. Geom., 33 (1991), 749-786. 

[9]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Diff. Equ., 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.

[10]

X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differ. Geom., 44 (1996), 262-311. 

[11]

L. C. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Commun. Pure Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903.

[12]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differ. Geom., 33 (1991), 635-681. 

[13]

H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.

[14]

Y. Giga, Surface Evolution Equations, Birkhäuser Verlag, Basel, 2006.

[15]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9486-0.

[16]

J. E. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J., 35 (1986), 45-71.  doi: 10.1512/iumj.1986.35.35003.

[17]

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differ. Geom., 38 (1993), 417-461. 

[18]

T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108 (1994). doi: 10.1090/memo/0520.

[19]

K. Kasai and Y. Tonegawa, A general regularity theory for weak mean curvature flow, Calc. Var. Partial Differ. Equ., 50 (2014), 1-68.  doi: 10.1007/s00526-013-0626-4.

[20]

L. Kim and Y. Tonegawa, On the mean curvature flow of grain boundaries, Ann. Inst. Fourier, 67 (2017), 43-142.  doi: 10.5802/aif.3077.

[21]

C. LiuN. Sato and Y. Tonegawa, On the existence of mean curvature flow with transport term, Interfaces Free Bound., 12 (2010), 251-277.  doi: 10.4171/IFB/234.

[22]

C. LiuN. Sato and Y. Tonegawa, Two-phase flow problem coupled with mean curvature flow, Interfaces Free Bound., 14 (2012), 185-203.  doi: 10.4171/IFB/279.

[23]

C. Liu and N. J. Walkington, An Eulerian description of fluids containing visco-elastic particles, Arch. Ration. Mech. Anal., 159 (2001), 229-252.  doi: 10.1007/s002050100158.

[24]

S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differ. Equ., 3 (1995), 253-271.  doi: 10.1007/BF01205007.

[25]

N. G. Meyers and W. P. Ziemmer, Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. Math., 99 (1977), 1345-1360.  doi: 10.2307/2374028.

[26]

L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Commun. Pure Appl. Math., 38 (1985), 679-684.  doi: 10.1002/cpa.3160380515.

[27]

L. Mugnai and M. Röger, The Allen-Cahn action functional in higher dimensions, Interfaces Free Bound., 10 (2008), 45-78.  doi: 10.4171/IFB/179.

[28]

L. Mugnai and M. Röger, Convergence of perturbed Allen-Cahn equations to forced mean curvature flow, Indiana Univ. Math. J., 60 (2011), 41-75.  doi: 10.1512/iumj.2011.60.3949.

[29]

M. Röger and R. Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254 (2006), 675-714.  doi: 10.1007/s00209-006-0002-6.

[30]

J. RubinsteinP. Sternberg and J. B. Keller, Fast reaction, slow diffusion and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.  doi: 10.1137/0149007.

[31]

L. Simon, Lectures on geometric measure theory, Proceedings of the Center for Mathematical Analysis, Australian National University, Vol. 3, Australian National University, 1983.

[32]

H. M. Soner, Motion of a set by the curvature of its boundary, J. Differ. Equ., 101 (1993), 313-372.  doi: 10.1006/jdeq.1993.1015.

[33]

H. M. Soner, Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling, Arch. Ration. Mech. Anal., 131 (1995), 139-197.  doi: 10.1007/BF00386194.

[34]

H. M. Soner, Ginzburg-Landau equation and motion by mean curvature. I. Convergence, J. Geom. Anal., 7 (1997), 437-475.  doi: 10.1007/BF02921628.

[35]

K. Takasao, Existence of weak solution for volume preserving mean curvature flow via phase field method, Indiana Univ. Math. J., 66 (2017), 2015-2035.  doi: 10.1512/iumj.2017.66.6183.

[36]

K. Takasao and Y. Tonegawa, Existence and regularity of mean curvature flow with transport term in higher dimensions, Math. Ann., 364 (2016), 857-935.  doi: 10.1007/s00208-015-1237-5.

[37]

Y. Tonegawa, A second derivative Hölder estimate for weak mean curvature flow, Adv. Calc. Var., 7 (2014), 91-138.  doi: 10.1515/acv-2013-0104.

[38]

Y. Tonegawa, Brakke's Mean Curvature Flow, SpringerBriefs in Mathematics, Springer Singapore, 2019. doi: 10.1007/978-981-13-7075-5.

[39]

W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-1015-3.

show all references

References:
[1]

W. Allard, On the first variation of a varifold, Ann. Math., 95 (1972), 417-491.  doi: 10.2307/1970868.

[2]

S. M. Allen and J. W. Cahn, A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta. Metal., 27 (1979), 1085-1095. 

[3]

F. J. AlmgrenJ. E. Taylor and L. H. Wang, Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31 (1993), 387-438.  doi: 10.1137/0331020.

[4]

L. BertiniP. Buttà and A. Pisante, Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound, Arch. Ration. Mech. Anal., 224 (2017), 659-707.  doi: 10.1007/s00205-017-1086-3.

[5] K. A. Brakke, The Motion of a Surface by Its Mean Curvature, Princeton University Press, Princeton, N.J., 1978. 
[6]

L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Diff. Equ., 90 (1991), 211-237.  doi: 10.1016/0022-0396(91)90147-2.

