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

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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

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

Received October 2012 Revised November 2012 Published May 2013

Abstract
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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.
[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.
[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.
[4]	T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws,, Mem. American Mathematical Society, 56 (1985).
[5]	T.-P. Liu, Interactions of nonlinear hyperbolic waves,, in, (1991), 171.
[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.
[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).
[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.
[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.
[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.

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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.
[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.
[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.
[4]	T.-P. Liu, Nonlinear stability of shock waves for viscous conservation laws,, Mem. American Mathematical Society, 56 (1985).
[5]	T.-P. Liu, Interactions of nonlinear hyperbolic waves,, in, (1991), 171.
[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.
[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).
[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.
[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.
[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.

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