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

Sun-Ho Choi; Sun-Ho Choi

doi:10.3934/nhm.2013.8.465

Networks and Heterogeneous Media

2013, Volume 8, Issue 2: 465-479. doi: 10.3934/nhm.2013.8.465

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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

Received: 01 October 2012 Revised: 01 November 2012
Primary: 35Q30, 35B40; Secondary: 35C15.

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 $t>0$.
- Compressible Navier-Stokes equation,
- Green function,
- asymptotic behavior,
- long wave short wave decomposition,
- weighted energy estimate
Citation: Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation[J]. Networks and Heterogeneous Media, 2013, 8(2): 465-479. doi: 10.3934/nhm.2013.8.465

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Abstract

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 $t>0$.

References

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