American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Domain patterns and hysteresis in phase-transforming solids: Analysis and numerical simulations of a sharp interface dissipative model via phase-field approximation
  • NHM Home
  • This Issue
  • Next Article
    General constrained conservation laws. Application to pedestrian flow modeling
June  2013, 8(2): 465-479. doi: 10.3934/nhm.2013.8.465

Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation

Sun-Ho Choi 1,

1. 

Department of Mathematics, National University of Singapore, Singapore 117543, Singapore

Received  October 2012 Revised  November 2012 Published  May 2013

The purpose of this paper is to study asymptotic behaviors of the Green function of the linearized compressible Navier-Stokes equation. Liu, T.-P. and Zeng, Y. obtained a point-wise estimate for the Green function of the linearized compressible Navier-Stokes equation in [Comm. Pure Appl. Math. 47, 1053--1082 (1994)] and [Mem. Amer. Math. Soc. 125 (1997), no. 599]. In this paper, we propose a new methodology to investigate point-wise behavior of the Green function of the compressible Navier-Stokes equation. This methodology consists of complex analysis method and weighted energy estimate which was originally proposed by Liu, T.-P. and Yu, S.-H. in [Comm. Pure Appl. Math., 57, 1543--1608 (2004)] for the Boltzmann equation. We will apply this methodology to get a point-wise estimate of the Green function for large $t>0$.
Keywords: weighted energy estimate., Green function, Compressible Navier-Stokes equation, asymptotic behavior, long wave short wave decomposition.
Mathematics Subject Classification: Primary: 35Q30, 35B40; Secondary: 35C1.
Citation: Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465
References:
[1]

I.-L. Chern and T.-P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws,, Comm. Math. Phys., 110 (1987), 503.  doi: 10.1007/BF01212425.  Google Scholar

[2]

I.-L. Chern and T.-P. Liu, Erratum: "Convergence to difision waves of solutions for viscous conservation laws,", Comm. Math. Phys., 120 (1989), 525.   Google Scholar

[3]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, Proc. Roy. SOC. Edinburgh Sect. A, 106 (1987), 169.  doi: 10.1017/S0308210500018308.  Google Scholar

[4]

T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws,, Mem. American Mathematical Society, 56 (1985).   Google Scholar

[5]

T.-P. Liu, Interactions of nonlinear hyperbolic waves,, in, (1991), 171.   Google Scholar

[6]

T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation,, Comm. Pure Appl. Math., 57 (2004), 1543.  doi: 10.1002/cpa.20011.  Google Scholar

[7]

T.-P. Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws,, Mem. Amer. Math. Soc., 125 (1997).   Google Scholar

[8]

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofuid dynamics,, Japan J. Appl. Math., 1 (1984), 435.  doi: 10.1007/BF03167068.  Google Scholar

[9]

S. Zheng and W. Shen, Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems,, Scientia Sinica Ser. A, 30 (1987), 1133.   Google Scholar

[10]

Y. Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous 1-D flow,, Comm. Pure Appl. Math., 47 (1994), 1053.  doi: 10.1002/cpa.3160470804.  Google Scholar

show all references

References:
[1]

I.-L. Chern and T.-P. Liu, Convergence to diffusion waves of solutions for viscous conservation laws,, Comm. Math. Phys., 110 (1987), 503.  doi: 10.1007/BF01212425.  Google Scholar

[2]

I.-L. Chern and T.-P. Liu, Erratum: "Convergence to difision waves of solutions for viscous conservation laws,", Comm. Math. Phys., 120 (1989), 525.   Google Scholar

[3]

S. Kawashima, Large-time behaviour of solutions to hyperbolic-parabolic systems of conservation laws and applications,, Proc. Roy. SOC. Edinburgh Sect. A, 106 (1987), 169.  doi: 10.1017/S0308210500018308.  Google Scholar

[4]

T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws,, Mem. American Mathematical Society, 56 (1985).   Google Scholar

[5]

T.-P. Liu, Interactions of nonlinear hyperbolic waves,, in, (1991), 171.   Google Scholar

[6]

T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation,, Comm. Pure Appl. Math., 57 (2004), 1543.  doi: 10.1002/cpa.20011.  Google Scholar

[7]

T.-P. Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws,, Mem. Amer. Math. Soc., 125 (1997).   Google Scholar

[8]

T. Umeda, S. Kawashima and Y. Shizuta, On the decay of solutions to the linearized equations of electromagnetofuid dynamics,, Japan J. Appl. Math., 1 (1984), 435.  doi: 10.1007/BF03167068.  Google Scholar

[9]

S. Zheng and W. Shen, Global solutions to the Cauchy problem of quasilinear hyperbolic parabolic coupled systems,, Scientia Sinica Ser. A, 30 (1987), 1133.   Google Scholar

[10]

Y. Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous 1-D flow,, Comm. Pure Appl. Math., 47 (1994), 1053.  doi: 10.1002/cpa.3160470804.  Google Scholar

[1]

Linglong Du, Haitao Wang. Pointwise wave behavior of the Navier-Stokes equations in half space. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1349-1363. doi: 10.3934/dcds.2018055
[2]

Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263
[3]

Xinhua Zhao, Zilai Li. Asymptotic behavior of spherically or cylindrically symmetric solutions to the compressible Navier-Stokes equations with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1421-1448. doi: 10.3934/cpaa.2020052
[4]

Feimin Huang, Xiaoding Shi, Yi Wang. Stability of viscous shock wave for compressible Navier-Stokes equations with free boundary. Kinetic & Related Models, 2010, 3 (3) : 409-425. doi: 10.3934/krm.2010.3.409
[5]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[6]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92
[7]

Xingni Tan, Fuqi Yin, Guihong Fan. Random exponential attractor for stochastic discrete long wave-short wave resonance equation with multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3153-3170. doi: 10.3934/dcdsb.2020055
[8]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[9]

Anhui Gu, Boling Guo, Bixiang Wang. Long term behavior of random Navier-Stokes equations driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2495-2532. doi: 10.3934/dcdsb.2020020
[10]

Shujuan Lü, Zeting Liu, Zhaosheng Feng. Hermite spectral method for Long-Short wave equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 941-964. doi: 10.3934/dcdsb.2018255
[11]

Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic & Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010
[12]

Bo-Qing Dong, Juan Song. Global regularity and asymptotic behavior of modified Navier-Stokes equations with fractional dissipation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 57-79. doi: 10.3934/dcds.2012.32.57
[13]

Tomás Caraballo, Xiaoying Han. A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1079-1101. doi: 10.3934/dcdss.2015.8.1079
[14]

Anhui Gu, Kening Lu, Bixiang Wang. Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 185-218. doi: 10.3934/dcds.2019008
[15]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[16]

Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861-869. doi: 10.3934/cpaa.2005.4.861
[17]

Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441
[18]

Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46
[19]

Boris Haspot, Ewelina Zatorska. From the highly compressible Navier-Stokes equations to the porous medium equation -- rate of convergence. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3107-3123. doi: 10.3934/dcds.2016.36.3107
[20]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences