July  2016, 36(7): 4051-4062. doi: 10.3934/dcds.2016.36.4051

Remarks on nonlinear elastic waves in the radial symmetry in 2-D

1. 

School of Mathematical Sciences, Fudan University, Shanghai 200433

Received  June 2015 Revised  November 2015 Published  March 2016

In this paper, we first give the explicit variational structure of the nonlinear elastic waves for isotropic, homogeneous, hyperelastic materials in 2-D. Based on this variational structure, we suggest a null condition which is a kind of structural condition on the nonlinearity in order to stop the formation of finite time singularities of local smooth solutions. In the radial symmetric case, inspired by Alinhac's work on 2-D quasilinear wave equations [S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001) 597--618], we show that such null condition can ensure the global existence of smooth solutions with small initial data.
Citation: Dongbing Zha. Remarks on nonlinear elastic waves in the radial symmetry in 2-D. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 4051-4062. doi: 10.3934/dcds.2016.36.4051
References:
[1]

R. Agemi, Global existence of nonlinear elastic waves, Invent. Math., 142 (2000), 225-250. doi: 10.1007/s002220000084.

[2]

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math., 145 (2001), 597-618. doi: 10.1007/s002220100165.

[3]

S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and their Applications, 17, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2.

[4]

D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205.

[5]

P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity, vol. 20 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1988.

[6]

P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916. doi: 10.1080/03605309308820955.

[7]

M. E. Gurtin, Topics in Finite Elasticity, vol. 35 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.

[8]

L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, in Pseudodifferential operators (Oberwolfach, 1986), vol. 1256 of Lecture Notes in Math., Springer, Berlin, 1987, 214-280. doi: 10.1007/BFb0077745.

[9]

A. Hoshiga, The initial value problems for quasi-linear wave equations in two space dimensions with small data, Adv. Math. Sci. Appl., 5 (1995), 67-89.

[10]

F. John, Formation of singularities in elastic waves, in Trends and applications of pure mathematics to mechanics (Palaiseau, 1983), vol. 195 of Lecture Notes in Phys., Springer, Berlin, 1984, 194-210. doi: 10.1007/3-540-12916-2_58.

[11]

F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math., 41 (1988), 615-666. doi: 10.1002/cpa.3160410507.

[12]

F. John, Nonlinear Wave Equations, Formation of Singularities, vol. 2 of University Lecture Series, American Mathematical Society, Providence, RI, 1990, doi: 10.1090/ulect/002.

[13]

S. Katayama, Global existence for systems of nonlinear wave equations in two space dimensions, Publ. Res. Inst. Math. Sci., 29 (1993), 1021-1041. doi: 10.2977/prims/1195166427.

[14]

S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, 1986, 293-326.

[15]

S. Klainerman, On the work and legacy of Fritz John, 1934-1991, Comm. Pure Appl. Math., 51 (1998), 991-1017. doi: 10.1002/(SICI)1097-0312(199809/10)51:9/10<991::AID-CPA3>3.0.CO;2-T.

[16]

S. Klainerman, Long time behaviour of solutions to nonlinear wave equations, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, 1209-1215.

[17]

S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in $3$D, Comm. Pure Appl. Math., 49 (1996), 307-321. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H.

[18]

Z. Lei, Global well-posedness of incompressible elastodynamics in 2D,, , (). 

[19]

Zhen, Lei and T. C. Sideris and Yi, Zhou, Almost global existence for $2$-D incompressible isotropic elastodynamics, Trans. Amer. Math. Soc., 367 (2015), 8175-8197. doi: 10.1090/tran/6294.

[20]

W. Peng and D. Zha, Lifespan of classical solutions to the Cauchy problem for nonlinear elastic wave equations in 2-D,, preprint., (). 

[21]

T. C. Sideris, The null condition and global existence of nonlinear elastic waves, Invent. Math., 123 (1996), 323-342. doi: 10.1007/s002220050030.

[22]

T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2), 151 (2000), 849-874. doi: 10.2307/121050.

[23]

T. C. Sideris and B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 58 (2005), 750-788. doi: 10.1002/cpa.20049.

[24]

T. C. Sideris and B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics, Comm. Pure Appl. Math., 60 (2007), 1707-1730. doi: 10.1002/cpa.20196.

[25]

A. S. Tahvildar-Zadeh, Relativistic and nonrelativistic elastodynamics with small shear strains, Ann. Inst. H. Poincaré Phys. Théor., 69 (1998), 275-307.

