Convergence to strong nonlinear rarefaction waves for global smooth solutions of p-system with relaxation

Previous Article
A spectral characterization of exponential stability for linear time-invariant systems on time scales
DCDS Home
This Issue
Next Article
Generalized quasilinearization method for semilinear hyperbolic problems

September 2003, 9(5): 1243-1262. doi: 10.3934/dcds.2003.9.1243

Convergence to strong nonlinear rarefaction waves for global smooth solutions of $p-$system with relaxation

1.	Laboratory of Mathematical Physics, Wuhan Institute of Physics and Mathematics, The Chinese Academy of Sciences, P. O. Box 71010, Wuhan 430071, China, China

Received July 2002 Revised December 2002 Published June 2003

Abstract
Related Papers

This paper is concerned with the large time behavior of global smooth solutions to the Cauchy problem of the $p-$system with relaxation. Former results in this direction indicate that such a problem possesses a global smooth solution provided that the first derivative of the solutions with respect to the space variable $x$ are sufficiently small. Under the same small assumption on the global smooth solution, we show that it converges to the corresponding nonlinear rarefaction wave and in our analysis, we do not ask the rarefaction wave to be weak and the initial error can also be chosen arbitrarily large.

Keywords: large initial error., p−system with relaxation, Strong nonlinear rarefaction waves.

Mathematics Subject Classification: 35L45, 35L6.

Citation: Huijiang Zhao, Yinchuan Zhao. Convergence to strong nonlinear rarefaction waves for global smooth solutions of $p-$system with relaxation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1243-1262. doi: 10.3934/dcds.2003.9.1243

[1]	Min Ding, Hairong Yuan. Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2911-2943. doi: 10.3934/dcds.2018125
[2]	Mina Jiang, Changjiang Zhu. Convergence rates to nonlinear diffusion waves for $p$-system with nonlinear damping on quadrant. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 887-918. doi: 10.3934/dcds.2009.23.887
[3]	Christian Rohde, Wenjun Wang, Feng Xie. Hyperbolic-hyperbolic relaxation limit for a 1D compressible radiation hydrodynamics model: superposition of rarefaction and contact waves. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2145-2171. doi: 10.3934/cpaa.2013.12.2145
[4]	Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035
[5]	Iryna Egorova, Johanna Michor, Gerald Teschl. Rarefaction waves for the Toda equation via nonlinear steepest descent. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2007-2028. doi: 10.3934/dcds.2018081
[6]	Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251
[7]	Eun Heui Kim, Charis Tsikkou. Two dimensional Riemann problems for the nonlinear wave system: Rarefaction wave interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6257-6289. doi: 10.3934/dcds.2017271
[8]	Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic & Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004
[9]	Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075
[10]	Jishan Fan, Shuxiang Huang, Fucai Li. Global strong solutions to the planar compressible magnetohydrodynamic equations with large initial data and vacuum. Kinetic & Related Models, 2017, 10 (4) : 1035-1053. doi: 10.3934/krm.2017041
[11]	Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[12]	Masahito Ohta. Strong instability of standing waves for nonlinear Schrödinger equations with a partial confinement. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1671-1680. doi: 10.3934/cpaa.2018080
[13]	Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315
[14]	Hualin Zheng. Stability of a superposition of shock waves with contact discontinuities for the Jin-Xin relaxation system. Kinetic & Related Models, 2015, 8 (3) : 559-585. doi: 10.3934/krm.2015.8.559
[15]	Renhui Wan. The 3D liquid crystal system with Cannone type initial data and large vertical velocity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5521-5539. doi: 10.3934/dcds.2017240
[16]	Zhong Tan, Zheng-An Yao. The existence and asymptotic behavior of the evolution p-Laplacian equations with strong nonlinear sources. Communications on Pure & Applied Analysis, 2004, 3 (3) : 475-490. doi: 10.3934/cpaa.2004.3.475
[17]	Paschalis Karageorgis. Small-data scattering for nonlinear waves with potential and initial data of critical decay. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 87-106. doi: 10.3934/dcds.2006.16.87
[18]	Jun-ichi Segata. Initial value problem for the fourth order nonlinear Schrödinger type equation on torus and orbital stability of standing waves. Communications on Pure & Applied Analysis, 2015, 14 (3) : 843-859. doi: 10.3934/cpaa.2015.14.843
[19]	Mikhail Kovalyov. On the nature of large and rogue waves. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3061-3093. doi: 10.3934/dcds.2014.34.3061
[20]	Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences