Generalized Riemann problem for Euler system Dedicated to Professor LI TaTsien on the Occasion of His 80th Birthday

This article is a survey on the progress in the study of the generalized Riemann problems for MD Euler system. A new result on generalized Riemann problems for Euler systems containing all three main nonlinear waves(shock, rarefaction wave and contact discontinuity) is also introduced.     

A formulation of the fractional Noether-type theorem for multidimensional Lagrangians

This paper presents the Euler–Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses well-known notion of the Riemann–Liouville fractional derivative.     

带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解的极限

研究了带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解的极限.由于非齐次项的影响,带有源项的广义Chaplygin气体磁流体Euler方程组Riemann解不再是自相似的.当压力和磁感强度同时消失时,它的解会收敛到零压流输运方程组的Riemann解,解中会出现δ-激波和真空现象.同时研究还得到了仅当磁感强度消失时,它的解会收敛到非齐次广义Chaplygin气体Euler方程组的Riemann解,并且解中只出现δ-激波.     

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     

On relationship between convergence ball of euler iteration in banach spaces and its dynamical behavior on Riemann spheres

The relationship between the convergence ball of the Euler iteration in Banach Spaces and its exclusive fixed points on Riemann spheres is investigated. By using an exclusive fixed point of the Euler iteration, the convergence ball is determined accurately for a class of operators whose derivatives satisfy some generalized Lipschitz condition on Banach spaces.     

Riemann problem for the relativistic Chaplygin Euler equations

The relativistic Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are five kinds of Riemann solutions, in which four only contain different contact discontinuities and the other involves delta shock waves. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

Delta Shocks and Vacuum States in Vanishing Pressure Limits of Solutions to the Relativistic Euler Equations

The Riemann problems for the Euler system of conservation laws of energy and momentum in special relativity as pressure vanishes are considered. The Riemann solutions for the pressureless relativistic Euler equations are obtained constructively. There are two kinds of solutions, the one involves delta shock wave and the other involves vacuum. The authors prove that these two kinds of solutions are the limits of the solutions as pressure vanishes in the Euler system of conservation laws of energy and momentum in special relativity.     

一类扩展Euler和的表示问题

应用Parseval定理和Nielsen广义多重对数函数的性质,给出了非线性扩展Euler和的Riemann Zeta函数表示.对来自于实验数学中的扩展Euler和∑n=1∞H2n/n2的经验公式给出了严格的理论证明.此方法也适用于求其它扩展Euler和的计算问题.     

Riemann problem with Delta initial data for the relativistic Chaplygin Euler equations

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the relativistic Chaplygin Euler equations. Under the generalized Rankine-Hugoniot conditions and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, we obtain the stability of generalized solutions by making use of the perturbation of the initial data     

Riemann problem with delta initial data for the isentropic relativistic Chaplygin Euler equations

In this paper, we study the Riemann problem with the initial data containing the Dirac delta function for the isentropic relativistic Chaplygin Euler equations. Under suitably generalized Rankine–Hugoniot relation and entropy condition, we constructively obtain the global existence of generalized solutions including delta shock waves that explicitly exhibit four kinds of different structures. Moreover, it can be found that the solutions constructed here are stable for the perturbation of the initial data.     

Riemann problem for the isentropic relativistic Chaplygin Euler equations

This paper studies the Riemann problem of the isentropic relativistic Euler equations for a Chaplygin gas. The solutions exactly include five kinds. The first four consist of different contact discontinuities while the rest involves delta-shock waves. Under suitable generalized Rankine?CHugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.     

实轴上的双解析函数Riemann边值逆问题

提出了一类实轴上的双解析函数Riemann边值逆问题.先消去参变未知函数,再采用易于推广的矩阵形式记法,可把问题转化为两个实轴上的解析函数Riemann边值问题.利用经典的Riemann边值问题理论,讨论了该问题正则型情况的解法,得到了它的可解性定理.     

Euler and number theory

We give an account of the most important results obtained by Euler in number theory, including the main contribution of Euler, application of analysis to problems of number theory. We note an important role played in modern number theory by the function that was introduced by Euler and is called the Riemann zeta function. We also discuss Euler’s works in other fields of science such as function theory and theory of music, as well as the relationship between music and mathematics.     

Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

Abstract Euler–Poisson–Darboux equation with nonlocal condition

We find conditions for the unique solvability of nonlocal problems for abstract differential equation of the Euler–Poisson–Darboux equation. Nonlocal conditions contain either Erdelyi–Kober operator or Riemann–Liouville fractional integration operator.     

Singular and selfsimilar solutions for Euler equations with phase transitions

Riemann problems for the full set of Euler equations for two phases with phase transition are considered. Based on the assumptions across the phase boundary kinetic relations to describe the mass transfer between the phase are derived from the second law of thermodynamics. Self-similar as well as singular solutions can be constructed. For both cases the structure of the solution is discussed.     

The limits of Riemann solutions for the isentropic Euler system with extended Chaplygin gas

The Riemann problem for the isentropic Euler system with the state equation for the extended Chaplygin gas is considered, and the Riemann solutions are constructed completely for all the cases. The limiting relations of Riemann solutions for the isentropic Euler system with the state equation from the extended Chaplygin gas to the Chaplygin gas are derived in detail when the corrected term tends to zero. The formation of delta shock wave solution and two-contact-discontinuity solution is investigated during the process of taking the limit.     

High resolution multi-moment finite volume method for supersonic combustion on unstructured grids

In this study, we present a novel numerical model for simulating detonation waves on unstructured grids. In contrast to the conventional finite volume method (FVM), two types of moment comprising the volume-integrated average (VIA) and the point value (PV) at the cell vertex are treated as the evolution variables for the reacting Euler equations. The VIA is computed based on a finite volume formulation of the flux form where the conventional Riemann problem is solved by the HLLC Riemann solver. The PV is updated in a point-wise manner by using the differential formulation where the Roe solver is used to compute the differential Riemann problems. In order to increase the accuracy around discontinuities, numerical oscillations and dissipations are reduced using the boundary variation diminishing algorithm. Convergence tests demonstrated that the proposed model could achieve third-order accuracy with unstructured grids for reacting Euler equations. The high resolution property of the proposed method was verified based on simulations of several detonation wave propagation problems in two and three dimensions. In particular, the current model could resolve the cellular structures with fewer degrees of freedom for the unstable oblique detonation wave problem. These fine structures may be smoothed out by the conventional FVM due to the excessive amount of numerical dissipation errors. Importantly, a simulation of stiff detonation waves showed that the proposed method could capture the correct position of the reaction front whereas the conventional FVMs produced spurious phenomena. Thus, the proposed model can obtain highly accurate solutions for detonation problems on unstructured grids, which is highly advantageous for real applications involving complex geometrical configurations.     

Local Smooth Solutions to the 3-Dimensional Isentropic Relativistic Euler equations

The authors consider the local smooth solutions to the isentropic relativistic Euler equations in （3＋1）-dimensional space-time for both non-vacuum and vacuum cases. The local existence is proved by symmetrizing the system and applying the Friedrichs- Lax-Kato theory of symmetric hyperbolic systems. For the non-vacuum case, according to Godunov, firstly a strictly convex entropy function is solved out, then a suitable sym- metrizer to symmetrize the system is constructed. For the vacuum case, since the coefficient matrix blows-up near the vacuum, the authors use another symmetrization which is based on the generalized Riemann invariants and the normalized velocity.