带有一般线性耗散项的p -方程组解的衰减估计

数学物理学报 ›› 2014, Vol. 34 ›› Issue (4): 802-815.

带有一般线性耗散项的p -方程组解的衰减估计

蒋盼盼|王泽军

南京航空航天大学数学系南京 211106

收稿日期:2012-10-12 修回日期:2013-12-20 出版日期:2014-08-25 发布日期:2014-08-25
基金资助:
国家自然科学基金青年基金(10901082) 和中国博士后基金(20090450149) 资助

Decay Estimates of Solutions for the p-System withGeneral Linear Dissipation

JIANG Pan-Pan, WANG Ze-Jun

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106

Received:2012-10-12 Revised:2013-12-20 Online:2014-08-25 Published:2014-08-25
Supported by:
国家自然科学基金青年基金(10901082) 和中国博士后基金(20090450149) 资助

摘要/Abstract

摘要：

该文主要研究带有一般的线性耗散项的p -方程组Cauchy 问题解的衰减性. 该文采用Fourier 变换的方法, 利用基本解方法构造Cauchy 问题的解, 证明了当初始值有紧支集、耗散项系数满足一定条件时p -方程组Cauchy 问题的解任意阶导数具有衰减性. 该文的讨论可以推广到更加一般的2×2 带耗散项的双曲型方程组的情形.

关键词: 一般线性耗散项, p -方程组, 解的衰减性, Fourier 变换, Green 函数

Abstract:

In this paper, the decay property of solutions for the p-system with general linear dissipation are considered. The main methods the author used are the Fourier analysis method in combination with the fundamental solution used in the Cauchy problem for partial differential equations. Under some suitable conditions on the initial data and the dissipative coefficients, the author gives a detailed description for the decay property of the solutions for the p-system with general linear dissipation. In addition, the discussion can be extended to more general 2×2 hyperbolic equations with the same kind of dissipation.

Key words: General linear dissipation, p-System, The degradation of solutions, Fourier transform, Green function

中图分类号:

35L60

蒋盼盼, 王泽军. 带有一般线性耗散项的p -方程组解的衰减估计[J]. 数学物理学报, 2014, 34(4): 802-815.

JIANG Pan-Pan, WANG Ze-Jun. Decay Estimates of Solutions for the p-System withGeneral Linear Dissipation[J]. Acta mathematica scientia,Series A, 2014, 34(4): 802-815.

参考文献

[1] Dafermos C M. Hyperbolic Conservation Laws in Continuum Physics. 北京: 清华大学出版社, 2005

[2] Courant R, Friedrichs K O. Suppersonic Flow and Shock Waves. New York: Interscience Publishers Inc, 1948

[3] Smoller J. Shock Waves and Reaction-Diffusion Equations. Michigan: Springer-Verlag, 1983

[4] Nishida T. Global solution for an initial-boundary-value problem of a quasilinear hyperbolic system. Proc Jap Acad, 1968, 44: 642--646

[5] 刘法贵. 一维可压流体动力学方程组整体经典解. 郑州: 黄河水利出版社, 2005: 20--49

[6] Smoller J A, Johnson J L. Global solutions for an extended class of hyperbolic systems of conservation laws. Arch Rational Mech Anal, 1969, 32: 169--189

[7]丁夏畦, 张同, 王靖华等. 拟线性双曲型守恒律组的整体解的研究. 中国科学A辑, 1973, 3: 239--254

[8] 丁夏畦, 陈贵强, 罗佩珠. 等熵气体动力学方程组Lax-Friedrichs格式的收敛性(II). 数学物理学报, 1988, 8(1): 61--94

[9]丁夏畦, 陈贵强, 罗佩珠. 等熵气体动力学方程组Lax-Friedrichs格式的收敛性(I). 数学物理学报, 1987, 7(4): 467--481

[10] 陈贵强. 等熵气体动力学方程组Lax-Friedrichs格式的收敛性(III). 数学物理学报, 1988, 8(3): 243--276

[11] Wang J H, Li C Z. Global resolvability and formation of singularities for quasilinear hyperbolic systems with dissipative terms. Chin Ann of Math, 1988, 9A(5): 509--523 (in Chinese)

[12] Lin L W, Zheng Y S. Existence and nonexistence of global smooth solutions for quasilinear hyperbolic systems.
Chin Ann of Math, 1988, 9B(3): 372--377

[13] Zheng Y S. Global smooth solutions for systems of gas dynamics with the dissipations. Acta Math Sci, 1987, 7(4): 383--390

[14] Dafermos C M, Hsiao L. Hyperbolic systems of balance laws with inhomogeneity and dissipation. Indiana Univ Math Jour, 1982, 31(4): 471--491

[15] Dafermos C M. Hyperbolic system of balance laws with weak dissipation. J Hyper Differ Equ, 2006, 3: 505--527

[16] Li Ta-tsien. Global Classical Solutions for Quasilinear Hyperbolic Systems. New York: John Wiley & Sons, 1994: 139--157

[17] Liu T P, Wang W K. The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi-dimensions. J Commun Math Phys, 1998, 196: 145--173

[18] Kobayashi T, Shibata Y. Decay estimates of solutions for the equations of motion of compressible viscous and heat-conductive gases in an exterior domain in R³. Comm Math Phys, 1999, 200: 621--659

[19] Xu H M, Wang W K. Pointwise estimate of solutions of isentropic Navier - Stokes equations in even space dimensions. Acta Math Sci, 2001, 21B(3): 417--427

[20] Wang W K, Yang T. The pointwise estimates of solutions of Euler equations with damping in multi-dimensions. J Differ Equ, 2001, 173: 410--450

[21] Wang W K, Wang Z J. The pointwise estimates to solutions for 1-dimensional linear thermoelastic system with second sound. J Math Anal Appl, 2007, 326: 1061--1075