# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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.  doi: 10.1007/s002220000084.  Google Scholar [2] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I,, Invent. Math., 145 (2001), 597.  doi: 10.1007/s002220100165.  Google Scholar [3] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations,, Progress in Nonlinear Differential Equations and their Applications, (1995).  doi: 10.1007/978-1-4612-2578-2.  Google Scholar [4] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data,, Comm. Pure Appl. Math., 39 (1986), 267.  doi: 10.1002/cpa.3160390205.  Google Scholar [5] P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity, vol. 20 of Studies in Mathematics and its Applications,, North-Holland Publishing Co., (1988).   Google Scholar [6] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions,, Comm. Partial Differential Equations, 18 (1993), 895.  doi: 10.1080/03605309308820955.  Google Scholar [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), (1981).   Google Scholar [8] L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations,, in Pseudodifferential operators (Oberwolfach, (1986), 214.  doi: 10.1007/BFb0077745.  Google Scholar [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.   Google Scholar [10] F. John, Formation of singularities in elastic waves,, in Trends and applications of pure mathematics to mechanics (Palaiseau, (1983), 194.  doi: 10.1007/3-540-12916-2_58.  Google Scholar [11] F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances,, Comm. Pure Appl. Math., 41 (1988), 615.  doi: 10.1002/cpa.3160410507.  Google Scholar [12] F. John, Nonlinear Wave Equations, Formation of Singularities, vol. 2 of University Lecture Series,, American Mathematical Society, (1990).  doi: 10.1090/ulect/002.  Google Scholar [13] S. Katayama, Global existence for systems of nonlinear wave equations in two space dimensions,, Publ. Res. Inst. Math. Sci., 29 (1993), 1021.  doi: 10.2977/prims/1195166427.  Google Scholar [14] S. Klainerman, The null condition and global existence to nonlinear wave equations,, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, (1984), 293.   Google Scholar [15] S. Klainerman, On the work and legacy of Fritz John, 1934-1991,, Comm. Pure Appl. Math., 51 (1998), 991.  doi: 10.1002/(SICI)1097-0312(199809/10)51:9/10<991::AID-CPA3>3.0.CO;2-T.  Google Scholar [16] S. Klainerman, Long time behaviour of solutions to nonlinear wave equations,, in Proceedings of the International Congress of Mathematicians, (1983), 1209.   Google Scholar [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.  doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H.  Google Scholar [18] Z. Lei, Global well-posedness of incompressible elastodynamics in 2D,, , ().   Google Scholar [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.  doi: 10.1090/tran/6294.  Google Scholar [20] W. Peng and D. Zha, Lifespan of classical solutions to the Cauchy problem for nonlinear elastic wave equations in 2-D,, preprint., ().   Google Scholar [21] T. C. Sideris, The null condition and global existence of nonlinear elastic waves,, Invent. Math., 123 (1996), 323.  doi: 10.1007/s002220050030.  Google Scholar [22] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves,, Ann. of Math. (2), 151 (2000), 849.  doi: 10.2307/121050.  Google Scholar [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.  doi: 10.1002/cpa.20049.  Google Scholar [24] T. C. Sideris and B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics,, Comm. Pure Appl. Math., 60 (2007), 1707.  doi: 10.1002/cpa.20196.  Google Scholar [25] A. S. Tahvildar-Zadeh, Relativistic and nonrelativistic elastodynamics with small shear strains,, Ann. Inst. H. Poincaré Phys. Théor., 69 (1998), 275.   Google Scholar [26] X. Wang, Global existence for the 2D incompressible isotropic elastodynamics for small initial data,, , ().   Google Scholar [27] Y. Zhou, Nonlinear Wave Equations (in Chinese),, Unpublished Lecture Notes, (2006).   Google Scholar

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##### References:
