一类双曲平均曲率流的对称与整体解

摘要/Abstract

摘要：

该文讨论一类图(Graph)的双曲平均曲率流, 它与Lagrangian型抛物平均曲率流密切相关. 首先研究该平均曲率流的对称及其对称约化方程, 得到若干常微分方程, 进而讨论解的存在性; 最后, 研究具有一般形式的双曲平均曲率流的整体BV解、光滑解的爆破和整体存在性等.

关键词: 双曲平均曲率流, 对称, 整体解, 解的爆破

Abstract:

In this paper, we consider a class of one dimensional hyperbolic mean curvature flow, which is related to the Lagrangian parabolic mean curvature flow. First we derive the point Lie symmetries of this flow and obtain several ordinary differential equations. The existence of solutions is investigated. Finally, the global BV solutions, the lifespan for classical solutions and global existence of smooth solutions for this hyperbolic mean curvature flow are studied in detail.

Key words: Hyperbolic mean curvature flow, Symmetry, Global solution, Blow up of solution

中图分类号:

O175.2

何春蕾,刘子慧. 一类双曲平均曲率流的对称与整体解[J]. 数学物理学报, 2022, 42(4): 1089-1102.

Chunlei He,Zihui Liu. Symmetries and Global Solutions for a Class of Hyperbolic Mean Curvature Flow[J]. Acta mathematica scientia,Series A, 2022, 42(4): 1089-1102.

参考文献 34

1	Begley T , Moore K . On short time existence of Lagrangian mean curvature flow. Math Ann, 2017, 367: 1473- 1515 doi: 10.1007/s00208-016-1420-3
2	Bressan A . Vanishing viscosity solutions of nonlinear hyperbolic systems. Annals of Math, 2005, 161: 223- 342 doi: 10.4007/annals.2005.161.223
3	Chau A , Chen J Y , He W Y . Lagrangian mean curvature flow for entire Lipschitz graphs. Calc Var, 2012, 44: 199- 220 doi: 10.1007/s00526-011-0431-x
4	Chau A , Chen J Y , Yuan Y . Lagrangian mean curvature flow for entire Lipschitz graphs Ⅱ. Math Ann, 2013, 357: 165- 183 doi: 10.1007/s00208-013-0897-2
5	Chen J Y , Pang C . Uniqueness of unbounded solutions of the Lagrangian mean curvature flow equation for graphs. C R Acad Sci Paris Ser I, 2009, 347: 1031- 1034 doi: 10.1016/j.crma.2009.06.020
6	Chou K S , Kwong Y C . General initial data for a class of parabolic equations including the curve shortening problem. Discrete and Continuous Dynamical Systems, 2020, 40: 2963- 2986 doi: 10.3934/dcds.2020157
7	Duan S S , He C L , Huang S J . Hyperbolic mean curvature flow for Lagrangian graphs: One dimensional case. Journal of Geometry and Physics, 2020, 157: 103853 doi: 10.1016/j.geomphys.2020.103853
8	Gage M , Hamilton R S . The heat equation shrinking convex plane curves. J Differential Geom, 1986, 23: 69- 96
9	Ginder E , Svadlenka K . Wave-type threshold dynamics and the hyperbolic mean curvature flow. Japan J Indust Appl Math, 2016, 33: 501- 523 doi: 10.1007/s13160-016-0221-0
10	Grayson M A . The heat equation shrinks embedded plane curves to round points. J Differential Geom, 1987, 26: 285- 314
11	He C L , Kong D X , Liu K F . Hyperbolic mean curvature flow. J Differential Equations, 2009, 246: 373- 390 doi: 10.1016/j.jde.2008.06.026
12	He C L , Huang S J , Xing X M . Self-similar solutions to the hyperbolic mean curvature flow. Acta Math Sci, 2017, 37: 657- 667 doi: 10.1016/S0252-9602(17)30028-0
13	Huisken G . Flow by mean curvature of convex surfaces into spheres. J Differential Geom, 1984, 20: 237- 266
14	Huisken G , Ilmanen T . The inverse mean curvature flow and the Riemannian Penrose inequality. J Differential Geom, 2001, 59: 353- 437
15	Kong D X. Cauchy Problem for Quasilinear Hyperbolic Systems. Tokyo: The Mathematical Society of Japan, 2000
16	Kong D X . Hyperbolic Geometric Flow//Proceedings of ICCM 2007. Beijing: Higher Education Press, 2007, 95- 110
17	Kong D X , Liu K F . Wave character of metrics and hyperbolic geometric flow. J Math Phys, 2007, 48: 103508 doi: 10.1063/1.2795839
18	Kong D X , Liu K F , Wang Z G . Hyperbolic mean curvature flow: evolution of plane curves. Acta Math Sci, 2009, 29B: 493- 514
19	Kong D X , Liu K F , Wang Y Z . Life-span of classical solutions to hyperbolic geometric flow in two space variables with slow decay initial data. Comm Partial Differential Equations, 2011, 36: 162- 184 doi: 10.1080/03605302.2010.513409
20	Kong D X , Liu K F , Xu D L . The hyperbolic geometric flow on Riemann surfaces. Comm Partial Differential Equations, 2009, 34: 553- 580 doi: 10.1080/03605300902768933
21	Kong D X , Wang Z G . Formation of singularities in the motion of plane curves under hyperbolic mean curvature flow. J Differential Equations, 2009, 247: 1694- 1719 doi: 10.1016/j.jde.2009.04.016
22	Kusumasari V . Hyperbolic mean curvature flow with an obstacle. Sci Rep Kanazawa Univ, 2018, 62: 1- 22
23	LeFloch P G , Smoczyk K . The hyperbolic mean curvature flow. J Math Pures Appl, 2008, 90: 591- 614 doi: 10.1016/j.matpur.2008.09.006
24	Li T T, Zhou Y. Nonlinear Wave Equations. Berlin: Springer, 2017
25	Nguyen T A , Yuan Y . A priori estimates for Lagrangian mean curvature flows. International Mathematics Research Notices, 2011, 2011: 4376- 4383
26	Olver P J. Applications of Lie Groups to Differential Equations. Berlin: Springer-Verlag, 1993
27	Smoczyk K . Longtime existence of the Lagrangian mean curvature flow. Calculus of Variations, 2004, 20: 25- 46
28	Smoczyk K , Wang M T . Mean curvature flows of Lagrangian submanifolds with convex potentials. J Differential Geometry, 2002, 62: 243- 257
29	Smoczyk K . Angle theorems for the Lagrangian mean curvature flow. Math Z, 2002, 240: 849- 883 doi: 10.1007/s002090100402
30	Xu C. Symmetries of geometric fows. 2010, ArXiv: 1001. 1394
31	Yau S T . Review of geometry and analysis. Asian J Math, 2000, 4: 235- 278 doi: 10.4310/AJM.2000.v4.n1.a16
32	Yuan Y. Special Lagrangian Equations//Chen J, Lu P, Lu Z, Zhang Z. Geometric Analysis. Boston: Birkhauser, 2020, 521-536
33	Yuan Y . A Bernstein problem for special Lagrangian equations. Invent Math, 2002, 150: 117- 125 doi: 10.1007/s00222-002-0232-0
34	Zhu X P. Lectures on Mean Curvature flows. Providence, RI: Amer Math Soc, 2002