[26]

X. Wang, Global existence for the 2D incompressible isotropic elastodynamics for small initial data,, , (). 

[27]

Y. Zhou, Nonlinear Wave Equations (in Chinese), Unpublished Lecture Notes, Fudan University, 2006.

show all references

References:
[1]

R. Agemi, Global existence of nonlinear elastic waves, Invent. Math., 142 (2000), 225-250. doi: 10.1007/s002220000084.

[2]

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math., 145 (2001), 597-618. doi: 10.1007/s002220100165.

[3]

S. Alinhac, Blowup for Nonlinear Hyperbolic Equations, Progress in Nonlinear Differential Equations and their Applications, 17, Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-1-4612-2578-2.

[4]

D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., 39 (1986), 267-282. doi: 10.1002/cpa.3160390205.

[5]

P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity, vol. 20 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam, 1988.

[6]

P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions, Comm. Partial Differential Equations, 18 (1993), 895-916. doi: 10.1080/03605309308820955.

[7]

M. E. Gurtin, Topics in Finite Elasticity, vol. 35 of CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.

[8]

L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations, in Pseudodifferential operators (Oberwolfach, 1986), vol. 1256 of Lecture Notes in Math., Springer, Berlin, 1987, 214-280. doi: 10.1007/BFb0077745.

[9]

A. Hoshiga, The initial value problems for quasi-linear wave equations in two space dimensions with small data, Adv. Math. Sci. Appl., 5 (1995), 67-89.

[10]

F. John, Formation of singularities in elastic waves, in Trends and applications of pure mathematics to mechanics (Palaiseau, 1983), vol. 195 of Lecture Notes in Phys., Springer, Berlin, 1984, 194-210. doi: 10.1007/3-540-12916-2_58.

[11]

F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances, Comm. Pure Appl. Math., 41 (1988), 615-666. doi: 10.1002/cpa.3160410507.

[12]

F. John, Nonlinear Wave Equations, Formation of Singularities, vol. 2 of University Lecture Series, American Mathematical Society, Providence, RI, 1990, doi: 10.1090/ulect/002.

[13]

S. Katayama, Global existence for systems of nonlinear wave equations in two space dimensions, Publ. Res. Inst. Math. Sci., 29 (1993), 1021-1041. doi: 10.2977/prims/1195166427.

[14]

S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, 1986, 293-326.

[15]

S. Klainerman, On the work and legacy of Fritz John, 1934-1991, Comm. Pure Appl. Math., 51 (1998), 991-1017. doi: 10.1002/(SICI)1097-0312(199809/10)51:9/10<991::AID-CPA3>3.0.CO;2-T.

[16]

S. Klainerman, Long time behaviour of solutions to nonlinear wave equations, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, 1209-1215.

[17]

S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in $3$D, Comm. Pure Appl. Math., 49 (1996), 307-321. doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H.

[18]

Z. Lei, Global well-posedness of incompressible elastodynamics in 2D,, , (). 

[19]

Zhen, Lei and T. C. Sideris and Yi, Zhou, Almost global existence for $2$-D incompressible isotropic elastodynamics, Trans. Amer. Math. Soc., 367 (2015), 8175-8197. doi: 10.1090/tran/6294.

[20]

W. Peng and D. Zha, Lifespan of classical solutions to the Cauchy problem for nonlinear elastic wave equations in 2-D,, preprint., (). 

[21]

T. C. Sideris, The null condition and global existence of nonlinear elastic waves, Invent. Math., 123 (1996), 323-342. doi: 10.1007/s002220050030.

[22]

T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2), 151 (2000), 849-874. doi: 10.2307/121050.

[23]

T. C. Sideris and B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics via the incompressible limit, Comm. Pure Appl. Math., 58 (2005), 750-788. doi: 10.1002/cpa.20049.

[24]

T. C. Sideris and B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics, Comm. Pure Appl. Math., 60 (2007), 1707-1730. doi: 10.1002/cpa.20196.

[25]

A. S. Tahvildar-Zadeh, Relativistic and nonrelativistic elastodynamics with small shear strains, Ann. Inst. H. Poincaré Phys. Théor., 69 (1998), 275-307.

[26]

X. Wang, Global existence for the 2D incompressible isotropic elastodynamics for small initial data,, , (). 

[27]

Y. Zhou, Nonlinear Wave Equations (in Chinese), Unpublished Lecture Notes, Fudan University, 2006.

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