双曲方程组与双曲—抛物方程组解的大时间状态估计
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摘要
在客观世界的发展过程中，不同物理量的变化将会有相互的制约或者促进作用，偏微分方程可以用来反映这种相互关系。对微分方程的研究可以更好的认知物理现象和自然界的客观规律，更好地指导生产和实践，因此对偏微分方程的研究一直是人们关心和研究的问题。
对于发展方程，如果方程的光滑解整体存在，就可以考虑方程解的大时间渐进行为，也就是解在充分大时间时的发展趋势。本文分别研究了一维拟线性双曲方程组与多维双曲-抛物方程组解的大时间状态行为。一维拟线性双曲系统反应了理想模型中空气动力学、浅水波理论、燃烧理论、非线性弹性碰撞力学、经典流体力学系统以及石油储存工程等物理和应用领域中相关物理量变化之间的相互制约关系。这些双曲方程组的经典解可以刻画流体系统或者连续系统中波的传播，对方程经典解的大时间状态估计能进一步了解方程的性质。如果考虑流体之间的粘性作用，那么描述物理量变化的系统就变成双曲-抛物耦合的方程组。其中最典型的是Navier-Stokes方程，它描叙了声学、浅水波理论等物理学方面一些物理量比仿质量、动量、能量的变化规律，并且这个方程在水利工程等方面也有广泛的应用，对这些微分方程的研究，在数学理论上和实际应用中都有非常重要的意义。
由于物理方程都是在实际中经过抽象和提炼出来的关系，所以方程组解的性态必须与物理现象相符合，只有能够体现物理运动发展规律的方程才是有实际价值的。一方面，双曲方程组的解反映着扰动的传播即波的运动特性。另一方面，波是以有限速度传播并且受惠更斯原理的影响。对于上面的两类方程，对方程组解的大时间状态估计，我们也期望得到的结果与这些物理现象相一致。
Green函数的方法被广泛地运用于研究方程(组)解的逐点估计，特别的，如果方程的光滑解整体存在，用这一方法可以得到关于解的最佳估计结果。本文就是利用Green函数的方法，分别讨论了一维拟线性双曲方程组解的逐点估计和偶数维可压缩等熵Navier-Stokes方程解的大时间状态估计。
对于一维空间一阶拟线性双曲方程组，其经典解的适定性理论已经研究的相当完善。由于方程组经典解具有局部适定性，往往利用特征线的方法讨论关于经典解适当范数的先验估计，从而得到经典解的有界性。我们知道拟线性双曲方程解的正则性会随着初始值及系数矩阵而提高，因此可以进一步讨论方程组解的高阶导数。拟线性双曲方程经典解的存在性与方程组的非线性性质密切相关。对于具有紧支集小初值的Cauchy问题，如果方程组满足线性退
化或者弱线性退化条件，其经典解整体存在.如果不满足这样的条件，其经典
解会在有限时间内爆破.因此要讨论解的大时间状态，所讨论的方程首先必须
满足(弱)线性退化条件.
对于偶数维空间Navier一Stokes方程，利用能量估计以及Fourie:分析的方
法，可以证明方程在非真空常状态附近的扰动问题存在性结论是清晰的.在大
于1的奇数维空间，解的大时间状态估计已经有了相应的结果，而且状态估
计所描述的结果与惠更斯原理相一致，对偶数维空间，由于惠更斯原理的影
响，波在传播的过程中，只有明显的波前，没有明显的波后，扰动具有长时间
的影响，所以对解的大时间渐进行为的研究有很大困难.
下面简单介绍本文的结构:
第一章，我们首先讨论了拟线性双曲方程组以及双曲一抛物方程与守恒
律方程之间的关系，以及在微分方程研究过程中人们所关心的问题.其次，我
们回顾了关于这两类方程的一些研究历史和现状，并且指出了运用Green函
数讨论方程组解的大时间状态估计的具体思路.最后，综合叙述了本文的主要
内容.
第二章，我们详细讨论了拟线性双曲方程组Cauchy间题经典解的逐点估
计问题，此时方程组满足弱线性退化条件和适配性条件，初始值在无穷远具
有适当的衰减，对于这样的方程，提高方程系数以及初始值的正则性，可以得
到关于方程组解的高阶导数相应估计.再利用这些估计来讨论经典解〔解的。
阶和1阶导数)的逐点估计.本章第一节，我们主要推导了波的分解公式，提
出了弱线性退化、适配性条件和正规化坐标系的概念，并且给出了在正规化坐
标系下一些关系式.第二节我们给出了本章的主要定理，包括解的存在性和解
的逐点估计结果.第三节，我们给出了相关范数.第四节，通过对这些范数的
讨论，得到方程组解及其高阶导数的一致先验估计，第五节分成两个部分，第
一部分证明了一些引理，这些引理对逐点估计的证明很重要，第二部分我们
讨论了解的逐点估计问题，得到方程组的经典解在整个时空空间(:全0)的精
确估计，特别的，在传统方法难以估计的各条特征线附近，我们也给出了解的
衰减估计.
第三章我们考虑了满足弱线性退化的齐次拟线性双曲方程组具有慢性衰
减的初始值cauchy问题经典解的存在区间估计，包括一般双曲方程但初始值
一阶导数衰减较快和角型方程组初始值衰减比较慢的情形，我们得到了关于
经典解存在区间的最佳估计.本章第一节，我们直接给出所要证明的结论，第
二节讨论了一般双曲系统经典解才存在区间，第三节讨论了对角型拟线性双
曲方程组经典解的存在区间.因为
Within the progress of development in the world, the change of different physical variables will restrict or promote each other. PDEs used to reflect this relationship. The researchs in PDEs help people to understand the physical phenomema or the laws in nature, and to guide the production and practices. So people paid attention to and researched in the study of PDEs all along.
For evolution equation, we can consider the large time behavior of solutions i.e. the tendency of the solutions as soon as the smooth solutions exist globally. Here we study the large time behavior of solutions to quasilinear hyperbolic systems in one dimension and hyperbolic-parabolic systems in multi-dimensions. Quasilinear hyperbolic systems reflect the restriction between different materials in gas dynamics, shallow water theory, combustion theory, nonliear elasticity, acoustics, classial fluid dynamics and petroleum resevoir engineering. Classical solutions to these systems will describe the wave propagation in fluid dynamics or continuum mechanics. We will know more about the character of the systems by studing the large time behavior of the classical solutions. The equations would be hyperbolic-parabolic systems if we consider viscidity. Naviable-Stokes equation is a typical example for hyperbolic-parabolic sytems. It models the progress of some physical variables like mass, momentum and energy in the
field of acoustics, shallow water theory. This equation has broad applications in water conservancy, too. It is very important to study these PDEs for the theorem in mathematics and the applications in realism.
Because the physical equations come from practices in the world, the behavior of the solutions must coincide with the physical phenomena. PDEs are worthiness when they describe the laws of the movememt in physics. On the one hand, solutions to hyperbolic systems reflect the wave propagation. On the other hand, waves propagation with finite speed and follow Huygens' principle. It is nature that we expect the behavior of solutions to these systems would coincide with these physical phenomena.
The method of Green function has been used to study pointwise estimates of solutions to all kinds of equations widely. In particular, we can get the sharp estimates of the solution by this method if the smooth solutions exist globally. In this paper, using the method of Green fuction, we consider pointwise estimates of solutions to quasilinear hyperbolic systems in one dimension and the large time behavior of solution to compressible isentropic Navier-Stokes equaation in even dimensions.
The well-posedness of the solutions for one order quasilinear hyperbolic system in one dimensionhas been researched completely. In general, we just need to get a priori estimates of the suitably norms about the solution by using characteristic, because the well-posedness of solution in local has been proved. In addition, the regularity of solution would increases as the regularity of initial data and the systems increases. So we also can study the higher order derivations of the solutions. It is well known that the existence of classical solutions to Cauchy problem for quasilinear hyperbolic systems linked closely to the nonlinearity of the system. If the systems satisfy linearly degenerate or weakly linearly degenerate, the classical solutions to Cauchy problem for quasilinear hyperbolic system will exist as the initial data with compact support is small enough. Otherwise, the classical solutions shall blowup in finite time. So we study the systems which satisfy (weakly) linearly degenerate when we consider the large time behavior of solutions.
