多孔介质中可压缩流体力学方程组经典解的整体存在性与破裂现象

利用拟线性双曲型方程组极值原理,改进了HSIAO Ling和D.Serre得到的关于多孔介质中可压缩流体力学方程组解的存在性结果,给出了其Cauchy问题的一个关于经典解整体存在和破裂的完整结果．这些结果说明强耗散有助于“小”解的光滑性．     

Two-Dimensional Riemann Problems for the Compressible Euler System

Riemann problems for the compressible Euler system in two space dimensions are complicated and difficult, but a viable alternative remains missing. The author lists merits of one-dimensional Riemann problems and compares them with those for the current two-dimensional Riemann problems, to illustrate their worthiness. Two-dimensional Riemann problems are approached via the methodology promoted by Andy Majda in the spirits of modern applied mathematics; that is, simplified model is built via asymptotic analysis, numerical simulation and theoretical analysis. A simplified model called the pressure gradient system is derived from the full Euler system via an asymptotic process. State-of-the-art numerical methods in numerical simulations are used to discern smallscale structures of the solutions, e.g., semi-hyperbolic patches. Analytical methods are used to establish the validity of the structure revealed in the numerical simulation. The entire process, used in many of Majda＇s programs, is shown here for the two-dimensional Riemann problems for the compressible Euler systems of conservation laws     

Riemann Problem for a Combustion Model System: the Z-N-D Solutions

The Riemann problem for a combustion model system with special kind of viscosity and chemical reaction is considered and the existence of the Riemann problem is proved. The limit of the Riemann solution as vanished viscosity is also investigated.     

耗散等熵气体动力学方程组广义Riemann问题

The paper concerns with generalized Riemann problem for isentropic flow with dissipation, and show that if the similarity solution to Riemann problem is composed of a backward centered rarefaction wave and a forward centered rarefaction wave, then general-ized Riemann problem admits a unique global solution on t ≥ 0. This solution is composed of backward centered wave and a forward centered wave with the origin as their center and then continuous for t>0.     

一维非线性弹性力学方程组的Riemann问题

本文研究了一维非线性弹性力学方程组的Riemann问题.根据左右状态所处的相对位置,分情况构造了问题的唯一整体解.由于激波条件退化,系统的基本波除了稀疏波和激波还包含退化激波.     

Riemann solutions without an intermediate constant state for a system of two conservation laws

We present a class of systems consisting of two conservation laws in one spatial dimension that share an intriguing property: they admit structurally stable Riemann solutions without the standard constant state. This striking phenomenon emerges in sharp contrast to what is known for strictly hyperbolic systems of conservation laws, in which the existence of constant states is necessary for the structural stability of Riemann solutions. We prove that, together, coincidence of characteristic speeds and a certain amount of genuine nonlinearity are sufficient to trigger the aforementioned phenomenon. The proof revolves about the presence of a singular point in the coincidence set that organizes the construction of our Riemann solutions.     

一类非线性双曲型方程Goursat问题的奇摄动

本文研究一类非线性奇摄动问题：在适当的假设下，讨论了相应问题解的存在性和唯一性，构造了其解的形式渐近展开式，并证明了它的一致有效性。     

无穷直线上的Riemann边值问题解的稳定性

讨论了无穷直线X上的R iem ann边值问题[1]的解当X轴发生光滑摄动时解的存在性和稳定性问题,并给出了相应的误差估计.     

Riemann问题的间断有限元

用时空全离散间断零次有限元对Riemarm问题进行了数值求解，没有出现振荡，很好的模拟了稀疏波的逐渐稀疏化和激波的剧烈变化。     

The Non-selfsimilar Riemann Problem for 2-D Zero-Pressure Flow in Gas Dynamics

The non-selfsimilar Riemann problem for two-dimensional zero-pressure flow in gas dynamics with two constant states separated by a convex curve is considered.By means of the generalized Rankine-Hugoniot relation and the generalized characteristic analysis method,the global solution involving delta shock wave and vacuum is constructed.The explicit solution for a special case is also given.     

Wave interactions and stability of the Riemann solutions for the chromatography equations

We prove that the Riemann solutions are stable for the chromatography system under the local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the wave interactions by applying the method of characteristic analysis. It is noteworthy that both the propagation directions of the shock wave S and rarefaction wave R are unchanged when they interact with the contact discontinuity J. Moreover, the global structures and large time asymptotic behaviors of the perturbed Riemann solutions are constructed and analyzed case by case.     

