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On some Liouville type theorems for the compressible Navier-Stokes equations
Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems
1. | School of Mathematics, Taiyuan University of Technology, Shanxi, 030024, China |
2. | Department of Mathematics, Shanghai University, Shanghai 200444, China |
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.
doi: 10.1007/BF01209629. |
[2] |
G. Carbou, B. Hanouzet and R .Natalini, Semilinear behavior of totally linearly degenerate hyperbolic systems with relaxation,, J. Differential Equations, 246 (2009), 291.
doi: 10.1016/j.jde.2008.05.015. |
[3] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Applied Mathematical Sciences, (1976).
|
[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. 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.
doi: 10.1016/j.jde.2006.12.020. |
[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.
doi: 10.1007/s00220-006-0124-z. |
[7] |
D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems,, Comm. Partial Differemtial Equations, 28 (2003), 1203.
doi: 10.1081/PDE-120021192. |
[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).
doi: 10.1063/1.2158435. |
[9] |
P. D. Lax, Hyperbolic systems of conservation laws $\mbox{I\!I}$,, Comm. Pure Appl. Math., 10 (1957), 537.
doi: 10.1002/cpa.3160100406. |
[10] |
T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems,, Research in Applied Mathematics, (1994).
|
[11] |
T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems,, Duke University Mathematics Series V, (1985).
|
[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.
doi: 10.1080/03605309408821055. |
[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.
doi: 10.1016/0362-546X(95)00228-N. |
[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.
doi: 10.1016/j.matpur.2012.10.011. |
[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.
doi: 10.1002/mma.797. |
[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).
doi: 10.1063/1.2890393. |
[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.
doi: 10.3934/dcds.2009.23.381. |
[18] |
A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables,, Volume 53, (1984).
doi: 10.1007/978-1-4612-1116-7. |
[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).
doi: 10.1063/1.3591133. |
[20] |
B. L. Rozdestvenkii and N. N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics,, Translated mathematical monographs 55, (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.
doi: 10.1016/j.na.2010.03.029. |
[22] |
Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, Chin.Ann.Math., 25 (2004), 37.
doi: 10.1142/S0252959904000044. |
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.
doi: 10.1007/BF01209629. |
[2] |
G. Carbou, B. Hanouzet and R .Natalini, Semilinear behavior of totally linearly degenerate hyperbolic systems with relaxation,, J. Differential Equations, 246 (2009), 291.
doi: 10.1016/j.jde.2008.05.015. |
[3] |
R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves,, Applied Mathematical Sciences, (1976).
|
[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. 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.
doi: 10.1016/j.jde.2006.12.020. |
[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.
doi: 10.1007/s00220-006-0124-z. |
[7] |
D. X. Kong and T. Yang, Asymptotic behavior of global classical solutions of quasilinear hyperbolic systems,, Comm. Partial Differemtial Equations, 28 (2003), 1203.
doi: 10.1081/PDE-120021192. |
[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).
doi: 10.1063/1.2158435. |
[9] |
P. D. Lax, Hyperbolic systems of conservation laws $\mbox{I\!I}$,, Comm. Pure Appl. Math., 10 (1957), 537.
doi: 10.1002/cpa.3160100406. |
[10] |
T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems,, Research in Applied Mathematics, (1994).
|
[11] |
T. T. Li and W. C. Yu, Boundary Value Problems for Quasilinear Hyperbolic Systems,, Duke University Mathematics Series V, (1985).
|
[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.
doi: 10.1080/03605309408821055. |
[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.
doi: 10.1016/0362-546X(95)00228-N. |
[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.
doi: 10.1016/j.matpur.2012.10.011. |
[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.
doi: 10.1002/mma.797. |
[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).
doi: 10.1063/1.2890393. |
[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.
doi: 10.3934/dcds.2009.23.381. |
[18] |
A. Majda, Compressible Fluid Flow and System of Conservation Laws in Several Space Variables,, Volume 53, (1984).
doi: 10.1007/978-1-4612-1116-7. |
[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).
doi: 10.1063/1.3591133. |
[20] |
B. L. Rozdestvenkii and N. N. Janenko, Systems of Quasilinear Equations and Their Applications to Gas Dynamics,, Translated mathematical monographs 55, (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.
doi: 10.1016/j.na.2010.03.029. |
[22] |
Y. Zhou, Global classical solutions to quasilinear hyperbolic systems with weak linear degeneracy,, Chin.Ann.Math., 25 (2004), 37.
doi: 10.1142/S0252959904000044. |
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