-
Previous Article
Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces
- DCDS Home
- This Issue
-
Next Article
On the Cauchy problem of a three-component Camassa--Holm equations
A note on quasilinear wave equations in two space dimensions
1. | School of Mathematical Sciences, Fudan University, Shanghai 200433, China |
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, Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.
doi: 10.1007/BF01231301. |
[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] |
D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, vol. 41 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993. |
[6] | |
[7] |
L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26, Springer-Verlag, Berlin, 1997. |
[8] |
A. Hoshiga, The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space, Funkcial. Ekvac., 49 (2006), 357-384.
doi: 10.1619/fesi.49.357. |
[9] |
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. |
[10] |
S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, 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. |
[11] |
H. Lindblad and I. Rodnianski, The global stability of Minkowski space-time in harmonic gauge, Ann. of Math. (2), 171 (2010), 1401-1477.
doi: 10.4007/annals.2010.171.1401. |
[12] |
J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z., 256 (2007), 521-549.
doi: 10.1007/s00209-006-0083-2. |
[13] |
T. C. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.
doi: 10.1353/ajm.1997.0014. |
[14] |
T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2), 151 (2000), 849-874.
doi: 10.2307/121050. |
[15] |
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. |
[16] |
T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal., 33 (2001), 477-488.
doi: 10.1137/S0036141000378966. |
[17] |
S. Yang, Global solutions of nonlinear wave equations in time dependent inhomogeneous media, Arch. Ration. Mech. Anal., 209 (2013), 683-728.
doi: 10.1007/s00205-013-0631-y. |
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, Temps de vie des solutions régulières des équations d'Euler compressibles axisymétriques en dimension deux, Invent. Math., 111 (1993), 627-670.
doi: 10.1007/BF01231301. |
[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] |
D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, vol. 41 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993. |
[6] | |
[7] |
L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26, Springer-Verlag, Berlin, 1997. |
[8] |
A. Hoshiga, The existence of global solutions to systems of quasilinear wave equations with quadratic nonlinearities in 2-dimensional space, Funkcial. Ekvac., 49 (2006), 357-384.
doi: 10.1619/fesi.49.357. |
[9] |
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. |
[10] |
S. Klainerman and T. C. Sideris, On almost global existence for nonrelativistic wave equations in 3D, 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. |
[11] |
H. Lindblad and I. Rodnianski, The global stability of Minkowski space-time in harmonic gauge, Ann. of Math. (2), 171 (2010), 1401-1477.
doi: 10.4007/annals.2010.171.1401. |
[12] |
J. Metcalfe and C. D. Sogge, Global existence of null-form wave equations in exterior domains, Math. Z., 256 (2007), 521-549.
doi: 10.1007/s00209-006-0083-2. |
[13] |
T. C. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422.
doi: 10.1353/ajm.1997.0014. |
[14] |
T. C. Sideris, Nonresonance and global existence of prestressed nonlinear elastic waves, Ann. of Math. (2), 151 (2000), 849-874.
doi: 10.2307/121050. |
[15] |
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. |
[16] |
T. C. Sideris and S.-Y. Tu, Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal., 33 (2001), 477-488.
doi: 10.1137/S0036141000378966. |
[17] |
S. Yang, Global solutions of nonlinear wave equations in time dependent inhomogeneous media, Arch. Ration. Mech. Anal., 209 (2013), 683-728.
doi: 10.1007/s00205-013-0631-y. |
[1] |
Minggang Cheng, Soichiro Katayama. Systems of semilinear wave equations with multiple speeds in two space dimensions and a weaker null condition. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022092 |
[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] |
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 |
[4] |
Nakao Hayashi, Seishirou Kobayashi, Pavel I. Naumkin. Nonlinear dispersive wave equations in two space dimensions. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1377-1393. doi: 10.3934/cpaa.2015.14.1377 |
[5] |
Dongbing Zha, Yi Zhou. The lifespan for quasilinear wave equations with multiple propagation speeds in four space dimensions. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1167-1186. doi: 10.3934/cpaa.2014.13.1167 |
[6] |
M. Petcu, Roger Temam, D. Wirosoetisno. Existence and regularity results for the primitive equations in two space dimensions. Communications on Pure and Applied Analysis, 2004, 3 (1) : 115-131. doi: 10.3934/cpaa.2004.3.115 |
[7] |
Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471 |
[8] |
Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure and Applied Analysis, 2002, 1 (2) : 191-219. doi: 10.3934/cpaa.2002.1.191 |
[9] |
Jason Metcalfe, Christopher D. Sogge. Global existence for high dimensional quasilinear wave equations exterior to star-shaped obstacles. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1589-1601. doi: 10.3934/dcds.2010.28.1589 |
[10] |
Nathan Glatt-Holtz, Mohammed Ziane. The stochastic primitive equations in two space dimensions with multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 801-822. doi: 10.3934/dcdsb.2008.10.801 |
[11] |
Kunio Hidano, Dongbing Zha. Remarks on a system of quasi-linear wave equations in 3D satisfying the weak null condition. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1735-1767. doi: 10.3934/cpaa.2019082 |
[12] |
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 |
[13] |
Ning-An Lai, Yi Zhou. Blow up for initial boundary value problem of critical semilinear wave equation in two space dimensions. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1499-1510. doi: 10.3934/cpaa.2018072 |
[14] |
Akihiro Shimomura. Modified wave operators for the coupled wave-Schrödinger equations in three space dimensions. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1571-1586. doi: 10.3934/dcds.2003.9.1571 |
[15] |
Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 |
[16] |
Jason Metcalfe, Jacob Perry. Global solutions to quasilinear wave equations in homogeneous waveguides with Neumann boundary conditions. Communications on Pure and Applied Analysis, 2012, 11 (2) : 547-556. doi: 10.3934/cpaa.2012.11.547 |
[17] |
Norisuke Ioku. Some space-time integrability estimates of the solution for heat equations in two dimensions. Conference Publications, 2011, 2011 (Special) : 707-716. doi: 10.3934/proc.2011.2011.707 |
[18] |
G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279 |
[19] |
Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control and Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031 |
[20] |
Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5383-5405. doi: 10.3934/dcdsb.2020348 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]