For the isentropic Navier-Stokes equation in even dimensions, we can prove that it is well-posed to the perturbation in a neighborhood of any non-vacuum constant state using the energy method and Fourier analysis. Furthermore, we have obtained the large time behavior of solutions in odd but larger than one space dimensions. Fortunately, it coincides with weak Huygens' principle. Because the affect of Huygens' pri
对于发展方程，如果方程的光滑解整体存在，就可以考虑方程解的大时间渐进行为，也就是解在充分大时间时的发展趋势。本文分别研究了一维拟线性双曲方程组与多维双曲-抛物方程组解的大时间状态行为。一维拟线性双曲系统反应了理想模型中空气动力学、浅水波理论、燃烧理论、非线性弹性碰撞力学、经典流体力学系统以及石油储存工程等物理和应用领域中相关物理量变化之间的相互制约关系。这些双曲方程组的经典解可以刻画流体系统或者连续系统中波的传播，对方程经典解的大时间状态估计能进一步了解方程的性质。如果考虑流体之间的粘性作用，那么描述物理量变化的系统就变成双曲-抛物耦合的方程组。其中最典型的是Navier-Stokes方程，它描叙了声学、浅水波理论等物理学方面一些物理量比仿质量、动量、能量的变化规律，并且这个方程在水利工程等方面也有广泛的应用，对这些微分方程的研究，在数学理论上和实际应用中都有非常重要的意义。
由于物理方程都是在实际中经过抽象和提炼出来的关系，所以方程组解的性态必须与物理现象相符合，只有能够体现物理运动发展规律的方程才是有实际价值的。一方面，双曲方程组的解反映着扰动的传播即波的运动特性。另一方面，波是以有限速度传播并且受惠更斯原理的影响。对于上面的两类方程，对方程组解的大时间状态估计，我们也期望得到的结果与这些物理现象相一致。
Green函数的方法被广泛地运用于研究方程(组)解的逐点估计，特别的，如果方程的光滑解整体存在，用这一方法可以得到关于解的最佳估计结果。本文就是利用Green函数的方法，分别讨论了一维拟线性双曲方程组解的逐点估计和偶数维可压缩等熵Navier-Stokes方程解的大时间状态估计。
对于一维空间一阶拟线性双曲方程组，其经典解的适定性理论已经研究的相当完善。由于方程组经典解具有局部适定性，往往利用特征线的方法讨论关于经典解适当范数的先验估计，从而得到经典解的有界性。我们知道拟线性双曲方程解的正则性会随着初始值及系数矩阵而提高，因此可以进一步讨论方程组解的高阶导数。拟线性双曲方程经典解的存在性与方程组的非线性性质密切相关。对于具有紧支集小初值的Cauchy问题，如果方程组满足线性退
化或者弱线性退化条件，其经典解整体存在.如果不满足这样的条件，其经典
解会在有限时间内爆破.因此要讨论解的大时间状态，所讨论的方程首先必须
满足(弱)线性退化条件.
对于偶数维空间Navier一Stokes方程，利用能量估计以及Fourie:分析的方
法，可以证明方程在非真空常状态附近的扰动问题存在性结论是清晰的.在大
于1的奇数维空间，解的大时间状态估计已经有了相应的结果，而且状态估
计所描述的结果与惠更斯原理相一致，对偶数维空间，由于惠更斯原理的影
响，波在传播的过程中，只有明显的波前，没有明显的波后，扰动具有长时间
的影响，所以对解的大时间渐进行为的研究有很大困难.
下面简单介绍本文的结构:
第一章，我们首先讨论了拟线性双曲方程组以及双曲一抛物方程与守恒
律方程之间的关系，以及在微分方程研究过程中人们所关心的问题.其次，我
们回顾了关于这两类方程的一些研究历史和现状，并且指出了运用Green函
数讨论方程组解的大时间状态估计的具体思路.最后，综合叙述了本文的主要
内容.
第二章，我们详细讨论了拟线性双曲方程组Cauchy间题经典解的逐点估
计问题，此时方程组满足弱线性退化条件和适配性条件，初始值在无穷远具
有适当的衰减，对于这样的方程，提高方程系数以及初始值的正则性，可以得
到关于方程组解的高阶导数相应估计.再利用这些估计来讨论经典解〔解的。
阶和1阶导数)的逐点估计.本章第一节，我们主要推导了波的分解公式，提
出了弱线性退化、适配性条件和正规化坐标系的概念，并且给出了在正规化坐
标系下一些关系式.第二节我们给出了本章的主要定理，包括解的存在性和解
的逐点估计结果.第三节，我们给出了相关范数.第四节，通过对这些范数的
讨论，得到方程组解及其高阶导数的一致先验估计，第五节分成两个部分，第
一部分证明了一些引理，这些引理对逐点估计的证明很重要，第二部分我们
讨论了解的逐点估计问题，得到方程组的经典解在整个时空空间(:全0)的精
确估计，特别的，在传统方法难以估计的各条特征线附近，我们也给出了解的
衰减估计.
第三章我们考虑了满足弱线性退化的齐次拟线性双曲方程组具有慢性衰
减的初始值cauchy问题经典解的存在区间估计，包括一般双曲方程但初始值
一阶导数衰减较快和角型方程组初始值衰减比较慢的情形，我们得到了关于
经典解存在区间的最佳估计.本章第一节，我们直接给出所要证明的结论，第
二节讨论了一般双曲系统经典解才存在区间，第三节讨论了对角型拟线性双
曲方程组经典解的存在区间.因为
Within the progress of development in the world, the change of different physical variables will restrict or promote each other. PDEs used to reflect this relationship. The researchs in PDEs help people to understand the physical phenomema or the laws in nature, and to guide the production and practices. So people paid attention to and researched in the study of PDEs all along.
For evolution equation, we can consider the large time behavior of solutions i.e. the tendency of the solutions as soon as the smooth solutions exist globally. Here we study the large time behavior of solutions to quasilinear hyperbolic systems in one dimension and hyperbolic-parabolic systems in multi-dimensions. Quasilinear hyperbolic systems reflect the restriction between different materials in gas dynamics, shallow water theory, combustion theory, nonliear elasticity, acoustics, classial fluid dynamics and petroleum resevoir engineering. Classical solutions to these systems will describe the wave propagation in fluid dynamics or continuum mechanics. We will know more about the character of the systems by studing the large time behavior of the classical solutions. The equations would be hyperbolic-parabolic systems if we consider viscidity. Naviable-Stokes equation is a typical example for hyperbolic-parabolic sytems. It models the progress of some physical variables like mass, momentum and energy in the
field of acoustics, shallow water theory. This equation has broad applications in water conservancy, too. It is very important to study these PDEs for the theorem in mathematics and the applications in realism.
Because the physical equations come from practices in the world, the behavior of the solutions must coincide with the physical phenomena. PDEs are worthiness when they describe the laws of the movememt in physics. On the one hand, solutions to hyperbolic systems reflect the wave propagation. On the other hand, waves propagation with finite speed and follow Huygens' principle. It is nature that we expect the behavior of solutions to these systems would coincide with these physical phenomena.
The method of Green function has been used to study pointwise estimates of solutions to all kinds of equations widely. In particular, we can get the sharp estimates of the solution by this method if the smooth solutions exist globally. In this paper, using the method of Green fuction, we consider pointwise estimates of solutions to quasilinear hyperbolic systems in one dimension and the large time behavior of solution to compressible isentropic Navier-Stokes equaation in even dimensions.
The well-posedness of the solutions for one order quasilinear hyperbolic system in one dimensionhas been researched completely. In general, we just need to get a priori estimates of the suitably norms about the solution by using characteristic, because the well-posedness of solution in local has been proved. In addition, the regularity of solution would increases as the regularity of initial data and the systems increases. So we also can study the higher order derivations of the solutions. It is well known that the existence of classical solutions to Cauchy problem for quasilinear hyperbolic systems linked closely to the nonlinearity of the system. If the systems satisfy linearly degenerate or weakly linearly degenerate, the classical solutions to Cauchy problem for quasilinear hyperbolic system will exist as the initial data with compact support is small enough. Otherwise, the classical solutions shall blowup in finite time. So we study the systems which satisfy (weakly) linearly degenerate when we consider the large time behavior of solutions.
For the isentropic Navier-Stokes equation in even dimensions, we can prove that it is well-posed to the perturbation in a neighborhood of any non-vacuum constant state using the energy method and Fourier analysis. Furthermore, we have obtained the large time behavior of solutions in odd but larger than one space dimensions. Fortunately, it coincides with weak Huygens' principle. Because the affect of Huygens' pri
引文
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[37]Kong D-X., Breakdown of solutions to quasilinear hyperbolic systems with slow decay initial data, Chin. Nnn. of Math. 21B (2000), 1-28.
