Existence and regularity of mean curvature flow with transport term in higher dimensions. (English) Zbl 1351.53083
Authors’ abstract: Given an initial \(C^1\) hypersurface and a time-dependent vector field in a Sobolev space, we prove a global-time existence of a family of hypersurfaces which starts from the given hypersurface and which moves by the velocity equal to the mean curvature plus the given vector field. We show that the hypersurfaces are \(C^1\) for a short time and, even after some singularities occur, almost everywhere \(C^1\) away from the higher multiplicity region.
Reviewer: Jan Swoboda (München)
MSC:
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 35K55 | Nonlinear parabolic equations |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
References:
| [1] | Allard, W.: On the first variation of a varifold. Ann. Math. 95(2), 417-491 (1972) · Zbl 0252.49028 · doi:10.2307/1970868 |
| [2] | Allen, S.M., Cahn, J.W.: A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta. Metal. 27, 1085-1095 (1979) · doi:10.1016/0001-6160(79)90196-2 |
| [3] | Almgren, F.J., Taylor, J.E., Wang, L.-H.: Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31(2), 387-438 (1993) · Zbl 0783.35002 · doi:10.1137/0331020 |
| [4] | Ambrosio, L.: Geometric Evolution Problems, Distance Function and Viscosity Solutions. Calculus of Variations and Partial Differential Equations (Pisa, 1996), pp. 5-93. Springer, Berlin (1996) · Zbl 0956.35002 |
| [5] | Bellettini, G.: Lecture notes on mean curvature flow: barriers and singular perturbations. Edizioni Della Normale, 12, Scuola Normale Supriore, Pisa (2014) · Zbl 1312.53002 |
| [6] | Brakke, K.: The Motion of a Surface by Its Mean Curvature. Mathematical Notes 20. Princeton University Press, Princeton (1978) · Zbl 0386.53047 |
| [7] | Bronsard, L., Kohn, R.V.: Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics. J. Differ. Equ. 90(2), 211-237 (1991) · Zbl 0735.35072 · doi:10.1016/0022-0396(91)90147-2 |
| [8] | Bronsard, L., Stoth, B.: On the existence of high multiplicity interfaces. Math. Res. Lett. 3(1), 41-50 (1996) · Zbl 0853.35008 · doi:10.4310/MRL.1996.v3.n1.a4 |
| [9] | Chen, X.: Generation and propagation of interface in reaction-diffusion equations. J. Differ. Equ. 96(1), 116-141 (1992) · Zbl 0765.35024 · doi:10.1016/0022-0396(92)90146-E |
| [10] | Chen, X.-Y.: Dynamics of interfaces in reaction diffusion systems. Hiroshima Math. J. 21(1), 47-83 (1991) · Zbl 0784.35048 |
| [11] | Chen, Y.-G., Giga, Y., Goto, S.: Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33(3), 749-786 (1991) · Zbl 0696.35087 |
| [12] | Colding, T.H., Minicozzi II, W.P.: Minimal surfaces and mean curvature flow. Surveys in geometric analysis and relativity. Adv. Lect. Math. 20, 73-143 (2011) · Zbl 1261.53006 |
| [13] | de Mottoni, P., Schatzman, M.: Evolution géométric d’interfaces. C. R. Acad. Paris Sci. Sér. I Math. 309(7), 453-4558 (1989) · Zbl 0698.35078 |
| [14] | Ecker, K.: Regularity theory for mean curvature flow. In: Progress in Nonlinear Differential Equations and their Applications 57. Birkhäuser Boston Inc, Boston (2004) · Zbl 1058.53054 |
| [15] | Evans, L.C., Soner, H.M., Souganidis, P.E.: Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45(9), 1097-1123 (1992) · Zbl 0801.35045 · doi:10.1002/cpa.3160450903 |
| [16] | Evans, L.C., Spruck, J.: Motion of level sets by mean curvature I. J. Differ. Geom. 33(3), 635-681 (1991) · Zbl 0726.53029 |
| [17] | Evans, L.E., Spruck, J.: Motion of level sets by mean curvature II. Trans. Am. Math. Soc. 330(1), 321-332 (1992) · Zbl 0776.53005 · doi:10.1090/S0002-9947-1992-1068927-8 |
| [18] | Evans, L.C., Spruck, J.: Motion of level sets by mean curvature IV. J. Geom. Anal. 5(1), 77-114 (1995) · Zbl 0829.53040 · doi:10.1007/BF02926443 |
| [19] | Fife, P.C.: Dynamics of internal layers and diffusive interfaces. In: CBMS-NSF Reg. Conf. Ser. Applied Math., vol. 53. SIAM, Philadelphia (1988) · Zbl 0684.35001 |
| [20] | Giga, Y.: Surface Evolution Equations. A Level Set Approach. Monographs in Mathematics 99. Birkhäuser Verlag, Basel (2006) · Zbl 1096.53039 |
| [21] | Giga, Y., Goto, S.: Geometric evolution of phase-boundaries, On the evolution of phase boundaries (Minneapolis, MN, 1990-91). IMA Vol. Math. Appl. 43, 51-65 (1992) · Zbl 0771.35027 · doi:10.1007/978-1-4613-9211-8_3 |
| [22] | Giga, Y., Goto, S., Ishii, H., Sato, M.-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40(2), 443-470 (1991) · Zbl 0836.35009 · doi:10.1512/iumj.1991.40.40023 |
| [23] | Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983) · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0 |
| [24] | Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser, Basel (1984) · Zbl 0545.49018 · doi:10.1007/978-1-4684-9486-0 |
| [25] | Haslhofer, R., Kleiner, B.: Mean curvature flow of mean convex hypersurfaces. arXiv:1304.0926 · Zbl 1360.53069 |
| [26] | Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31(1), 285-299 (1990) · Zbl 0694.53005 |
| [27] | Hutchinson, J.E., Tonegawa, Y.: Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. PDE 10(1), 49-84 (2000) · Zbl 1070.49026 · doi:10.1007/PL00013453 |
| [28] | Ilmanen, T.: Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38(2), 417-461 (1993) · Zbl 0784.53035 |
| [29] | Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Am. Math. Soc. 108, 520 (1994) · Zbl 0798.35066 |
| [30] | Kasai, K., Tonegawa, Y.: A general regularity theory for weak mean curvature flow. Calc. Var. PDE 50(1), 1-68 (2014) · Zbl 1298.53063 · doi:10.1007/s00526-013-0626-4 |
| [31] | Kim, Y., Lai, M.-C., Peskin, C.: Numerical simulations of two-dimensional foam by the immersed boundary method. J. Comput. Phys. 229(13), 5194-5207 (2010) · Zbl 1346.76139 · doi:10.1016/j.jcp.2010.03.035 |
| [32] | Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasi-linear equations of parabolic type. Am. Math. Soc. (1968, Translated from Russian) · Zbl 1126.49010 |
| [33] | Liu, C., Sato, N., Tonegawa, Y.: On the existence of mean curvature flow with transport term. Interf. Free Bound 12(2), 251-277 (2010) · Zbl 1201.53076 · doi:10.4171/IFB/234 |
| [34] | Luckhaus, S., Sturzenhecker, T.: Implicit time discretization for the mean curvature flow equation. Calc. Var. PDE 3(2), 253-271 (1995) · Zbl 0821.35003 · doi:10.1007/BF01205007 |
| [35] | Mantegazza, C.: Lecture Notes on Mean Curvature Flow. Progress in Mathematics, vol. 290. Birkhäuser/Springer Basel AG, Basel (2011) · Zbl 1230.53002 · doi:10.1007/978-3-0348-0145-4 |
| [36] | Meyers, N.G., Ziemmer, W.P.: Integral inequalities of Poincaré and Wirtinger type for BV functions. Am. J. Math. 99(6), 1345-1360 (1977) · Zbl 0416.46025 · doi:10.2307/2374028 |
| [37] | Mugnai, L., Röger, M.: Convergence of perturbed Allen-Cahn equations to forced mean curvature flow. Indiana Univ. Math. J. 60(1), 41-75 (2011) · Zbl 1316.35150 · doi:10.1512/iumj.2011.60.3949 |
| [38] | Röger, M., Schätzle, R.: On a modified conjecture of De Giorgi. Math. Z. 254(4), 675-714 (2006) · Zbl 1126.49010 · doi:10.1007/s00209-006-0002-6 |
| [39] | Rubinstein, J., Sternberg, P., Keller, J.B.: Fast reaction, slow diffusion and curve shortening. SIAM J. Appl. Math. 49(1), 116-133 (1989) · Zbl 0701.35012 · doi:10.1137/0149007 |
| [40] | Sato, N.: A simple proof of convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature. Indiana Univ. Math. J. 57(4), 1743-1751 (2008) · Zbl 1160.35444 · doi:10.1512/iumj.2008.57.3283 |
| [41] | Simon, L.: Lectures on Geometric Measure Theory. Proc. Centre Math. Anal. Austral. Nat. Univ. 3 (1983) · Zbl 0546.49019 |
| [42] | Soner, H.M.: Convergence of the phase-field equations to the Mullins-Sekerka problem with kinetic undercooling. Arch. Rational Mech. Anal. 131(2), 139-197 (1995) · Zbl 0829.73010 · doi:10.1007/BF00386194 |
| [43] | Soner, H.M.: Ginzburg-Landau equation and motion by mean curvature. Converg. J. Geom. Anal. I. 7(3), 437-475 (1997) · Zbl 0935.35060 · doi:10.1007/BF02921628 |
| [44] | Takasao, K.: Gradient estimates and existence of mean curvature flow with transport term. Differ. Integral Equ. 26(1-2), 141-154 (2013) · Zbl 1299.35183 |
| [45] | Tonegawa, Y.: Integrality of varifolds in the singular limit of reaction-diffusion equations. Hiroshima Math. J. 33(3), 323-341 (2003) · Zbl 1059.35061 |
| [46] | Tonegawa, Y.: A second derivative Hölder estimate for weak mean curvature flow. Adv. Calc. Var. 7(1), 91-138 (2014) · Zbl 1283.53064 · doi:10.1515/acv-2013-0104 |
| [47] | White, B.: Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Angew. Math. 488, 1-35 (1997) · Zbl 0874.58007 |
| [48] | White, B.: Evolution of curves and surfaces by mean curvature. In: Proceedings of the ICM, 1 (Beijing, 2002), pp. 525-538. Higher Ed. Press, Beijing (2002) · Zbl 1036.53045 |
| [49] | Ziemer, W.P.: Weakly Differentiable Functions. Springer, Berlin (1989) · Zbl 0692.46022 · doi:10.1007/978-1-4612-1015-3 |
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