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Remarks on nonlinear elastic waves in the radial symmetry in 2-D
1. | School of Mathematical Sciences, Fudan University, Shanghai 200433 |
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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