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

A note on quasilinear wave equations in two space dimensions

Dongbing Zha 1,

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.
Keywords: global existence., Quasilinear wave equations, null condition, two space dimensions.
Mathematics Subject Classification: Primary: 35L05, 35L15; Secondary: 35L7.
Citation: Dongbing Zha. A note on quasilinear wave equations in two space dimensions. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Helms, Private communication via E-mail, Oct. 2014. Google Scholar

[7]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26, Springer-Verlag, Berlin, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. C. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422. doi: 10.1353/ajm.1997.0014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Agemi, Global existence of nonlinear elastic waves, Invent. Math., 142 (2000), 225-250. 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), 597-618. doi: 10.1007/s002220100165.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

J. Helms, Private communication via E-mail, Oct. 2014. Google Scholar

[7]

L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26, Springer-Verlag, Berlin, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

T. C. Sideris, Delayed singularity formation in 2D compressible flow, Amer. J. Math., 119 (1997), 371-422. doi: 10.1353/ajm.1997.0014.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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