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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), 503-517. 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), 525-527.  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), 169-194. 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 "Nonlinear Analysis" (Taipei, 1989), World Sci. Publ., Teaneck, New Jersey, (1991), 171-183.  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), 1543-1608. 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), 435-457. 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), 1133-1149.  Google Scholar

[10]

Y. Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous 1-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082. 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), 503-517. 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), 525-527.  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), 169-194. 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 "Nonlinear Analysis" (Taipei, 1989), World Sci. Publ., Teaneck, New Jersey, (1991), 171-183.  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), 1543-1608. 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), 435-457. 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), 1133-1149.  Google Scholar

[10]

Y. Zeng, $L^1$ asymptotic behavior of compressible, isentropic, viscous 1-D flow, Comm. Pure Appl. Math., 47 (1994), 1053-1082. doi: 10.1002/cpa.3160470804.  Google Scholar

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