用L^2能量法研究粘性守恒律解的渐近性态

本文讨论单个粘性守恒律方程与具有粘性的p方程组的Cauchy问题。根据初始材料的不同情形,其相应的Reimann问题以疏散波,激波或它们的迭加为弱解。本文的目的是指出Cauchy问题的解将分别趋于疏散波,激波或它们的迭加。本文基本方法是能量积分法。文中综述了现有的成果,也提出了一些未解决的问题。     

二阶偏微分方程的广义Cauchy问题

本文应用Ｇｒｅｅｎ公式求解偏微分方程的广义Ｃａｕｃｈｙ问题，得到了解的积分表达式。并将一般Ｃａｕｃｈｙ问题归结为广义Ｃａｕｃｈｙ问题的特例，从而用本方法统一处理了各类Ｃａｕｃｈｙ问题。     

关于一类非线性双曲方程组的二维Cauchy问题

本文通过引进新的广义解定义，对一类非线性双曲方程组的二维Ｃａｕｃｈｙ问题，证明了解的存在唯一性．并且，解可能含δ波．     

Banach空间中Cauchy问题解存在定理的注记

本文讨论Ｂａｎａｃｈ空间中的Ｃａｕｃｈｙ问题解的存在性定理。证明了Ｃａｕｃｈｙ方程右端函数ｆ（·）为ｋ－集压缩映象时，Ｃａｕｃｈｙ问题的解必存在，从而，将通常要求ｆ（·）满足Ｌｉｐ条件或绝对连续这两种条件统一起来，并加以推广。     

用Lebesgue－Stieltjes积分定义的双曲型方程广义解

该文用Ｌｅｂｅｓｇｕｅ－Ｓｔｉｅｌｔｊｅｓ积分给出一个双曲型方程组广义解的新定义，在这个意义下证明了Ｃａｕｃｈｙ问题整体广义解的存在性．这种解自然地包含了δ－激波．     

一般耦合Maxwell－SchrOdinger方程的Cauchy问题（1）

本文研究了一般耦合Ｍａｘｗｅｌｌ－Ｓｃｈｒｏｄｉｎｇｅｒ方程的Ｃａｕｃｈｙ问题，采用正则化方法，借助于推广的Ｈ．Ｐｅｃｈｅｒ引理及其它分析工具，在光滑解的框架下，建立了任意维空间中一般耦合Ｍａｘｗｅｌｌ－Ｓｃｈｒｏｄｉｎｇｅｒ方程的Ｃａｕｃｈｙ问题的局部适定性．     

复Banach空间中C-R方程的全纯解

二重复数是复数的一种推广，在其上的全纯映照族对应于Ｃ２上满足复Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯映照族．可以证明，这样的映照族本质上是由二个单复变数的全纯函数的直乘积所组成的族．本文证明：即使在Ｂａｎａｃｈ空间中，Ｃａｕｃｈｙ－Ｒｉｅｍａｎｎ方程的全纯解，具有同样的性质．     

三维Wigner-Poisson方程组的Cauchy问题

考虑三维 Ｗｉｇｎｅｒ－Ｐｏｉｓｓｏｎ方程组的 Ｃａｕｃｈｙ问题，将 ＷＰ问题转化为等价的 Ｓｃｈｒｏｄｉｎｇｅｒ－Ｐｏｉｓｓｏｎ问题．采用有限区域序列上的解的逼近方法，通过对逼近解建立与区域无关的先验估计，证明了 Ｃａｕｃｈｙ问题解的存在性、唯一性和逼近解的收敛性     

用Lebesgue—Stieltjes积分定义的双曲型方程广义解

该文用Ｌｅｂｅｓｇｕｅ－Ｓｔｉｅｌｔｊｅｓ积分给出一个双曲型方程组广义解的新定义，在这个意义下证明了Ｃａｕｃｈｙ问题整体广义解的存在性。这种解自然地包含了δ－激波。     

含奇异项的半线性抛物方程组的Cauchy问题

本文考察含奇异项的半线性抛物方程组的Ｃａｕｃｈｙ问题,算出了该问题的猝灭临界指标和猝灭临界维数．     

Asymptotic behavior of solutions for the 1-D isentropic Navier–Stokes–Korteweg equations with free boundary

In this paper, we consider the problem with a gas–gas free boundary for the one dimensional isentropic compressible Navier–Stokes–Korteweg system. For shock wave, asymptotic profile of the problem is shown to be a shifted viscous shock profile, which is suitably away from the boundary, and prove that if the initial data around the shifted viscous shock profile and its strength are sufficiently small, then the problem has a unique global strong solution, which tends to the shifted viscous shock profile as time goes to infinity. Also, we show the asymptotic stability toward rarefaction wave without the smallness on the strength if the initial data around the rarefaction wave are sufficiently small.     

