American Institute of Mathematical Sciences

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August  2015, 35(8): 3327-3342. doi: 10.3934/dcds.2015.35.3327

Existence and regularity of solutions in nonlinear wave equations

Geng Chen 1, and  Yannan Shen 2,

1. 

School of Mathematics, Georgia Institute of Technology, Atlanta, Ga. 30332
2. 

Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX, 75080, United States

Received  September 2014 Revised  October 2014 Published  February 2015

In this paper, we study the global existence and regularity of Hölder continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.
Keywords: Hölder continuity., singularity, large data, existence, Nonlinear wave equations.
Mathematics Subject Classification: Primary: 35L05; Secondary: 35L6.
Citation: Geng Chen, Yannan Shen. Existence and regularity of solutions in nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3327-3342. doi: 10.3934/dcds.2015.35.3327
References:
[1]

A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. doi: 10.1137/050623036.

[2]

A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.

[3]

A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pures Appl. (9), 94 (2010), 68-92. doi: 10.1016/j.matpur.2010.02.005.

[4]

A. Bressan and Y. Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys., 266 (2006), 471-497. doi: 10.1007/s00220-006-0047-8.

[5]

A. Bressan, P. Zhang and Y. Zheng, Asymptotic variational wave equations, Arch. Ration. Mech. Anal., 183 (2007), 163-185. doi: 10.1007/s00205-006-0014-8.

[6]

G. Chen, Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations, J. Hyperbolic Differ. Equ., 8 (2011), 671-690. doi: 10.1142/S0219891611002536.

[7]

G. Chen, R. Pan and S. Zhu, Singularity formation for compressible Euler equations, submitted, available at arXiv:1408.6775.

[8]

G. Chen and R. Young, Smooth waves and gradient blowup for the inhomogeneous wave equations, J. Differential Equations, 252 (2012), 2580-2595. doi: 10.1016/j.jde.2011.09.004.

[9]

G. Chen, R. Young and Q. Zhang, Shock formation in the compressible Euler equations and related systems, J. Hyperbolic Differ. Equ., 10 (2013), 149-172. doi: 10.1142/S0219891613500069.

[10]

G. Chen, P. Zhang and Y. Zheng, Energy conservative solutions to a nonlinear wave system of nematic liquid crystals, Comm. Pure Appl. Anal., 12 (2013), 1445-1468. doi: 10.3934/cpaa.2013.12.1445.

[11]

G. Chen and Y. Zheng, Existence and singularity to a wave system of nematic liquid crystals, J. Math. Anal. Appl., 398 (2013), 170-188. doi: 10.1016/j.jmaa.2012.08.048.

[12]

C. M. Dafermos, Hyperbolic Conservations Laws in Continuum Physics, (third edition), Springer-Verlag, Heidelberg, 2010. doi: 10.1007/978-3-642-04048-1.

[13]

H. Holden and X. Raynaud, Global semigroup of conservative solutions of the nonlinear variational wave equation, Arch. Ration. Mech. Anal., 201 (2011), 871-964. doi: 10.1007/s00205-011-0403-5.

[14]

R. Glassey, J. Hunter and Y. Zheng, Singularities of a variational wave equation, J. Differential Equations, 129 (1996), 49-78. doi: 10.1006/jdeq.1996.0111.

[15]

R. Glassey, J. Hunter and Y. Zheng, Singularities and Oscillations in a Nonlinear Variational Wave Equation, Singularities and Oscillations, The IMA Volumes in Mathematics and its Applications, 91 (1997), 37-60. doi: 10.1007/978-1-4612-1972-9_3.

[16]

J. K. Hunter and R. H. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.

[17]

J. K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: I. global existence of weak solutions, Arch. Rat. Mech. Anal., 129 (1995), 305-353. doi: 10.1007/BF00379259.

[18]

J. K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: II. the zero viscosity and dispersion limits, Arch. Rat. Mech. Anal., 129 (1995), 355-383. doi: 10.1007/BF00379260.

[19]

P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Physics, 5 (1964), 611-613. doi: 10.1063/1.1704154.

[20]

P. Lax and J. Glimm, Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws, Memoirs of the American Mathematical Society, 101, American Mathematical Society, Providence, R.I. 1970.

[21]

H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096. doi: 10.1002/cpa.3160450902.

[22]

P. Zhang and Y. Zheng, Weak solutions to a nonlinear variational wave equation, Arch. Ration. Mech. Anal., 166 (2003), 303-319. doi: 10.1007/s00205-002-0232-7.

[23]

P. Zhang and Y. Zheng, Weak solutions to a nonlinear variational wave equation with general data, Ann. I. H. Poincaré, 22 (2005), 207-226. doi: 10.1016/j.anihpc.2004.04.001.

[24]

P. Zhang and Y. Zheng, Conservative solutions to a system of variational wave equations of nematic liquid crystals, Arch. Ration. Mech. Anal., 195 (2010), 701-727. doi: 10.1007/s00205-009-0222-0.

[25]

P. Zhang and Y. Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl. Math., 65 (2012), 683-726. doi: 10.1002/cpa.20380.

show all references

References:
[1]

A. Bressan and A. Constantin, Global solutions of the Hunter-Saxton equation, SIAM J. Math. Anal., 37 (2005), 996-1026. doi: 10.1137/050623036.

