弹性力学方程解的整体适定性
|
摘要
本文为一篇综述文章,主要回顾中外数学家在可压缩和不可压缩弹性力学方程平衡态附近经典解的整体适定性方面所取得的关键研究成果.由于这里所涉及的研究思想和方法与研究拟线性波动方程相应问题的思想和方法密切相关,因此也将回顾拟线性波动方程的一些相应问题的理论和研究方法.本文将尽可能简单明了地指出各研究课题的关键困难及克服它们的基本想法,并对其中大部分关键成果给予更为直截了当的证明.本文还将提出几个公开问题并简单讨论其困难所在,以期向更年轻的专家学者抛砖引玉.
This is a survey paper. We mainly revisit those highlighted results on the global well-posedness of classical solutions near equilibrium for systems of compressible and incompressible elasticity. Due to the fact that the methods and ideas used for elasticity are closely related to those used for quasilinear wave equations, we will also revisit some corresponding highlighted results on quasilinear wave equations. We will try to pinpoint the key difficulties of those problems and the key methods to overcome them, and present direct and simple proofs as many as we can. We will also raise some open questions and make a few comments to attract the attention of young researchers.
This is a survey paper. We mainly revisit those highlighted results on the global well-posedness of classical solutions near equilibrium for systems of compressible and incompressible elasticity. Due to the fact that the methods and ideas used for elasticity are closely related to those used for quasilinear wave equations, we will also revisit some corresponding highlighted results on quasilinear wave equations. We will try to pinpoint the key difficulties of those problems and the key methods to overcome them, and present direct and simple proofs as many as we can. We will also raise some open questions and make a few comments to attract the attention of young researchers.
引文
1 Sideris T C.Nonresonance and global existence of prestressed nonlinear elastic waves.Ann of Math(2),2000,151:849-874
2 Agemi R.Global existence of nonlinear elastic waves.Invent Math,2000,142:225-250
3 Sideris T C,Thomases B.Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit.Comm Pure Appl Math,2005,58:750-788
4 Sideris T C,Thomases B.Global existence for three-dimensional incompressible isotropic elastodynamics.Comm Pure Appl Math,2007,60:1707-1730
5 Lei Z.Global well-posedness of incompressible elastodynamics in two dimensions.Comm Pure Appl Math,2016,69:2072-2106
6 Klainerman S.The null condition and global existence to nonlinear wave equations.Lect Appl Math,1986,23:293-326
7 Christodoulou D.Global solutions of nonlinear hyperbolic equations for small initial data.Comm Pure Appl Math,1986,39:267-282
8 Alinhac S.The null condition for quasilinear wave equations in two space dimensions,I.Invent Math,2001,145:597-618
9 Lin F H.Some analytical issues for elastic complex fluids.Comm Pure Appl Math,2012,65:893-919
10 Cai Y,Lei Z,Lin F,et al.Vanishing viscosity limit for the incompressible viscoelasticity in two dimensions.ArXiv:1703.01176,2017
11 Lei Z,Liu C,Zhou Y.Global solutions for incompressible viscoelastic fluids.Arch Ration Mech Anal,2008,188:371-398
12 Lei Z.Global existence and incompressible limit for systems of viscoelastic fluids.PhD Thesis.Shanghai:Fudan University,2006
13 Li T T,Chen Y M.Global Classical Solutions for Nonlinear Evolution Equations.Pitman Monographs and Surveys in Pure and Applied Mathematics,vol.45.Harlow:Longman Scientific&Technical,1992
14 Klainerman S.Uniform decay estimates and the Lorentz invariance of the classical wave equation.Comm Pure Appl Math,1985,38:321-332
15 Sideris T C.The null condition and global existence of nonlinear elastic waves.Invent Math,1996,123:323-342
16 Sideris T C,Thomases B.Local energy decay for solutions of multi-dimensional isotropic symmetric hyperbolic systems.J Hyperbolic Differ Equ,2006,3:673-690
17 Lei Z,Wang F.Uniform bound of the highest energy for the three dimensional incompressible elastodynamics.Arch Ration Mech Anal,2015,216:593-622
18 Wang X.Global existence for the 2D incompressible isotropic elastodynamics for small initial data.Ann Henri Poincaré,2017,18:1213-1267
19 Lei Z,Sideris T C,Zhou Y.Almost global existence for 2-D incompressible isotropic elastodynamics.Trans Amer Math Soc,2015,367:8175-8197
20 Alinhac S.Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions,II.Acta Math,1999,182:1-23
21 Alinhac S.Blowup of small data solutions for a quasilinear wave equation in two space dimensions.Ann of Math(2),1999,149:97-127
22 Alinhac S.Rank 2 singular solutions for quasilinear wave equations.Int Math Res Not IMRN,2000,2000:955-984
23 Alinhac S.The null condition for quasilinear wave equations in two space dimensions,II.Amer J Math,2001,123:1071-1101
24 John F.Blow-up for quasi-linear wave equations in three space dimensions.Comm Pure Appl Math,1981,34:29-51
25 John F.Formation of singularities in elastic waves.In:Lecture Notes in Physics,vol.195.New York:Springer-Verlag,1984,194-210
26 John F.Almost global existence of elastic waves of finite amplitude arising from small initial disturbances.Comm Pure Appl Math,1988,41:615-666
27 John F,Klainerman S.Almost global existence to nonlinear wave equations in three space dimensions.Comm Pure Appl Math,1984,37:443-455
28 Katayama S.Global existence for systems of nonlinear wave equations in two space dimensions,II.Publ Res Inst Math Sci,1995,31:645-665
29 Klainerman S,Sideris T C.On almost global existence for nonrelativistic wave equations in 3D.Comm Pure Appl Math,1996,49:307-321
30 Ogden R.Nonlinear Elastic Deformations.Mathematics and Its Applications.New York:Halsted Press,1984
31 Sideris T C,Tu S Y.Global existence for systems of nonlinear wave equations in 3D with multiple speeds.SIAM JMath Anal,2001,33:477-488
32 Tahvildar-Zadeh A S.Relativistic and nonrelativistic elastodynamics with small shear strains.Ann Inst H Poincaré Phys Théor,1998,69:275-307
33 Yokoyama K.Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions.J Math Soc Japan,2000,52:609-632
34 Yin S,Cui X.Long-time existence of nonlinear inhomogeneous compressible elastic waves.ArXiv:1707.00504,2017
35 Wang F.Uniform bound of Sobolev norms of solutions to 3D nonlinear wave equations with null condition.J Differential Equations,2014,256:4013-4032
36 Alinhac S.Geometric Analysis of Hyperbolic Differential Equations:An Introduction.Cambridge:Cambridge University Press,2010
1)由于不可压缩条件的限制,此时的弹性力学方程仍然是非线性的.
2 )在不可压缩情形时需要用到文献[11, 12]中发现的恒等式.
3)i、j和k也可以取值于{0, 1, 2,..., n},?0=?t.请读者自行对证明作适当的改动.
4 )这里对尺度变换算子作了少许改动以使后面的陈述及运算更简便.
5)众所周知,二维情形Hardy不等式∥r-1f∥L2∥?f∥L2是不成立的.
6 )系数Cijk中的指标一定是成对出现的lmn.
7)在广义能量Es的定义中只需要一个时空导数.
2 Agemi R.Global existence of nonlinear elastic waves.Invent Math,2000,142:225-250
3 Sideris T C,Thomases B.Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit.Comm Pure Appl Math,2005,58:750-788
4 Sideris T C,Thomases B.Global existence for three-dimensional incompressible isotropic elastodynamics.Comm Pure Appl Math,2007,60:1707-1730
5 Lei Z.Global well-posedness of incompressible elastodynamics in two dimensions.Comm Pure Appl Math,2016,69:2072-2106
6 Klainerman S.The null condition and global existence to nonlinear wave equations.Lect Appl Math,1986,23:293-326
7 Christodoulou D.Global solutions of nonlinear hyperbolic equations for small initial data.Comm Pure Appl Math,1986,39:267-282
8 Alinhac S.The null condition for quasilinear wave equations in two space dimensions,I.Invent Math,2001,145:597-618
9 Lin F H.Some analytical issues for elastic complex fluids.Comm Pure Appl Math,2012,65:893-919
10 Cai Y,Lei Z,Lin F,et al.Vanishing viscosity limit for the incompressible viscoelasticity in two dimensions.ArXiv:1703.01176,2017
11 Lei Z,Liu C,Zhou Y.Global solutions for incompressible viscoelastic fluids.Arch Ration Mech Anal,2008,188:371-398
12 Lei Z.Global existence and incompressible limit for systems of viscoelastic fluids.PhD Thesis.Shanghai:Fudan University,2006
13 Li T T,Chen Y M.Global Classical Solutions for Nonlinear Evolution Equations.Pitman Monographs and Surveys in Pure and Applied Mathematics,vol.45.Harlow:Longman Scientific&Technical,1992
14 Klainerman S.Uniform decay estimates and the Lorentz invariance of the classical wave equation.Comm Pure Appl Math,1985,38:321-332
15 Sideris T C.The null condition and global existence of nonlinear elastic waves.Invent Math,1996,123:323-342
16 Sideris T C,Thomases B.Local energy decay for solutions of multi-dimensional isotropic symmetric hyperbolic systems.J Hyperbolic Differ Equ,2006,3:673-690
17 Lei Z,Wang F.Uniform bound of the highest energy for the three dimensional incompressible elastodynamics.Arch Ration Mech Anal,2015,216:593-622
18 Wang X.Global existence for the 2D incompressible isotropic elastodynamics for small initial data.Ann Henri Poincaré,2017,18:1213-1267
19 Lei Z,Sideris T C,Zhou Y.Almost global existence for 2-D incompressible isotropic elastodynamics.Trans Amer Math Soc,2015,367:8175-8197
20 Alinhac S.Blowup of small data solutions for a class of quasilinear wave equations in two space dimensions,II.Acta Math,1999,182:1-23
21 Alinhac S.Blowup of small data solutions for a quasilinear wave equation in two space dimensions.Ann of Math(2),1999,149:97-127
22 Alinhac S.Rank 2 singular solutions for quasilinear wave equations.Int Math Res Not IMRN,2000,2000:955-984
23 Alinhac S.The null condition for quasilinear wave equations in two space dimensions,II.Amer J Math,2001,123:1071-1101
24 John F.Blow-up for quasi-linear wave equations in three space dimensions.Comm Pure Appl Math,1981,34:29-51
25 John F.Formation of singularities in elastic waves.In:Lecture Notes in Physics,vol.195.New York:Springer-Verlag,1984,194-210
26 John F.Almost global existence of elastic waves of finite amplitude arising from small initial disturbances.Comm Pure Appl Math,1988,41:615-666
27 John F,Klainerman S.Almost global existence to nonlinear wave equations in three space dimensions.Comm Pure Appl Math,1984,37:443-455
28 Katayama S.Global existence for systems of nonlinear wave equations in two space dimensions,II.Publ Res Inst Math Sci,1995,31:645-665
29 Klainerman S,Sideris T C.On almost global existence for nonrelativistic wave equations in 3D.Comm Pure Appl Math,1996,49:307-321
30 Ogden R.Nonlinear Elastic Deformations.Mathematics and Its Applications.New York:Halsted Press,1984
31 Sideris T C,Tu S Y.Global existence for systems of nonlinear wave equations in 3D with multiple speeds.SIAM JMath Anal,2001,33:477-488
32 Tahvildar-Zadeh A S.Relativistic and nonrelativistic elastodynamics with small shear strains.Ann Inst H Poincaré Phys Théor,1998,69:275-307
33 Yokoyama K.Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions.J Math Soc Japan,2000,52:609-632
34 Yin S,Cui X.Long-time existence of nonlinear inhomogeneous compressible elastic waves.ArXiv:1707.00504,2017
35 Wang F.Uniform bound of Sobolev norms of solutions to 3D nonlinear wave equations with null condition.J Differential Equations,2014,256:4013-4032
36 Alinhac S.Geometric Analysis of Hyperbolic Differential Equations:An Introduction.Cambridge:Cambridge University Press,2010
1)由于不可压缩条件的限制,此时的弹性力学方程仍然是非线性的.
2 )在不可压缩情形时需要用到文献[11, 12]中发现的恒等式.
3)i、j和k也可以取值于{0, 1, 2,..., n},?0=?t.请读者自行对证明作适当的改动.
4 )这里对尺度变换算子作了少许改动以使后面的陈述及运算更简便.
5)众所周知,二维情形Hardy不等式∥r-1f∥L2∥?f∥L2是不成立的.
6 )系数Cijk中的指标一定是成对出现的lmn.
7)在广义能量Es的定义中只需要一个时空导数.