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November  2014, 34(11): 4735-4749. doi: 10.3934/dcds.2014.34.4735

Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems

Cunming Liu 1, and  Jianli Liu 2,

1. 

School of Mathematics, Taiyuan University of Technology, Shanxi, 030024, China
2. 

Department of Mathematics, Shanghai University, Shanghai 200444, China

Received  July 2013 Revised  March 2014 Published  May 2014

In this paper we consider the existence and stability of traveling wave solutions to Cauchy problem of diagonalizable quasilinear hyperbolic systems. Under the appropriate small oscillation assumptions on the initial traveling waves, we derive the stability result of the traveling wave solutions, especially for intermediate traveling waves. As the important examples, we will apply the results to some systems arising in fluid dynamics and elementary particle physics.
Keywords: linearly degenerate, stability., traveling wave solution, classical solution, Diagonalizable quasilinear hyperbolic system.
Mathematics Subject Classification: Primary: 35C07, 37C75; Secondary: 35L4.
Citation: Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
References:
[1]

B. M. Barbashov, V. V. Nesterenko and A. M. Chervyakov, General solutions of nonlinear equations in the geometric theory of the relativistic string, Commun. Math. Phys., 84 (1982), 471-481. doi: 10.1007/BF01209629.  Google Scholar

[2]

G. Carbou, B. Hanouzet and R .Natalini, Semilinear behavior of totally linearly degenerate hyperbolic systems with relaxation, J. Differential Equations, 246 (2009), 291-319. doi: 10.1016/j.jde.2008.05.015.  Google Scholar

[3]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol. 21. Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[4]

W. R. Dai and D. X. Kong, Asymptotic behavior of global classical solutions of general quasilinear hyperbolic systems with weakly linear degeneracy, Chinese Annals of Mathematics, 27B (2006), 263-286. Google Scholar

[5]

W. R. Dai and D. X. Kong, Global existence and asymptotic behavior of classical solutions of quasilnear hyperbolic systems with linear degenerate characteristic fields, J.Differential Equations, 235 (2007), 127-165. doi: 10.1016/j.jde.2006.12.020.  Google Scholar

[6]

D. X. Kong, Q. Zhang and Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space $R^{1+n}$, Commun. Math. Phys., 269 (2007), 153-174. doi: 10.1007/s00220-006-0124-z.  Google Scholar

[7]

D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Partial Differemtial Equations, 28 (2003), 1203-1220. doi: 10.1081/PDE-120021192.  Google Scholar

[8]

D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $R^{2+n}$, Journal Math. Phy, 47 (2006), 013503,16pp. doi: 10.1063/1.2158435.  Google Scholar

[9]

P. D. Lax, Hyperbolic systems of conservation laws $\mbox{I\!I}$, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[10]

T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics, Masson/ John Wiley, Paris, 1994.  Google Scholar

[11]

T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985.  Google Scholar

[12]

T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Differential Equations, 19 (1994), 1263-1317. doi: 10.1080/03605309408821055.  Google Scholar

[13]

T. T. Li, Y. Zhou and D. X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Analysis, 28 (1997), 1299-1332. doi: 10.1016/0362-546X(95)00228-N.  Google Scholar

[14]

C. M. Liu and P. Qu, Existence and stability of traveling wave solutions to first-order quasilinear hyperbolic systems, J. Math. Pures Appl., 100 (2013), 34-68. doi: 10.1016/j.matpur.2012.10.011.  Google Scholar

[15]

J. L. Liu and Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Meth. Appl. Sci., 30 (2007), 479-500. doi: 10.1002/mma.797.  Google Scholar

[16]

J. L. Liu and Y. Zhou, Initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space, J. Math. Phys., 49 (2008), 043507, 26pp. doi: 10.1063/1.2890393.  Google Scholar

[17]

J. L. Liu and Y. Zhou, The initial-boundary value problem on a strip for the equation of time-like extremal surfaces in Minkowski space, Discrete Contin. Dyn. Syst., 23 (2009), 381-397. doi: 10.3934/dcds.2009.23.381.  Google Scholar

[18]

A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, Volume 53, Applied Mathematical Sciences, Springer, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[19]

Y. J. Peng and Y. F. Yang, Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations, Journal of Mathematical Physics, 52 (2011), 053702, 23pp. doi: 10.1063/1.3591133.  Google Scholar

[20]

B. L. Rozdestvenkii and N. N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics, Translated mathematical monographs 55, American Math. Soc., Providence, RI, 1981. Google Scholar

[21]

Z. Q. Shao, A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Nonlinear Analysis, 73 (2010), 600-613. doi: 10.1016/j.na.2010.03.029.  Google Scholar

[22]

Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chin.Ann.Math., 25 (2004), 37-56. doi: 10.1142/S0252959904000044.  Google Scholar

show all references

References:
[1]

B. M. Barbashov, V. V. Nesterenko and A. M. Chervyakov, General solutions of nonlinear equations in the geometric theory of the relativistic string, Commun. Math. Phys., 84 (1982), 471-481. doi: 10.1007/BF01209629.  Google Scholar

[2]

G. Carbou, B. Hanouzet and R .Natalini, Semilinear behavior of totally linearly degenerate hyperbolic systems with relaxation, J. Differential Equations, 246 (2009), 291-319. doi: 10.1016/j.jde.2008.05.015.  Google Scholar

[3]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Applied Mathematical Sciences, Vol. 21. Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[4]

W. R. Dai and D. X. Kong, Asymptotic behavior of global classical solutions of general quasilinear hyperbolic systems with weakly linear degeneracy, Chinese Annals of Mathematics, 27B (2006), 263-286. Google Scholar

[5]

W. R. Dai and D. X. Kong, Global existence and asymptotic behavior of classical solutions of quasilnear hyperbolic systems with linear degenerate characteristic fields, J.Differential Equations, 235 (2007), 127-165. doi: 10.1016/j.jde.2006.12.020.  Google Scholar

[6]

D. X. Kong, Q. Zhang and Q. Zhou, The dynamics of relativistic strings moving in the Minkowski space $R^{1+n}$, Commun. Math. Phys., 269 (2007), 153-174. doi: 10.1007/s00220-006-0124-z.  Google Scholar

[7]

D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems, Comm. Partial Differemtial Equations, 28 (2003), 1203-1220. doi: 10.1081/PDE-120021192.  Google Scholar

[8]

D. X. Kong, Q. Y. Sun and Y. Zhou, The equation for time-like extremal surfaces in Minkowski space $R^{2+n}$, Journal Math. Phy, 47 (2006), 013503,16pp. doi: 10.1063/1.2158435.  Google Scholar

[9]

P. D. Lax, Hyperbolic systems of conservation laws $\mbox{I\!I}$, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar

[10]

T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, Research in Applied Mathematics, Masson/ John Wiley, Paris, 1994.  Google Scholar

[11]

T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems, Duke University Mathematics Series V, 1985.  Google Scholar

[12]

T. T. Li, Y. Zhou and D. X. Kong, Weak linear degeneracy and global classical solutions for general quasilinear hyperbolic systems, Comm. Partial Differential Equations, 19 (1994), 1263-1317. doi: 10.1080/03605309408821055.  Google Scholar

[13]

T. T. Li, Y. Zhou and D. X. Kong, Global classical solutions for general quasilinear hyperbolic systems with decay initial data, Nonlinear Analysis, 28 (1997), 1299-1332. doi: 10.1016/0362-546X(95)00228-N.  Google Scholar

[14]

C. M. Liu and P. Qu, Existence and stability of traveling wave solutions to first-order quasilinear hyperbolic systems, J. Math. Pures Appl., 100 (2013), 34-68. doi: 10.1016/j.matpur.2012.10.011.  Google Scholar

[15]

J. L. Liu and Y. Zhou, Asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems, Math. Meth. Appl. Sci., 30 (2007), 479-500. doi: 10.1002/mma.797.  Google Scholar

[16]

J. L. Liu and Y. Zhou, Initial-boundary value problem for the equation of time-like extremal surfaces in Minkowski space, J. Math. Phys., 49 (2008), 043507, 26pp. doi: 10.1063/1.2890393.  Google Scholar

[17]

J. L. Liu and Y. Zhou, The initial-boundary value problem on a strip for the equation of time-like extremal surfaces in Minkowski space, Discrete Contin. Dyn. Syst., 23 (2009), 381-397. doi: 10.3934/dcds.2009.23.381.  Google Scholar

[18]

A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables, Volume 53, Applied Mathematical Sciences, Springer, New York, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[19]

Y. J. Peng and Y. F. Yang, Well-posedness and long-time behavior of Lipschitz solutions to generalized extremal surface equations, Journal of Mathematical Physics, 52 (2011), 053702, 23pp. doi: 10.1063/1.3591133.  Google Scholar

[20]

B. L. Rozdestvenkii and N. N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics, Translated mathematical monographs 55, American Math. Soc., Providence, RI, 1981. Google Scholar

[21]

Z. Q. Shao, A note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems, Nonlinear Analysis, 73 (2010), 600-613. doi: 10.1016/j.na.2010.03.029.  Google Scholar

[22]

Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy, Chin.Ann.Math., 25 (2004), 37-56. doi: 10.1142/S0252959904000044.  Google Scholar

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