[2]

A. Bressan and A. Constantin, Global conservative solutions to the Camassa-Holm equation, Arch. Rat. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.

[3]

A. Bressan, H. Holden and X. Raynaud, Lipschitz metric for the Hunter-Saxton equation, J. Math. Pures Appl. (9), 94 (2010), 68-92. doi: 10.1016/j.matpur.2010.02.005.

[4]

A. Bressan and Y. Zheng, Conservative solutions to a nonlinear variational wave equation, Comm. Math. Phys., 266 (2006), 471-497. doi: 10.1007/s00220-006-0047-8.

[5]

A. Bressan, P. Zhang and Y. Zheng, Asymptotic variational wave equations, Arch. Ration. Mech. Anal., 183 (2007), 163-185. doi: 10.1007/s00205-006-0014-8.

[6]

G. Chen, Formation of singularity and smooth wave propagation for the non-isentropic compressible Euler equations, J. Hyperbolic Differ. Equ., 8 (2011), 671-690. doi: 10.1142/S0219891611002536.

[7]

G. Chen, R. Pan and S. Zhu, Singularity formation for compressible Euler equations, submitted, available at arXiv:1408.6775.

[8]

G. Chen and R. Young, Smooth waves and gradient blowup for the inhomogeneous wave equations, J. Differential Equations, 252 (2012), 2580-2595. doi: 10.1016/j.jde.2011.09.004.

[9]

G. Chen, R. Young and Q. Zhang, Shock formation in the compressible Euler equations and related systems, J. Hyperbolic Differ. Equ., 10 (2013), 149-172. doi: 10.1142/S0219891613500069.

[10]

G. Chen, P. Zhang and Y. Zheng, Energy conservative solutions to a nonlinear wave system of nematic liquid crystals, Comm. Pure Appl. Anal., 12 (2013), 1445-1468. doi: 10.3934/cpaa.2013.12.1445.

[11]

G. Chen and Y. Zheng, Existence and singularity to a wave system of nematic liquid crystals, J. Math. Anal. Appl., 398 (2013), 170-188. doi: 10.1016/j.jmaa.2012.08.048.

[12]

C. M. Dafermos, Hyperbolic Conservations Laws in Continuum Physics, (third edition), Springer-Verlag, Heidelberg, 2010. doi: 10.1007/978-3-642-04048-1.

[13]

H. Holden and X. Raynaud, Global semigroup of conservative solutions of the nonlinear variational wave equation, Arch. Ration. Mech. Anal., 201 (2011), 871-964. doi: 10.1007/s00205-011-0403-5.

[14]

R. Glassey, J. Hunter and Y. Zheng, Singularities of a variational wave equation, J. Differential Equations, 129 (1996), 49-78. doi: 10.1006/jdeq.1996.0111.

[15]

R. Glassey, J. Hunter and Y. Zheng, Singularities and Oscillations in a Nonlinear Variational Wave Equation, Singularities and Oscillations, The IMA Volumes in Mathematics and its Applications, 91 (1997), 37-60. doi: 10.1007/978-1-4612-1972-9_3.

[16]

J. K. Hunter and R. H. Saxton, Dynamics of director fields, SIAM J. Appl. Math., 51 (1991), 1498-1521. doi: 10.1137/0151075.

[17]

J. K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: I. global existence of weak solutions, Arch. Rat. Mech. Anal., 129 (1995), 305-353. doi: 10.1007/BF00379259.

[18]

J. K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: II. the zero viscosity and dispersion limits, Arch. Rat. Mech. Anal., 129 (1995), 355-383. doi: 10.1007/BF00379260.

[19]

P. Lax, Development of singularities of solutions of nonlinear hyperbolic partial differential equations, J. Math. Physics, 5 (1964), 611-613. doi: 10.1063/1.1704154.

[20]

P. Lax and J. Glimm, Decay of Solutions of Systems of Nonlinear Hyperbolic Conservation Laws, Memoirs of the American Mathematical Society, 101, American Mathematical Society, Providence, R.I. 1970.

[21]

H. Lindblad, Global solutions of nonlinear wave equations, Comm. Pure Appl. Math., 45 (1992), 1063-1096. doi: 10.1002/cpa.3160450902.

[22]

P. Zhang and Y. Zheng, Weak solutions to a nonlinear variational wave equation, Arch. Ration. Mech. Anal., 166 (2003), 303-319. doi: 10.1007/s00205-002-0232-7.

[23]

P. Zhang and Y. Zheng, Weak solutions to a nonlinear variational wave equation with general data, Ann. I. H. Poincaré, 22 (2005), 207-226. doi: 10.1016/j.anihpc.2004.04.001.

[24]

P. Zhang and Y. Zheng, Conservative solutions to a system of variational wave equations of nematic liquid crystals, Arch. Ration. Mech. Anal., 195 (2010), 701-727. doi: 10.1007/s00205-009-0222-0.

[25]

P. Zhang and Y. Zheng, Energy conservative solutions to a one-dimensional full variational wave system, Comm. Pure Appl. Math., 65 (2012), 683-726. doi: 10.1002/cpa.20380.

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