[38]Kong D-X., Formation and propagation of singularities for 2×2 quasilinear hyperbolic systems. Transactions of the A.M.S., Vol 354, no 8 (2002), 3155-3179.
[39]Kong D-X., and Yang Tong, Asymptotic behavior of the global classical solutions of general quasilinear hyperbolic systems, Comm. in Partial Diff. Eqns., 28 (2003), 1203- 1220
[40]P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conference Series in Applied Mathematics, SIAM, 1973.
[41]P.D. Lax, Hyperbolic systems of conservation laws Ⅱ Comm. Pure Appl. Math. 10 (1957), 537-566.
[42]Li T-T., Global classial solutions for quasilinear hyperbolic systems, Rearch in Applied Mathemaics 32,Wiley-Masson (1994).
[43]Li T-T., and Kong D-X., Initial value problem for general quasilinear hyperbolic sys- tems with characteristic with constant multiplicity, J. Partial Differential Equations 10,299-322(1997).
[44]Li T-T., and Kong D-X., Global classical discontinuous solutions to a class of generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws. Comm. in P.D.E. Vol 24(5,6) (1999), 801-820.
[45]Li T-T., Zhou Y.,and Kong D-X., Weak linear degeneracy and global classical solutions for general quasi-linear hyperbolic systems, Commus Partial diff. Eqns. 19 (1994), 1263-1317.
[46]Li T-T., Zhou Y., and Kong D-X., Global classial solutions for general quasilinear non-strictly hyperbolic systems with decay initial data, Nonlinear Studies 3, (1996), 203-229.
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[53]Liu T-P., and Wang W.K., The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi- dimension, Commun. Math. Phys.,196 (1998), 145-173.
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[55]A. Majda, Compressible Fluid Flow and System of Conservation laws in Several Space Variables, Appled Mathematical Sciences 53, Springer-Verlag, 1984.
[56]K.Nishihara, Wang Weike and Yang Tong, L_p- convergence rate to nonlinear diffusion waves for p-system with damping, J. Diff. Equations, 161 (2000), 191-218.
[57]J. Rauch, BV estimates fail for most quasilinear hyperbolic systems in dimension greater that one, Comm. Pure Appl. Phs., 106 (1986), 481-184.
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bridge University Press 1996.
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[60]T.C. Sideris, Nonexistence of Global Solutions to Semilinear Wave Equatuions in High Dimensions, J. Diff. Equations. 52 (1984), 378-406.
[61]T.C. Sideris, Formation of Singularities in Solutions to Nonlinear Hyperbolic Equations, Arch. Rat. Mech. Anal. 86 4 (1984), 369-381.
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[66]Wang Weike and Yang Tong, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J.Diff.Equations, 171 (2001), 410-450.
[67]Z.P. Xin, Some Current Topics in Nonlinear Conservation Laes, Studies in Advanced Mathematics, vol. 15 (2000).
[68]Z.P. Xin, On the linearized stability of viscous shock profiles for system of conservation laws, J. Diff. Equations, 100 (1992), 119-136.
[69]Xu H. M. and Wang W.K., Pointwise estimates of solutions of Isentropic Navier-Stokes equations in even space-dimensions. Acta Mathematic Scientia, 2001, 21B (3): 417-427.
[70]Wang W.K. and Yang X.F. Pointwise estimates of solutions to Cauchy problem for quasilinear hyperbolic systems. Chin. Ann. of Math., 2003, 24B(4):457-468.
[71]Wang W.K. and Yang X.F. The decay rate of solutions to isentropic Navier-Stokes equations in even space dimension. ( Submitted )
[72]Yang X.F. Lower bounds on life-span of classical solutions to quasiliner hyperbolic systems. ( Submitted )
[73]Zheng Yongshu, Globally smooth solutions to Cauchy problem of a quasilinear hyperbolic system arising in Biology, Mathemathica Acta Scientia, Vol 21(B) no 4,(2001)460- 468.
