American Institute of Mathematical Sciences

Advanced
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

Dongbing Zha 1,

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.
Keywords: null condition, variational structure, Nonlinear elastic waves in 2-D, radial symmetry, global existence..
Mathematics Subject Classification: Primary: 35L52; Secondary: 35Q7.
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.

[1]

Tian Ma, Shouhong Wang. Global structure of 2-D incompressible flows. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 431-445. doi: 10.3934/dcds.2001.7.431
[2]

Hideo Kubo. Global existence for exterior problems of semilinear wave equations with the null condition in $2$D. Evolution Equations and Control Theory, 2013, 2 (2) : 319-335. doi: 10.3934/eect.2013.2.319
[3]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[4]

José R. Quintero. Nonlinear stability of solitary waves for a 2-d Benney--Luke equation. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 203-218. doi: 10.3934/dcds.2005.13.203
[5]

Kunio Hidano, Kazuyoshi Yokoyama. Global existence and blow up for systems of nonlinear wave equations related to the weak null condition. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022058
[6]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407
[7]

Shirshendu Chowdhury, Debanjana Mitra, Michael Renardy. Null controllability of the incompressible Stokes equations in a 2-D channel using normal boundary control. Evolution Equations and Control Theory, 2018, 7 (3) : 447-463. doi: 10.3934/eect.2018022
[8]

Huicheng Yin, Lin Zhang. The global stability of 2-D viscous axisymmetric circulatory flows. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5065-5083. doi: 10.3934/dcds.2017219
[9]

Lingbing He. On the global smooth solution to 2-D fluid/particle system. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 237-263. doi: 10.3934/dcds.2010.27.237
[10]

H. T. Banks, R.C. Smith. Feedback control of noise in a 2-D nonlinear structural acoustics model. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 119-149. doi: 10.3934/dcds.1995.1.119
[11]

Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555
[12]

Orlando Lopes. Uniqueness and radial symmetry of minimizers for a nonlocal variational problem. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2265-2282. doi: 10.3934/cpaa.2019102
[13]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333
[14]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777
[15]

Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917
[16]

Mingwen Fei, Huicheng Yin. Nodal solutions of 2-D critical nonlinear Schrödinger equations with potentials vanishing at infinity. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2921-2948. doi: 10.3934/dcds.2015.35.2921
[17]

Nusret Balci, Ciprian Foias, M. S Jolly, Ricardo Rosa. On universal relations in 2-D turbulence. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1327-1351. doi: 10.3934/dcds.2010.27.1327
[18]

Jean-françois Coulombel, Paolo Secchi. Uniqueness of 2-D compressible vortex sheets. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1439-1450. doi: 10.3934/cpaa.2009.8.1439
[19]

Zhenjie Li, Chunqin Zhou. Radial symmetry of nonnegative solutions for nonlinear integral systems. Communications on Pure and Applied Analysis, 2022, 21 (3) : 837-844. doi: 10.3934/cpaa.2021201
[20]

Yizhao Qin, Yuxia Guo, Peng-Fei Yao. Energy decay and global smooth solutions for a free boundary fluid-nonlinear elastic structure interface model with boundary dissipation. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1555-1593. doi: 10.3934/dcds.2020086

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (123)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy