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.     

Stability of the Riemann solutions for a Chaplygin gas

In this paper, we study the structural stability of solutions to the Riemann problem for one-dimensional isentropic Chaplygin gas. We perturb the Riemann initial data by taking three piecewise constant states and construct the global structure. By letting the perturbed parameter ε$\epsilon$ tend to zero, we prove that the Riemann solutions are stable under the local small perturbations of the Riemann initial data even when the initial perturbed density depending on the parameter but with no mass concentration limit.     

THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE ONE-DIMENSIONAL CHAPLYGIN GAS EQUATIONS

In this article,we study the Riemann problem with delta initial data for the one-dimensional Chaplygin gas equations.Under the generalized Rankine-Hugoniot conditions and the entropy condition,we const...     

一类非严格双曲系统的Riemann解的稳定性

在Riemann初值的小扰动意义下,对于一类非严格双曲系统证明Riemann解是稳定的.通过详细分析基本波的相互作用,利用特征分析方法研究扰动的Riemann解的全局结构以及解的大时间性态.     

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.     

A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT AXISYMMETRIC VESSELS

We are concerned with the derivation and analysis of one-dimensional hyperbolic systems of conservation laws modelling fluid flows such as the blood flow through compliant axisymmetric vessels.Early models derived are nonconservative and/or nonhomogeneous with measure source terms,which are endowed with infinitely many Riemann solutions for some Riemann data.In this paper,we derive a one-dimensional hyperbolic system that is conservative and homogeneous.Moreover,there exists a unique global Riemann solution for the Riemann problem for two vessels with arbitrarily large Riemann data,under a natural stability entropy criterion.The Riemann solutions may consist of four waves for some cases.The system can also be written as a 3×3 system for which strict hyperbolicity fails and the standing waves can be regarded as the contact discontinuities corresponding to the second family with zero eigenvalue.     

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.     

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 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.     

THE GENERALIZED RIEMANN PROBLEM FOR A SCALAR NONCONVEX COMBUSTION MODEL-THE PERTURBATION ON INITIAL BINDING ENERGY

In this article,we study the generalized Riemann problem for a scalar nonconvex Chapman-Jouguet combustion model in a neighborhood of the origin (t ＞ 0) on the (x,t) plane.We focus our attention to the...     

Shadow wave solution for the generalized Langmuir isotherm in chromatography

The shadow wave solution of the Riemann problem for the chromatography system under the mixed competitive-cooperative generalized Langmuir isotherm is constructed. It is shown that this shadow wave solution is weakly unique in the sense that all the entropy shadow wave solutions have the same distributional limit which is exactly the corresponding delta shock wave solution of the Riemann problem. Furthermore, the generalized Riemann problem with the delta type initial data is also considered and the existence and weak uniqueness of a solution are obtained in the framework of shadow wave solution.     

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 problem for the nonisentropic Baer–Nunziato model of two-phase flows

The Riemann problem for the well-known Baer–Nunziato model of two-phase flows is solved. The system consists of seven partial differential equations with nonconservative terms. The most challenging problem is that this model possesses a double eigenvalue. Although characteristic speeds coincide, the curves of composite waves associated with different characteristic fields can be still constructed. They will also be incorporated into composite wave curves to form solutions of the Riemann problem. Solutions of the Riemann problem will be constructed when initial data are in supersonic regions, subsonic regions, or in both kinds of regions. A unique solution and solutions with resonance are also obtained.     

Wave interactions and stability of the Riemann solutions for a scalar conservation law with a discontinuous flux function

This paper is devoted to studying the interactions of elementary waves for a model of a scalar conservation law with a flux function involving discontinuous coefficients. In order to cover all the situations completely, we take the initial data as three piecewise constant states and the middle region is regarded as the perturbed region with small distance. It is proved that the Riemann solutions are stable under the local small perturbations of the Riemann initial data by letting the perturbed parameter tend to zero. The proof is based on the detailed analysis of the interactions of stationary wave discontinuities with shock waves and rarefaction waves. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

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 Data for Zero-Pressure Gas Dynamics

In this paper, the Riemann problem with delta initial data for the one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics is considered. Under the generalized Rankine-Hugoniot conditions and the entropy condition, we constructively obtained the global existence of generalized solutions which contains delta-shock. Moreover, the author obtains the stability of generalized solutions by making use of the perturbation of the initial data.     

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     

Interactions of delta shock waves for the Aw-Rascle traffic model with split delta functions

This paper is concerned with the interactions of $\delta$-shock waves for the Aw-Rascle traffic model with split delta functions. 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 $\varepsilon\rightarrow0$.     

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.     

A phase decomposition approach and the Riemann problem for a model of two-phase flows

We present a phase decomposition approach to deal with the generalized Rankine–Hugoniot relations and then the Riemann problem for a model of two-phase flows. By investigating separately the jump relations for equations in conservative form in the solid phase, we show that the volume fractions can change only across contact discontinuities. Then, we prove that the generalized Rankine–Hugoniot relations are reduced to the usual form. It turns out that shock waves and rarefaction waves remain on one phase only, and the contact waves serve as a bridge between the two phases. By decomposing Riemann solutions into each phase, we show that Riemann solutions can be constructed for large initial data. Furthermore, the Riemann problem admits a unique solution for an appropriate choice of initial data.