一类几何流方程周期解的爆破
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摘要
研究双曲平均曲率流中一类几何流方程周期解的爆破问题.引入合适的黎曼不变量,将该方程化为对角型的一阶拟线性双曲型方程组.该方程组在Lax意义下不是真正非线性的.假设初值是周期的,且在一个周期内全变差很小,此外假设初值还满足一定的结构条件,可以证得该几何流方程的周期解必在有限时间内发生爆破,解的生命跨度估计可以给出.
This article considers the blow up problem for a class of nonlinear partial differential equations of geometric flow. By introducing the proper Riemann invariants, the equations can be reduced into a system of quasilinear hyperbolic equations in the diagonal form, which are not genuinely nonlinear in the sense of Lax.Under the assumptions that the initial data have small total variations in one period and some certain conditions are satisfied, the C~1 solutions can be proved to blow up in finite time. In addition, the life span of the C~1 classical solutions are derived.
This article considers the blow up problem for a class of nonlinear partial differential equations of geometric flow. By introducing the proper Riemann invariants, the equations can be reduced into a system of quasilinear hyperbolic equations in the diagonal form, which are not genuinely nonlinear in the sense of Lax.Under the assumptions that the initial data have small total variations in one period and some certain conditions are satisfied, the C~1 solutions can be proved to blow up in finite time. In addition, the life span of the C~1 classical solutions are derived.
引文
[1]He C L,Kong D X,Liu K F.Hyperbolic mean curvature flow[J].Differential Equations,2009,246:373-390.
[2]Huisken G,Ilmanen T.The inverse mean curvature flow and the Riemannian Penrose inequality[J].Differential Geom.,2001,59:353-437.
[3]Lefloch P G,Smoczyk K.The hyperbolic mean curvature flow[J].Math.Pures.Appl.,2008,90:591-614.
[4]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.
[5]Chou K S,Wo W F.On hyperbolic Gauss curvature flows[J].Differential Geometry,2011,89:455-485.
[6]Kong D X,Liu K F,Wang Z G.Hyperbolic mean curvature flow:Evolution of plane curves[J].Acta Math.Sci.,2009,29:493-514.
[7]Cesan Lamberto.Existence in the large of periodic solutions of hyperbolic partial differential equations[J].Archive for Rational Mechanics and Analysis,1965,20:170-190.
[8]Cheng K S.Formation of singularities for nonlinear hyperbolic partial differential equation[J].Contemp.Math.,1983,17:45-56.
[9]Glimm J,Lax P D.Decay of solutions of systems of nonlinear hyperbolic conservation laws[J].Mem.Am.Math.Soc.,1970,101:1-112.
[10]Klainerman S,Majda A.Formation of singularities for wave equations including the nonlinear vibrating string[J],Comm.Pure Appl.Math.,1980,33:241-263.
[11]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.
[12]Li T T.Global Classical Solutions for Quasilinear Hyperbolic Systems[M].Paris:Wiley-Masson,1994.
[13]Li T T,Kong D X.Blow up of periodic solutions to quasilinear hyperbolic systems[J].Nonlinear Anal.,1996,26:1779-1789.
[14]Lax P D.Hyperbolic systems of conservation laws II[J].Comm.Pure Appl.Math.,1957,10:537-556.
[15]Qu P,Xin Z P.Long time existence of entropy solutions to the one-dimensional non-isentropic Euler equations with periodic initial data[J].Arch.Rational Mech.Anal.,2015,216:221-259.
[16]Wang Z G.Hyperbolic mean curvature flow in Minkowski space[J].Nonlinear Analysis,2014,94:259-271.
[17]Wang Z G.Blow-up of periodic solutions to reducible quasilinear hyperbolic systems[J].Nonlinear Analysis,2010,73:704-712.
[18]Xiao J J.Some topics on hyperbolic conservation laws[D].Hong Kong:The Chinese University of Hong Kong,2008.
[2]Huisken G,Ilmanen T.The inverse mean curvature flow and the Riemannian Penrose inequality[J].Differential Geom.,2001,59:353-437.
[3]Lefloch P G,Smoczyk K.The hyperbolic mean curvature flow[J].Math.Pures.Appl.,2008,90:591-614.
[4]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.
[5]Chou K S,Wo W F.On hyperbolic Gauss curvature flows[J].Differential Geometry,2011,89:455-485.
[6]Kong D X,Liu K F,Wang Z G.Hyperbolic mean curvature flow:Evolution of plane curves[J].Acta Math.Sci.,2009,29:493-514.
[7]Cesan Lamberto.Existence in the large of periodic solutions of hyperbolic partial differential equations[J].Archive for Rational Mechanics and Analysis,1965,20:170-190.
[8]Cheng K S.Formation of singularities for nonlinear hyperbolic partial differential equation[J].Contemp.Math.,1983,17:45-56.
[9]Glimm J,Lax P D.Decay of solutions of systems of nonlinear hyperbolic conservation laws[J].Mem.Am.Math.Soc.,1970,101:1-112.
[10]Klainerman S,Majda A.Formation of singularities for wave equations including the nonlinear vibrating string[J],Comm.Pure Appl.Math.,1980,33:241-263.
[11]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.
[12]Li T T.Global Classical Solutions for Quasilinear Hyperbolic Systems[M].Paris:Wiley-Masson,1994.
[13]Li T T,Kong D X.Blow up of periodic solutions to quasilinear hyperbolic systems[J].Nonlinear Anal.,1996,26:1779-1789.
[14]Lax P D.Hyperbolic systems of conservation laws II[J].Comm.Pure Appl.Math.,1957,10:537-556.
[15]Qu P,Xin Z P.Long time existence of entropy solutions to the one-dimensional non-isentropic Euler equations with periodic initial data[J].Arch.Rational Mech.Anal.,2015,216:221-259.
[16]Wang Z G.Hyperbolic mean curvature flow in Minkowski space[J].Nonlinear Analysis,2014,94:259-271.
[17]Wang Z G.Blow-up of periodic solutions to reducible quasilinear hyperbolic systems[J].Nonlinear Analysis,2010,73:704-712.
[18]Xiao J J.Some topics on hyperbolic conservation laws[D].Hong Kong:The Chinese University of Hong Kong,2008.