[74]陈恕行，偏微分方程概论，人民教育出版社，1982．
[75]陈恕行，周忆，一阶双曲型方程组的衰减率，《数学学报》，Vol 43(2000)，No．2，589-598．
[76]谷超豪等，数学物理方法第二册，上海科学技术出版社，1960．
[77]齐民友，广义函数与数学物理方程，高等教育出版社，1989．
[78]齐民友，徐超江，线性偏微分算子引论(下册)，科学出版社，1992．
[79]齐民友，徐超江，王维克，现代偏微分算子引论，武汉大学出版社，1992．
[80]李大潜，《数学进展》，1964．
[81]李大潜，陈韵梅，非线性发展方程，科学出版社，1997．
[82]徐红梅，偶数维空间Navier-Stokes方程解的逐点估计，博士论文．
[83]A．堪特劳维兹，气体动力学原理c编非定常气体动力学的一维处理法．徐华舫译科学出版社，1988．
[84]虞锦国，赵彦淳，一阶拟线性双曲型方程组的解的正则性，《数学年刊》，Vol 6(1985)No．5，597-609．
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[86]周民强，调和分析讲义，北京大学出版社，1999．
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[35]Kong D-X., Cauchy Problem for Quasi-linear Hyperbolic Systems MSJ Memoirs Mathematical Society of Japan, Vol. 6, (2000).
[36]Kong D-X., Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data. Chinese Ann. Math. Ser. B 21, no. 4 (2000), 413-440.
[37]Kong D-X., Breakdown of solutions to quasilinear hyperbolic systems with slow decay initial data, Chin. Nnn. of Math. 21B (2000), 1-28.
[38]Kong D-X., Formation and propagation of singularities for 2×2 quasilinear hyperbolic systems. Transactions of the A.M.S., Vol 354, no 8 (2002), 3155-3179.
[39]Kong D-X., and Yang Tong, Asymptotic behavior of the global classical solutions of general quasilinear hyperbolic systems, Comm. in Partial Diff. Eqns., 28 (2003), 1203- 1220
[40]P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, Regional Conference Series in Applied Mathematics, SIAM, 1973.
[41]P.D. Lax, Hyperbolic systems of conservation laws Ⅱ Comm. Pure Appl. Math. 10 (1957), 537-566.
[42]Li T-T., Global classial solutions for quasilinear hyperbolic systems, Rearch in Applied Mathemaics 32,Wiley-Masson (1994).
[43]Li T-T., and Kong D-X., Initial value problem for general quasilinear hyperbolic sys- tems with characteristic with constant multiplicity, J. Partial Differential Equations 10,299-322(1997).
[44]Li T-T., and Kong D-X., Global classical discontinuous solutions to a class of generalized Riemann problem for general quasilinear hyperbolic systems of conservation laws. Comm. in P.D.E. Vol 24(5,6) (1999), 801-820.
[45]Li T-T., Zhou Y.,and Kong D-X., Weak linear degeneracy and global classical solutions for general quasi-linear hyperbolic systems, Commus Partial diff. Eqns. 19 (1994), 1263-1317.
[46]Li T-T., Zhou Y., and Kong D-X., Global classial solutions for general quasilinear non-strictly hyperbolic systems with decay initial data, Nonlinear Studies 3, (1996), 203-229.
[47]Li T-T., Zhou Y., and Kong D-X., Global classical solutions for general Quasi-linear hyperbolic systems with Decay initial data. Nonlinear, Analysis 28 (1997), 1299-1332.
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[50]Liu T-P., Development of singularities in the nonlinear waves for quasi-linear hyperbolic partial differential equation, J. Diff. Eqns., 33 (1979), 92-111.
[51]Liu T-P., Nonlinear stability of shock waves for viscous conservation laws, Amer. Math. Soc Memoir, No 328. Providence, R.I., 1985.
[52]Liu T-P., Pointwise Convergence to Shock Waves for Viscous Conservation Laws, Comm. Pure Appl. Math. (1997), 1113-1182.
[53]Liu T-P., and Wang W.K., The pointwise estimates of diffusion wave for the Navier-Stokes systems in odd multi- dimension, Commun. Math. Phys.,196 (1998), 145-173.
[54]Liu T-P., and Zeng Y.N., Large time behavior of solutions general quasilinear hyperbolic-parabolic systems of conservation laws, A M S memoirs, 1997, 599.
[55]A. Majda, Compressible Fluid Flow and System of Conservation laws in Several Space Variables, Appled Mathematical Sciences 53, Springer-Verlag, 1984.
[56]K.Nishihara, Wang Weike and Yang Tong, L_p- convergence rate to nonlinear diffusion waves for p-system with damping, J. Diff. Equations, 161 (2000), 191-218.
[57]J. Rauch, BV estimates fail for most quasilinear hyperbolic systems in dimension greater that one, Comm. Pure Appl. Phs., 106 (1986), 481-184.
[58]Denis Serre, Systems of conservation laws I, hyperbolic, entropies, shock waves, Cam-
bridge University Press 1996.
[59]Shizuta Y and Kawashima S., Systems of equations hyperbolic-parabolic type with applications to the discrete Boltzmann equation. Hokkaido Math Journal, 1985, 14: 249-275
[60]T.C. Sideris, Nonexistence of Global Solutions to Semilinear Wave Equatuions in High Dimensions, J. Diff. Equations. 52 (1984), 378-406.
[61]T.C. Sideris, Formation of Singularities in Solutions to Nonlinear Hyperbolic Equations, Arch. Rat. Mech. Anal. 86 4 (1984), 369-381.
[62]T.C. Sideris, Formation of Singularities in Three-Dimensional Copressible Fluids, Commum Math. Phys. 101 (1985), 475-485.
[63]J. Smoller, Shock waves and reaction diffusion equations, Springer-Verlag, Berlin, 1983.
[64]C.D. Sogge, Lectures on Nonlinear Wave Equations, International Press, Monograghs in Analysis, Volume Ⅱ, 1995.
[65]A. Szepessy and Z. Xin, Nonlinear stability of viscous shock waves, Arch. Rat. Mech. Anal., Vol.122, No. 1 (1993), 53-103.
[66]Wang Weike and Yang Tong, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J.Diff.Equations, 171 (2001), 410-450.
[67]Z.P. Xin, Some Current Topics in Nonlinear Conservation Laes, Studies in Advanced Mathematics, vol. 15 (2000).
[68]Z.P. Xin, On the linearized stability of viscous shock profiles for system of conservation laws, J. Diff. Equations, 100 (1992), 119-136.
[69]Xu H. M. and Wang W.K., Pointwise estimates of solutions of Isentropic Navier-Stokes equations in even space-dimensions. Acta Mathematic Scientia, 2001, 21B (3): 417-427.
[70]Wang W.K. and Yang X.F. Pointwise estimates of solutions to Cauchy problem for quasilinear hyperbolic systems. Chin. Ann. of Math., 2003, 24B(4):457-468.
[71]Wang W.K. and Yang X.F. The decay rate of solutions to isentropic Navier-Stokes equations in even space dimension. ( Submitted )
[72]Yang X.F. Lower bounds on life-span of classical solutions to quasiliner hyperbolic systems. ( Submitted )
[73]Zheng Yongshu, Globally smooth solutions to Cauchy problem of a quasilinear hyperbolic system arising in Biology, Mathemathica Acta Scientia, Vol 21(B) no 4,(2001)460- 468.
[74]陈恕行，偏微分方程概论，人民教育出版社，1982．
[75]陈恕行，周忆，一阶双曲型方程组的衰减率，《数学学报》，Vol 43(2000)，No．2，589-598．
[76]谷超豪等，数学物理方法第二册，上海科学技术出版社，1960．
[77]齐民友，广义函数与数学物理方程，高等教育出版社，1989．
[78]齐民友，徐超江，线性偏微分算子引论(下册)，科学出版社，1992．
[79]齐民友，徐超江，王维克，现代偏微分算子引论，武汉大学出版社，1992．
[80]李大潜，《数学进展》，1964．
[81]李大潜，陈韵梅，非线性发展方程，科学出版社，1997．
[82]徐红梅，偶数维空间Navier-Stokes方程解的逐点估计，博士论文．
[83]A．堪特劳维兹，气体动力学原理c编非定常气体动力学的一维处理法．徐华舫译科学出版社，1988．
[84]虞锦国，赵彦淳，一阶拟线性双曲型方程组的解的正则性，《数学年刊》，Vol 6(1985)No．5，597-609．
[85]张恭庆等，泛函分析讲义(上、下册)，北京大学出版社，1987．
[86]周民强，调和分析讲义，北京大学出版社，1999．