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.     

Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

In this paper,firstly,by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation,we construct parameterized delta-shock and constant density solutions,then we show that,as the flux perturbation vanishes,they converge to the delta-shock and vacuum state solutions of the zero-pressure flow,respectively.Secondly,we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure.Furthermore,we rigorously prove that,as the two-parameter flux perturbation vanishes,any Riemann solution containing two shock waves tends to a delta-shock solution to the zero-pressure flow;any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state.Finally,numerical results are given to present the formation processes of delta shock waves and vacuum states.     

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.     

Delta shocks and vacuum states in vanishing pressure limits of solutions to the relativistic Euler equations for polytropic gases

The Riemann solutions for the Euler system of conservation laws of energy and momentum in special relativity for polytropic gases are considered. It is rigorously proved that, as pressure vanishes, they tend to the two kinds of Riemann solutions to the corresponding pressureless relativistic Euler equations: the one includes a delta shock, which is formed by a weighted δ-measure, and the other involves vacuum state.     

THE VANISHING PRESSURE LIMIT OF SOLUTIONS TO THE SIMPLIFIED EULER EQUATIONS FOR ISENTROPIC FLUIDS

In this paper,the Riemann problem of the 1-D reduced model for the 2-D Euler equations is considered and the Riemann solutions are obtained.It is proved that,as the pressure vanishes,they converge to two kinds of Riemann solutions to the 1D reduced model for the 2-D transport equations:one contains δ-shocks,the other contains vacuum.     

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.     

Interactions of delta shock waves for the transport equations with split delta functions

This paper is concerned with the interactions of δ-shock waves and the vacuum states between the two contact discontinuities for the transport equations. The solutions are obtained constructively when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter ε tends to zero.     

Global existence of smooth solutions to two-dimensional compressible isentropic Euler equations for Chaplygin gases

In this paper we investigate the two-dimensional compressible isentropic Euler equations for Chaplygin gases. Under the assumption that the initial data is close to a constant state and the vorticity of the initial velocity vanishes, we prove the global existence of the smooth solution to the Cauchy problem for twodimensional flow of Chaplygin gases.     

Approaching Chaplygin pressure limit of solutions to the Aw–Rascle model

This paper studies the limit of solutions to the Aw–Rascle model as the pressure tends to the Chaplygin gas pressure. For concreteness, the pressure is taken as a modified Chaplygin gas pressure. Firstly, the Riemann problem for the Aw–Rascle model with the modified Chaplygin gas pressure is solved constructively. Secondly, it is shown that as the pressure tends to the Chaplygin gas pressure, some Riemann solutions containing a shock and a contact discontinuity tend to a delta-shock solution, whose propagation speed and strength are different from those of delta-shock solution to the Aw–Rascle model with a Chaplygin gas pressure. Besides, it is also proven that the rest Riemann solutions converge to a two-contact-discontinuity solution, which is exactly the solution to the Aw–Rascle model with a Chaplygin gas pressure. Thirdly, some numerical results are presented to exhibit the process of formation of delta-shocks.     

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.     

One-dimensional nonlinear chromatography system and delta-shock waves

The Riemann problem for the nonlinear chromatography system is considered. Existence and admissibility of δ-shock type solution in both variables are established for this system. By the interactions of δ-shock wave with elementary waves, the generalized Riemann problem for this system is presented, the global solutions are constructed, and the large time-asymptotic behavior of the solutions are analyzed. Moreover, by studying the limits of the solutions as perturbed parameter ${\varepsilon}$ tends to zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data.     

Riemann boundary value problems and reflection of shock for the Chaplygin gas

In this paper,we study two-dimensional Riemann boundary value problems of Euler system for the isentropic and irrotational Chaplygin gas with initial data being two constant states given in two sectors respectively,where one sector is a quadrant and the other one has an acute vertex angle.We prove that the Riemann boundary value problem admits a global self-similar solution,if either the initial states are close,or the smaller sector is also near a quadrant.Our result can be applied to solving the problem of shock reflection by a ramp.     

Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions

In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time‐asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter ε tends to zero. Copyright © 2014 John Wiley & Sons, Ltd.     

三维可压等熵Euler方程光滑解的整体存在性

研究了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.如果初值是一个常状态的小扰动并且初速度的旋度等于零,证明了三维可压等熵Euler方程Cauchy问题光滑解的整体存在性.     

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.     

Comparison of Riemann Solutions for Non-isentropic Modified and Pure Chaplygin Gas Dynamics

Acta Mathematicae Applicatae Sinica, English Series - In this paper, we investigate the Riemann solutions of the non-isentropic Euler equations for the modified Chaplygin gas and the pure Chaplygin...     

Zero dissipation limit to a Riemann solution consisting of two shock waves for the 1D compressible isentropic Navier-Stokes equations

We investigate the zero dissipation limit problem of the one-dimensional compressible isentropic Navier-Stokes equations with Riemann initial data in the case of the composite wave of two shock waves.It is shown that the unique solution to the Navier-Stokes equations exists for all time,and converges to the Riemann solution to the corresponding Euler equations with the same Riemann initial data uniformly on the set away from the shocks,as the viscosity vanishes.In contrast to previous related works,where either the composite wave is absent or the efects of initial layers are ignored,this gives the frst mathematical justifcation of this limit for the compressible isentropic Navier-Stokes equations in the presence of both composite wave and initial layers.Our method of proof consists of a scaling argument,the construction of the approximate solution and delicate energy estimates.     

Concentration and cavitation phenomena of Riemann solutions for the isentropic Euler system with the logarithmic equation of state

The analytical solutions of the Riemann problem for the isentropic Euler system with the logarithmic equation of state are derived explicitly for all the five different cases. The concentration and cavitation phenomena are observed and analyzed during the process of vanishing pressure in the Riemann solutions. It is shown that the solution consisting of two shock waves converges to a delta shock wave solution as well as the solution consisting of two rarefaction waves converges to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting situation.     

Van der Waals气体Euler方程的Riemann问题及基本波的相互作用

研究了修正的等熵Van der Waals气体动力学Euler方程Riemann问题及其基本波的相互作用.利用Maxwell提出的等面积法则,将Van der Waals气体状态方程修正为与实际相符,从而守恒律方程组从混合型转化为双曲型.利用广义特征线分析法,构造性地得到了Riemann问题的解是存在的.进一步,得到了基本波相互作用.