The singular solutions to a nonsymmetric system of Keyfitz-Kranzer type with initial data of Riemann type

The solutions to the Riemann problem for a nonsymmetric system of Keyfitz-Kranzer type are constructed explicitly when the initial data are located in the quarter phase plane. In particular, some singular hyperbolic waves are discovered when one of the Riemann initial data is located on the boundary of the quarter phase plane, such as the delta shock wave and some composite waves in which the contact discontinuity coincides with the shock wave or the wave back of rarefaction wave. The double Riemann problem for this system with three piecewise constant states is also considered when the delta shock wave is involved. Furthermore, the global solutions to the double Riemann problem are constructed through studying the interaction between the delta shock wave and the other elementary waves by using the method of characteristics. Some interesting nonlinear phenomena are discovered during the process of constructing solutions; for example, a delta shock wave is decomposed into a delta contact discontinuity and a shock wave.     

DECAY RATES OF PLANAR VISCOUS RAREFACTION WAVE FOR MULTI-DIMENSIONAL SCALAR CONSERVATION LAW WITH DEGENERATE VISCOSITY ON HALF SPACE

We investigate the decay rates of the planar viscous rarefaction wave of the initial-boundary value problem to scalar conservation law with degenerate viscosity in several dimensions on the half-line space, where the corresponding one-dimensional problem admits the rarefaction wave as an asymptotic state. The analysis is based on the standard L2-energy method and L1-estimate.     

广义Chaplygin气体磁流体力学方程组的Riemann问题

利用特征分析和相平面分析的方法,由Rankine-Hugoniot条件和稳定性条件,构造性地得到了一维等熵广义Chaplygin气体磁流体力学方程组的Riemann解的存在唯一性.同时,详细研究了疏散波曲线和激波曲线的性质.     

一类非凸粘性平衡律方程的粘性激波解的存在性与稳定性

本文首先利用几何奇异摄动方法，证明了粘性系数充分小时一类非凸粘性平衡律方程的粘性冲击波的存在性，推广了原来在非线性项严格凸的条件下得到的结果．进而，利用谱分析和上下解方法，证明了对固定的小的粘性系数此类波是全局渐近指数稳定的，推广了反应扩散方程中经典的全局稳定性结果．     

有限变形弹性杆中的几何非线性波

利用有限变形理论的Lagrange描述,借助非保守系统的Hamilton型变分原理,导出了描述弹性杆中几何非线性波的波动方程．为了使非线性波动方程有稳定的行波解,计及了粘性效应引入的耗散和横向惯性效应导致的几何弥散．运用多重尺度法将非线性波动方程简化为KdV-Bergers方程,这个方程在相平面上对应着异宿鞍-焦轨道,其解为振荡孤波解．如果略去粘性效应或横向惯性,方程将分别退化为KdV方程或Bergers方程,由此得到孤波解或冲击波解,它们在相平面上对应着同宿轨道或异宿轨道．     

GLOBAL DISCONTINUOUS SOLUTIONS TO A CLASS OF DISCONTINUOUS INITIAL VALUE PROBLEMS FOR THE SYSTEM OF ISENTROPIC FLOW AND APPLICATIONS~(***)

In this paper, the authors discuss a kind of discontinuous initial value problems for the system of one-dimensional isentropic flow and prove the existence or nonexistence of global discontinuous solutions only containing one shock in a class of piecewise continuous and piecewise smooth functions. As applications, various interaction problems of a typical shock with a rarefaction wave are considered.     

Vanishing pressure limits of Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas

The Riemann solutions to the isentropic relativistic Euler system for Chaplygin gas with a small parameter are considered. Unlike the polytropic or barotropic gas cases, we find that firstly, as the parameter decreases to a certain critical number, the two-shock solution converges to a delta shock wave solution of the same system. Moreover, as the parameter goes to zero, that is, the pressure vanishes, the solution is nothing but the delta shock wave solution to the zero-pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution tends to the vacuum solution to the zero-pressure relativistic system, and the solution containing one rarefaction wave and one shock wave tends to the contact discontinuity solution to the zero-pressure relativistic system as pressure vanishes.     

一维绝热流动中初等波的相互作用

<正> 一维绝热流动的守恒律组(在拉格朗日座标下)为其中u——速度、p——压强、v——比容、E=e+u~2/2,而在多方气体的情形e=pv/(γ-1),绝热指数γ为常数,γ>1.人们称它的某些特解为初等波,其中包括前、后向激波S、S;前、后向中心疏散波R、R和上、下跳接触间断T、T.它们的相互作用一方面在