[7]

W. K. BurtonN. Cabrera and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Philos. Trans. R. Soc. London. Ser. A., 243 (1951), 299-358.  doi: 10.1098/rsta.1951.0006.

[8]

Y. G. ChenY. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differ. Geom., 33 (1991), 749-786. 

[9]

X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Diff. Equ., 96 (1992), 116-141.  doi: 10.1016/0022-0396(92)90146-E.

[10]

X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation, J. Differ. Geom., 44 (1996), 262-311. 

[11]

L. C. EvansH. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Commun. Pure Appl. Math., 45 (1992), 1097-1123.  doi: 10.1002/cpa.3160450903.

[12]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differ. Geom., 33 (1991), 635-681. 

[13]

H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.

[14]

Y. Giga, Surface Evolution Equations, Birkhäuser Verlag, Basel, 2006.

[15]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. doi: 10.1007/978-1-4684-9486-0.

[16]

J. E. Hutchinson, Second fundamental form for varifolds and the existence of surfaces minimising curvature, Indiana Univ. Math. J., 35 (1986), 45-71.  doi: 10.1512/iumj.1986.35.35003.

[17]

T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Differ. Geom., 38 (1993), 417-461. 

[18]

T. Ilmanen, Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., 108 (1994). doi: 10.1090/memo/0520.

[19]

K. Kasai and Y. Tonegawa, A general regularity theory for weak mean curvature flow, Calc. Var. Partial Differ. Equ., 50 (2014), 1-68.  doi: 10.1007/s00526-013-0626-4.

[20]

L. Kim and Y. Tonegawa, On the mean curvature flow of grain boundaries, Ann. Inst. Fourier, 67 (2017), 43-142.  doi: 10.5802/aif.3077.

[21]

C. LiuN. Sato and Y. Tonegawa, On the existence of mean curvature flow with transport term, Interfaces Free Bound., 12 (2010), 251-277.  doi: 10.4171/IFB/234.

[22]

C. LiuN. Sato and Y. Tonegawa, Two-phase flow problem coupled with mean curvature flow, Interfaces Free Bound., 14 (2012), 185-203.  doi: 10.4171/IFB/279.

[23]

C. Liu and N. J. Walkington, An Eulerian description of fluids containing visco-elastic particles, Arch. Ration. Mech. Anal., 159 (2001), 229-252.  doi: 10.1007/s002050100158.

[24]

S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var. Partial Differ. Equ., 3 (1995), 253-271.  doi: 10.1007/BF01205007.

[25]

N. G. Meyers and W. P. Ziemmer, Integral inequalities of Poincaré and Wirtinger type for BV functions, Amer. J. Math., 99 (1977), 1345-1360.  doi: 10.2307/2374028.

[26]

L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Commun. Pure Appl. Math., 38 (1985), 679-684.  doi: 10.1002/cpa.3160380515.

[27]

L. Mugnai and M. Röger, The Allen-Cahn action functional in higher dimensions, Interfaces Free Bound., 10 (2008), 45-78.  doi: 10.4171/IFB/179.

[28]

L. Mugnai and M. Röger, Convergence of perturbed Allen-Cahn equations to forced mean curvature flow, Indiana Univ. Math. J., 60 (2011), 41-75.  doi: 10.1512/iumj.2011.60.3949.

[29]

M. Röger and R. Schätzle, On a modified conjecture of De Giorgi, Math. Z., 254 (2006), 675-714.  doi: 10.1007/s00209-006-0002-6.

[30]

J. RubinsteinP. Sternberg and J. B. Keller, Fast reaction, slow diffusion and curve shortening, SIAM J. Appl. Math., 49 (1989), 116-133.  doi: 10.1137/0149007.

[31]

L. Simon, Lectures on geometric measure theory, Proceedings of the Center for Mathematical Analysis, Australian National University, Vol. 3, Australian National University, 1983.

[32]

H. M. Soner, Motion of a set by the curvature of its boundary, J. Differ. Equ., 101 (1993), 313-372.  doi: 10.1006/jdeq.1993.1015.

[33]

H. M. Soner, Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling, Arch. Ration. Mech. Anal., 131 (1995), 139-197.  doi: 10.1007/BF00386194.

[34]

H. M. Soner, Ginzburg-Landau equation and motion by mean curvature. I. Convergence, J. Geom. Anal., 7 (1997), 437-475.  doi: 10.1007/BF02921628.

[35]

K. Takasao, Existence of weak solution for volume preserving mean curvature flow via phase field method, Indiana Univ. Math. J., 66 (2017), 2015-2035.  doi: 10.1512/iumj.2017.66.6183.

[36]

K. Takasao and Y. Tonegawa, Existence and regularity of mean curvature flow with transport term in higher dimensions, Math. Ann., 364 (2016), 857-935.  doi: 10.1007/s00208-015-1237-5.

[37]

Y. Tonegawa, A second derivative Hölder estimate for weak mean curvature flow, Adv. Calc. Var., 7 (2014), 91-138.  doi: 10.1515/acv-2013-0104.

[38]

Y. Tonegawa, Brakke's Mean Curvature Flow, SpringerBriefs in Mathematics, Springer Singapore, 2019. doi: 10.1007/978-981-13-7075-5.

[39]

W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, 1989. doi: 10.1007/978-1-4612-1015-3.

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