The Riemann Hilbert Problem for General Elliptic Complex Equations of Fourth Order

§１．　ＣｏｎｄｉｔｉｏｎｓｆｏｒＧｅｎｅｒａｌＥｌｌｉｐｔｉｃＣｏｍｐｌｅｘＥｑｕａｔｉｏｎｓｏｆＦｏｕｒｔｈＯｒｄｅｒ　　ＬｅｔＤｂｅａｂｏｕｎｄｅｄｄｏｍａｉｎ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｇｅｎｅｒａｌｅｌｌｉｐｔｉｃｃｏｍｐｌｅｘｅｑｕａｔｉｏｎｏｆｆｏｕｒｔｈｏｒｄｅｒｉｎｔｈｅｆｏｌｌｏｗｉｎｇｆｏｒｍｗｚ２ｚ２＝Ｆ（ｚ,ｗ,ｗｚ,ｗｚ,…,ｗｚ３,ｗｚ４,ｗｚ３ｚ,ｗｚ３ｚ,ｗｚ４）＋Ｇ（ｚ,ｗ,ｗｚ,ｗｚ,…,ｗｚ３）,Ｆ＝∑ｊ＋ｋ＝４（ｊ,ｋ）≠（２,２）Ｑｊ…     

气体动力学燃烧模型的广义Riemann问题

考察了在(x,t)平面上原点(t＞0)的邻域内气体动力学燃烧模型的广义Riemann问题．在改进的熵条件下构造了此问题的唯一解．它们是自相似ZND燃烧模型的极限．发现对某些情形,广义Riemann问题的解与相应的Riemann问题的解有本质的不同．特别地,扰动会使得相应Riemann问题的强爆轰波转化为由预压激波点燃的弱爆燃波．在一些情形,尽管相应的Riemann解中不含燃烧波,扰动后燃烧波会出现．这反映了未燃气体的不稳定性．     

非等熵Chaplygin气体极限黎曼解 关于扰动的依赖性

主要研究了非等熵Chaplygin气体黎曼问题初值扰动后解的结构,分析了经典的黎曼问题和扰动问题解的结构及极限结构,发现后者的极限解在δ质量权趋于零时不同于前者解的结构.该结果表明对非等熵Chaplygin气体而言,经典的黎曼问题与带δ初值的黎曼问题有着本质的区别.     

Perturbed Riemann Problem for a Scalar Chapman-Jouguet Combustion Model

The author considers the perturbed Riemann problem for a scalar ChapmanJouguet combustion model which comes from Majda’s model with a modified, bump-type ignition function proposed in the results of Lyng and Zumbrun in 2004. Under the entropy conditions, the unique solution in a neighborhood of the origin on the(x, t) plane(t > 0) is obtained. It is found that, for some cases, the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. The perturbation may transform a strong detonation into a weak deflagration in the neighborhood of the origin. Especially, it can be observed that burning happens although the corresponding Riemann solution does not contain combustion wave, which exhibits the instability for the unburnt state.     

Mixed initial boundary value problem for hyperbolic geometric flow on Riemann surfaces

The hyperbolic geometric flow equations is introduced recently by Kong and Liu motivated by Einstein equation and Hamilton Ricci flow. In this paper, we consider the mixed initial boundary value problem for hyperbolic geometric flow, and prove the global existence of classical solutions. The results show that, for any given initial metric on R² in certain class of metric, one can always choose suitable initial velocity symmetric tensor such that the solutions exist, and the scalar curvature corresponding to the solution metric gij keeps bounded. If the initial velocity tensor does not satisfy the certain conditions, the solutions will blow up at a finite time. Some special explicit solutions to the reduced equation are given.     

One-Dimensional Riemann Problem for Equations of Constant Pressure Fluid Dynamics with Measure Solutions by the Viscosity Method

The Riemann problem for the equations of constant pressure fluid dynamics was considered. Solutions of this problem were constructed by employing the viscosity vanishing approach. For some initial data, solutions can be viewed as bounded functions in L(R × R+) plus bounded linear functionals on C0(R× R+) with nonclassical waves as their supports. Vacuum regions appear so that uniqueness of Riemann solutions fails for some initial data.     

绝热流Chaplygin气体的Riemann问题

本文研究了绝热流Chaplygin气体动力学方程组,利用特征分析方法,在得到所有基本波的基础上,构造出Riemann问题的所有解.Riemann解由前向疏散波(激波)、后向疏散波(激波)、接触间断以及δ波构成.     

Global Strong Solutions for the Viscous,Micropolar, Compressible Flow

In this paper, we consider the viscous, micropolar, compressible flow in one dimension. We give the proof of existence and uniqueness of strong solutions for the initial boundary problem that vacuum can be allowed initially.     

Riemann problem for Chaplygin gas with combustion

We study the Riemann problem of the Chapman–Jouguet model for an ideal combustible Chaplygin gas. By analyzing the wave curves in the phase plane, we obtain constructively the unique solution of the Riemann problem under the global entropy conditions.