 [1] R. Agemi, Global existence of nonlinear elastic waves,, Invent. Math., 142 (2000), 225.  doi: 10.1007/s002220000084.  Google Scholar [2] S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I,, Invent. Math., 145 (2001), 597.  doi: 10.1007/s002220100165.  Google Scholar [3] S. Alinhac, Blowup for Nonlinear Hyperbolic Equations,, Progress in Nonlinear Differential Equations and their Applications, (1995).  doi: 10.1007/978-1-4612-2578-2.  Google Scholar [4] D. Christodoulou, Global solutions of nonlinear hyperbolic equations for small initial data,, Comm. Pure Appl. Math., 39 (1986), 267.  doi: 10.1002/cpa.3160390205.  Google Scholar [5] P. G. Ciarlet, Mathematical Elasticity. Vol. I: Three-dimensional Elasticity, vol. 20 of Studies in Mathematics and its Applications,, North-Holland Publishing Co., (1988).   Google Scholar [6] P. Godin, Lifespan of solutions of semilinear wave equations in two space dimensions,, Comm. Partial Differential Equations, 18 (1993), 895.  doi: 10.1080/03605309308820955.  Google Scholar [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), (1981).   Google Scholar [8] L. Hörmander, The lifespan of classical solutions of nonlinear hyperbolic equations,, in Pseudodifferential operators (Oberwolfach, (1986), 214.  doi: 10.1007/BFb0077745.  Google Scholar [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.   Google Scholar [10] F. John, Formation of singularities in elastic waves,, in Trends and applications of pure mathematics to mechanics (Palaiseau, (1983), 194.  doi: 10.1007/3-540-12916-2_58.  Google Scholar [11] F. John, Almost global existence of elastic waves of finite amplitude arising from small initial disturbances,, Comm. Pure Appl. Math., 41 (1988), 615.  doi: 10.1002/cpa.3160410507.  Google Scholar [12] F. John, Nonlinear Wave Equations, Formation of Singularities, vol. 2 of University Lecture Series,, American Mathematical Society, (1990).  doi: 10.1090/ulect/002.  Google Scholar [13] S. Katayama, Global existence for systems of nonlinear wave equations in two space dimensions,, Publ. Res. Inst. Math. Sci., 29 (1993), 1021.  doi: 10.2977/prims/1195166427.  Google Scholar [14] S. Klainerman, The null condition and global existence to nonlinear wave equations,, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, (1984), 293.   Google Scholar [15] S. Klainerman, On the work and legacy of Fritz John, 1934-1991,, Comm. Pure Appl. Math., 51 (1998), 991.  doi: 10.1002/(SICI)1097-0312(199809/10)51:9/10<991::AID-CPA3>3.0.CO;2-T.  Google Scholar [16] S. Klainerman, Long time behaviour of solutions to nonlinear wave equations,, in Proceedings of the International Congress of Mathematicians, (1983), 1209.   Google Scholar [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.  doi: 10.1002/(SICI)1097-0312(199603)49:3<307::AID-CPA4>3.0.CO;2-H.  Google Scholar [18] Z. Lei, Global well-posedness of incompressible elastodynamics in 2D,, , ().   Google Scholar [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.  doi: 10.1090/tran/6294.  Google Scholar [20] W. Peng and D. Zha, Lifespan of classical solutions to the Cauchy problem for nonlinear elastic wave equations in 2-D,, preprint., ().   Google Scholar [21] T. C. Sideris, The null condition and global existence of nonlinear elastic waves,, Invent. Math., 123 (1996), 323.  doi: 10.1007/s002220050030.  Google Scholar [22] T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves,, Ann. of Math. (2), 151 (2000), 849.  doi: 10.2307/121050.  Google Scholar [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.  doi: 10.1002/cpa.20049.  Google Scholar [24] T. C. Sideris and B. Thomases, Global existence for three-dimensional incompressible isotropic elastodynamics,, Comm. Pure Appl. Math., 60 (2007), 1707.  doi: 10.1002/cpa.20196.  Google Scholar [25] A. S. Tahvildar-Zadeh, Relativistic and nonrelativistic elastodynamics with small shear strains,, Ann. Inst. H. Poincaré Phys. Théor., 69 (1998), 275.   Google Scholar [26] X. Wang, Global existence for the 2D incompressible isotropic elastodynamics for small initial data,, , ().   Google Scholar [27] Y. Zhou, Nonlinear Wave Equations (in Chinese),, Unpublished Lecture Notes, (2006).   Google Scholar
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