[1]	马晶晶, 魏娜. 上半空间分数阶 $p$-Laplace 算子相关的抛物方程解的单调性[J]. 数学物理学报, 2025, 45(4): 1086-1099.
[2]	王啟明,邓雪梅. 含引力的定常 Euler 方程组球对称解的适定性[J]. 数学物理学报, 2025, 45(2): 359-370.
[3]	龚思梦, 张学耀, 郭真华. 变粘可压缩轴对称 Navier-Stokes 方程组全局强解的存在性[J]. 数学物理学报, 2024, 44(6): 1445-1475.
[4]	石金诚, 刘炎. 带不同幂次型非线性项的半线性三阶发展方程整体解的存在性与爆破[J]. 数学物理学报, 2024, 44(6): 1550-1562.
[5]	柴孟岑, 代玉霞. 拟凸度量空间中的自由拟共形映射[J]. 数学物理学报, 2024, 44(5): 1127-1135.
[6]	向延誉, 王爱平. 两个高阶正则拟微分算子积的对称性[J]. 数学物理学报, 2024, 44(2): 265-275.
[7]	代吉永芷, 王逸如, 黄水波. 混合局部-非局部椭圆方程奇异解的单调性和对称性[J]. 数学物理学报, 2024, 44(2): 453-464.
[8]	兰超, 范兴亚. 仿射对称空间 $SO^\ast(6)/SO(3,\mathbb{C})$ 上的离散序列[J]. 数学物理学报, 2023, 43(6): 1649-1658.
[9]	唐炎娟. 分数阶抛物方程整体解的径向对称性与单调性[J]. 数学物理学报, 2023, 43(5): 1409-1416.
[10]	王梓欢,王超. 一类双质子弱耦合碰撞系统的对称周期解[J]. 数学物理学报, 2023, 43(5): 1427-1439.
[11]	葛斌, 袁文硕. 全空间$\mathbb{R} ^N$上双相问题径向解的多解性[J]. 数学物理学报, 2023, 43(2): 433-446.
[12]	张明玉. 输运系数依赖温度的可压缩辐射流体解的整体存在性[J]. 数学物理学报, 2023, 43(2): 458-480.
[13]	林杰, 王天怡. 三维柱对称定常非齐次不可压 Euler 方程管道问题解的适定性及无穷远渐近速率[J]. 数学物理学报, 2023, 43(1): 219-237.
[14]	苏晓,王书彬. 四阶p-广义Benney-Luke方程的初值问题[J]. 数学物理学报, 2022, 42(6): 1744-1753.
[15]	魏均平,黄小梦,张贻民. 分数阶Kirchhoff型约束变分问题Schwarz对称极小解的存在性[J]. 数学物理学报, 2022, 42(6): 1719-1728.