Generalized δ-entropy condition to Riemann solution for Chaplygin gas in traffic flow

The Riemann problem for the Aw-Rascle model in the traffic flow with the Delta initial data for the Chaplygin gas is studied. The solutions are constructed globally under the generalized Rankine-Hugoniot relations, the δ-entropy conditions, and the generalized δ-entropy conditions. A new Delta wave, which is called a primary Delta wave, is defined in some solutions. The primary Delta wave satisfies the generalized Rankine- Hugoniot relations and the generalized δ-entropy conditions. It generates initially from the Delta initial data, which either evaluates a Delta wave, whose weight becomes stronger and stronger, or disappears at a finite time.

GLOBAL SOLUTIONS OF THE PERTURBED RIEMANN PROBLEM FOR THE CHROMATOGRAPHY EQUATIONS

The Riemann problem for the chromatography equations in a conservative form is considered. The global solution is obtained under the assumptions that the initial data are taken to be three piecewise constant states. The wave interaction problems are discussed in detail during the process of constructing global solutions to the perturbed Riemann problem. In addition, it can be observed that the Riemann solutions are stable under small perturbations of the Riemann initial data.

Centered simple waves for the two-dimensional pseudo-steady isothermal ?ow around a convex corner

The two-dimensional(2D) pseudo-steady isothermal flow, which is isentropic and irrotational, around a convex corner is studied. The self-similar solutions for the supersonic flow around the convex corner are constructed, where the properties of the centered simple wave are used for the 2D isentropic irrotational pseudo-steady Euler equations. The geometric procedures of the center simple waves are given. It is proven that the supersonic flow turns the convex corner by an incomplete centered expansion wave or an incomplete centered compression wave, depending on the conditions of the downstream state.

THE STABILITY OF THE DELTA WAVE TO PRESSURELESS EULER EQUATIONS WITH VISCOUS AND FLUX PERTURBATIONS

This paper is concerned with the pressureless Euler equations with viscous and flux perturbations.The existence of Riemann solutions to the pressureless Euler equations with viscous and flux perturbations is obtained.We show the stability of the delta wave of the pressureless Euler equations to the perturbations;that is,the limit solution of the pressureless Euler equations with viscous and flux perturbations is the delta wave solution of the pressureless Euler equations as the viscous and flux perturbation simultaneously vanish in the case u_(-)> u_(+).

Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday

December 9,2022 is the 90th birthday of Tong Zhang,a mathematician in Institute of Mathematics,Chinese Academy of Sciences where he was always working on the Riemann problem for gas dynamics in his mathematical life.To celebrate his 90th birthday and great contributions to this specifc feld,we organize this focused issue in the journal Communications on Applied Mathematics and Computation,since the Riemann problem has been proven to be a building block in all felds of theory,numerics and applications of hyperbolic conservation laws.

Limits of Riemann Solutions for Isentropic MHD in a Variable Cross-Section Duct as Magnetic Field Vanishes

The stability for magnetic field to the solution of the Riemann problem for the polytropic fluid in a variable cross-section duct is discussed.By the vanishing magnetic field method,the stable solutions are determined by comparing the limit solutions with the solutions of the Riemann problem for the polytropic fluid in a duct obtained by the entropy rate admissibility criterion.

Centered Waves for the Two-dimensional Pseudo-Steady van der Waals Gas Satisfied Maxwell's Law Around a Sharp Corner

In this paper,the authors study the centered waves for the two-dimensional(2D for short)pseudo-steady supersonic flow with van der Waals gas satisfied Maxwell's law around a sharp corner.In view of the initial value of the specific volume and the properties of van der Waals gas,the centered waves at the sharp corner are constructed by classification.It is shown that the supersonic incoming flow turns the sharp corner by a centered simple wave or a centered simple wave with right-contact discontinuity or a composite wave(jump-fan,fan-jump or fan-jump-fan),or a combination of waves and constant state.Moreover,the critical angle of the sharp corner corresponding to the appearance of the vacuum phenomenon is obtained.

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.

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.

The Riemann Problem for a Blood Flow Model in Arteries

In this paper,the Riemann solutions of a reduced 6×6 blood flow model in mediumsized to large vessels are constructed.The model is nonstrictly hyperbolic and non-conservative in nature,which brings two difficulties of the Riemann problem.One is the appearance of resonance while the other one is loss of uniqueness.The elementary waves include shock wave,rarefaction wave,contact discontinuity and stationary wave.The stationary wave is obtained by solving a steady equation.We construct the Riemann solutions especially when the steady equation has no solution for supersonic initial data.We also verify that the global entropy condition proposed by C.Dafermos can be used here to select the physical relevant solution.The Riemann solutions may contribute to the design of numerical schemes,which can apply to the complex blood flows.

The Expansion of a Wedge of Gas into Vacuum with Small Angle in Two-Dimensional Isothermal Flow

In this paper, the authors consider the expansion problem of a wedge of gas into vacuum for the two-dimensional Euler equations in isothermal flow. By the bootstrapping argument, they prove the global existence of the smooth solution through the direct method in the case 0 〈 θ 〈 -θ=arctan 1/（√2＋√5）, where θ is the half angle of the wedge. Furthermore, they get the uniform C^1,1 estimates of the solution to the expansion problem.

The Riemann Problem for Chaplygin Gas Flow in a Duct with Discontinuous Cross-Section

The fluid flows in a variable cross-section duct are nonconservative because of the source term.Recently,the Riemann problem and the interactions of the elementary waves for the compressible isentropic gas in a variable cross-section duct were studied.In this paper,the Riemann problem for Chaplygin gas flow in a duct with discontinuous cross-section is studied.The elementary waves include rarefaction waves,shock waves,delta waves and stationary waves.