# American Institute of Mathematical Sciences

• 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
May  2016, 36(5): 2855-2871. doi: 10.3934/dcds.2016.36.2855

## A note on quasilinear wave equations in two space dimensions

 1 School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received  June 2015 Revised  September 2015 Published  October 2015

In this paper, we give an alternative proof of Alinhac's global existence result for the Cauchy problem of quasilinear wave equations with both null conditions in two space dimensions[S. Alinhac, The null condition for quasilinear wave equations in two space dimensions I, Invent. Math. 145 (2001) 597--618]. The innovation in our proof is that when applying the vector fields method to do the generalized energy estimates, we don't employ the Lorentz boost operator and only use the general space-time derivatives, spatial rotation and scaling operator.
Citation: Dongbing Zha. A note on quasilinear wave equations in two space dimensions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2855-2871. doi: 10.3934/dcds.2016.36.2855
##### 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] J. Helms, Private communication via E-mail, Oct. 2014. [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] J. Helms, Private communication via E-mail, Oct. 2014. [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