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.     

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.     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

The asymptotic limits of Riemann solutions for the isentropic drift-flux model of compressible two-phase flows

The formation of vacuum state and delta shock wave are observed and studied in the limits of Riemann solutions for the one-dimensional isentropic drift-flux model of compressible two-phase flows by letting the pressure in the mixture momentum equation tend to zero. It is shown that the Riemann solution containing two rarefaction waves and one contact discontinuity turns out to be the solution containing two contact discontinuities with the vacuum state between them in the limiting situation. By comparison, it is also proved rigorously in the sense of distributions that the Riemann solution containing two shock waves and one contact discontinuity converges to a delta shock wave solution under this vanishing pressure limit.     

Interactions of delta shock waves for a class of nonstrictly hyperbolic system of conservation laws

In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter $\varepsilon$ tend to zero, we analyze the stability of Riemann solutions.     

The limits of Riemann solutions to the isentropic magnetogasdynamics

The objective of this note is to prove that the Riemann solutions of the isentropic magnetogasdynamics equations converge to the corresponding Riemann solutions of the transport equations by letting both the pressure and the magnetic field vanish. The delta shock wave can be obtained as the limit of two shock waves and the vacuum state can be obtained as the limit of two rarefaction waves. Moreover the relation between the speed of formation of singular density and those of the vanishing pressure and the vanishing magnetic field is discussed in detail.     

Structural stability of solutions to the Riemann problem for a scalar conservation law

The aim of this paper is to study the structural stability of solutions to the Riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients. It is proved that the Riemann solution is possibly instable when one of the Riemann initial data is at the vacuum. Furthermore, we point out that the Riemann solution is also possibly instable even when the Riemann initial data stay far away from vacuum. In order to deal with it, we perturb the Riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively. It is also proved that the Riemann solutions are unstable in some certain situations under the local small perturbations of the Riemann initial data by letting the perturbed parameter ε tend to zero. In addition, the interaction of the delta standing wave and the contact vacuum state is considered which appear in the Riemann solutions.     

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.     

Riemann problem for a geometrical optics system

The paper solves analytically the Riemann problem for a nonstrictly hyperbolic system of conservation laws arising in geometrical optics,in which the flux contains the nonconvex function possessing an infinite number of inflection points.Firstly,the generalized Rankine–Hugoniot relations and entropy condition of delta shock waves and left(right)-contact delta shock waves are proposed and clarified.Secondly,with the help of the convex hull,seven kinds of structures of Riemann solutions are obtained.The solutions fall into three broad categories with a series of geometric structures involving simultaneously contact discontinuities,vacuums and delta shock waves.Finally,numerical experiments confirm the theoretical analysis.     

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.     

Global structure of Riemann solutions to a system of two-dimensional hyperbolic conservation laws

The Riemann problem for a two-dimensional nonstrictly hyperbolic system of conservation laws is considered. Without the restriction that each jump of the initial data projects one planar elementary wave, ten topologically distinct solutions are obtained by applying the method of generalized characteristic analysis. Some of these solutions involve the nonclassical waves, i.e., the delta shock wave and the delta contact discontinuity, for which we explicitly give the expressions of their strengths, locations and propagation speeds. Moreover, we demonstrate that the nature of our solutions is identical with that of solutions to the corresponding one-dimensional Cauchy problem, which provides a verification that our construction produces the correct unique global solutions.     

扰动色谱方程黎曼解的极限-delta激波的行成和转换

本文主要讨论扰动色谱方程delta激波解的行成和转换,并讨论上述方程的黎曼问题.当扰动参数趋于零时,通过研究黎曼解的极限,我们可以观察到如下两个重要现象:激波和接触间断重合行成delta激波,一类激波(一个变量含有delta函数).     

Delta shock waves with Dirac delta function in both components for systems of conservation laws

We study a class of non-strictly and weakly hyperbolic systems of conservation laws which contain the equations of geometrical optics as a prototype. The Riemann problems are constructively solved. The Riemann solutions include two kinds of interesting structures. One involves a cavitation where both state variables tend to zero forming a singularity, the other is a delta shock wave in which both state variables contain Dirac delta function simultaneously. The generalized Rankine–Hugoniot relation and entropy condition are proposed to solve the delta shock wave. Moreover, with the limiting viscosity approach, we show all of the existence, uniqueness and stability of solution involving the delta shock wave. The generalized Rankine–Hugoniot relation is also confirmed. Then our theory is successfully applied to two typical systems including the geometric optics equations. Finally, we present the numerical results coinciding with the theoretical analysis.     

Numerical verification of delta shock waves for pressureless gas dynamics

The subject of this paper is theoretical analysis and numerical verification of delta shock wave existence for pressureless gas dynamic system. The existence of overcompressive delta shock wave solution in the framework of Colombeau generalized functions is proved. This result is verified numerically by specially designed procedure that is based on wave propagation method implemented in CLAWPACK. The method is coupled with dynamic refinement mesh. We also consider a strictly hyperbolic system obtained from the original one by perturbation and change of variables. The same numerical procedure is applied to the perturbed problem. The obtained numerical results in both cases confirm theoretical expectations.     

关于一个非线性几何光学中的非严格双曲守恒律系统的研究

研究了一个产生于非线性几何光学中的非严格双曲守恒律系统.该系统具有强非线性流函数项,且狄拉克激波可能同时出现在解的两个状态变量中.通过未知函数的一个变换,该系统的非线性流函数项得到弱化,从而其黎曼问题被完全解决.     

Interactions of delta shock waves and stability of Riemann solutions for nonlinear chromatography equations

This paper is devoted to studying the simplified nonlinear chromatography equations by introducing the change of state variables. The Riemann solutions containing delta shock waves are presented. In order to study wave interactions of delta shock waves with elementary waves, the global structure of solutions is constructed completely when the initial data are taken as three pieces of constants and the delta shock waves are included. In particular, the strength of delta shock wave is expressed explicitly and the delta contact discontinuity is discovered during the process of wave interactions. Moreover, by analyzing the limits of the solutions as the middle region vanishes, we observe that the Riemann solutions are stable for such a local small perturbation of the Riemann initial data.     

The Riemann problem with delta initial data for a class of coupled hyperbolic systems of conservation laws

In this paper, we solve the Riemann problem with the initial data containing Dirac delta functions for a class of coupled hyperbolic systems of conservation laws. Under suitably generalized Rankine–Hugoniot relation and entropy condition, the existence and uniqueness of solutions involving delta shock waves are proved. Further, four kinds of different structure for solutions are established uniquely.     

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.