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Stability and Unstability of the Standing Wave to Euler Equations

Part of: Hyperbolic equations and systems Equations of mathematical physics and other areas of application Partial differential equations Representations of solutions

Published online by Cambridge University Press:  18 January 2017

Xiuli Tang ,
Xiuqing Wang  and
Ganshan Yang
Xiuli Tang
Affiliation:
College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350117, China
Xiuqing Wang
Affiliation:
College of Mathematics and Computer Science, Yunnan Nationalities University, Kunming, Yunnan 650504, China
Ganshan Yang*
Affiliation:
College of Mathematics and Computer Science, Yunnan Nationalities University, Kunming, Yunnan 650504, China
*
*Corresponding author. Email:[email protected] (G. S. Yang)
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Abstract

In this paper, we first discuss the well-posedness of linearizing equations, and then study the stability and unstability of the 3-D compressible Euler Equation, by analysing the existence of saddle point. In addition, we give the existence of local solutions of the compressible Euler equation.

Keywords

Euler EquationsLie Groupwell-posednesssaddle point

MSC classification

Secondary: 35Q31: Euler equations 35L04: Initial-boundary value problems for first-order hyperbolic equations 35C07: Traveling wave solutions 35B35: Stability
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 4 , August 2017 , pp. 818 - 838
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Dafermos, C., Hyperbolic Conservation Laws in Continuum Physics, Second Edition, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, 2005.Google Scholar
[2] Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Math. Sciences, New York, 1984.Google Scholar
[3] Sideris, T. C., Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys., 101 (1985), pp. 475–485.CrossRefGoogle Scholar
[4] Makino, T., Blowing up solutiona of the Euler-Poisson equation for the evolution of gaseous stars, Transport Theory Statist. Phys., 21 (1992), pp. 615–624.Google Scholar
[5] Friedlander, Susan and Vishik, Misha M., Lax pair formulation for the Euler equation, Phys. Lett. A., 148 (1990), pp. 313–319.Google Scholar
[6] Deng, Y., Liu, T. P., Yang, T., and Yao, Z. A., Solutions of EulerCPoisson equations for gaseous stars, Arch. Rational Mech. Anal., 164 (2002), pp. 261–285.Google Scholar
[7] Li, T. H., Some special solutions of the Euler equation in RN , Commun. Pure Appl. Anal., 4 (2005), pp. 757–762.Google Scholar
[8] Guo, B. L., Yang, G. S. and Pu, X. K., Blow-up and global smooth solutions for incompressible three-dimensional Navier-Stokes equations, China Phys. Lett., 25 (2008), pp. 2115–2117.Google Scholar
[9] Ma, W. X. and Chen, M., Direct search for exact solutions to the nonlinear Schrödinger equation, Appl. Math. Comput., 215 (2009), pp. 2835–2842.Google Scholar
[10] Xing, Lü, Ma, Wenxiu, Yu, Jun and Khalique, Chaudry Masood, Solitary waves with the Madelung fluid description: A generalized derivative nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simulation, 31 (2016), pp. 40–46.Google Scholar
[11] Lü, Xing and Lin, Fuhong, Soliton excitations and shape-changing collisions in alphahelical proteins with interspine coupling at higher order, Commun. Nonlinear Sci. Numer. Simulation, 32 (2016), pp. 241–261.Google Scholar
[12] Lü, Xing,Madelung fluid description on a generalized mixed nonlinear Schrödinger equation, Nonlinear Dynamics, 81 (2015), pp. 239–247.Google Scholar
[13] Lü, Xing, Ma, Wenxiu and Khalique, Chaudry Masood, A direct bilinear Bäcklund transformation of a (2+1)-dimensional Korteweg-de Vries-like model, Appl.Math. Lett., 50 (2015), pp. 37–42.Google Scholar
[14] Lü, Xing , Ma, Wenxiu, Yu, Jun, Lin, Fuhong and Khalique, Chaudry Masood, Envelope bright- and dark-soliton solutions for the GerdjikovCIvanov model, Nonlinear Dynamics, 82 (2015), pp. 1211–1220.Google Scholar
[15] Guo, Yan, Jiang, Juhi and Jiang, Ning, Acoustic limit for the Boltzmann equation in optimal scaling, Commun. Pure Appl. Math., LXIII (2010), pp. 0337–0361.Google Scholar
[16] Song, Wenjing, Li, Hua, Yang, Ganshan, and Yuan, George Xianzhi, Nonhomogeneous boundary value problem for (I, J) similar solutions o incompressible two-dimensional Euler equations, J. Inequalities Applications, 2014 (2014), pp. 1–15.Google Scholar