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.     

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 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.     

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.     

A NOTE ON THE INTERACTIONS OF ELEMENTARY WAVES FOR THE AR TRAFFIC FLOW MODEL WITHOUT VACUUM

In this note, we consider the interactions of elementary waves for the traffic flow model proposed by Aw and Rascle when the vacuum is not involved. The solutions are obtained constructively and globally when the initial data consist of three pieces of constant states. Furthermore, it can be found that the Riemann solutions are stable with respect to such small perturbations of the initial data in this particular situation by investigating the limits of the solutions as the perturbed parameter ε goes to zero.     

Delta-shocks and vacuum states in the vanishing pressure limit of solutions to the isentropic Euler equations for modified Chaplygin gas

The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the isentropic Euler equations for a modified Chaplygin gas are analyzed as the double parameter pressure vanishes. Firstly, the Riemann problem of the isentropic Euler equations for a modified Chaplygin gas is solved analytically. Secondly, it is rigorously shown that, as the pressure vanishes, any two-shock Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a δ-shock solution to the transport equations, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the isentropic Euler equations for a modified Chaplygin gas tends to a two-contact-discontinuity solution to the transport equations, the nonvacuum intermediate state between the two rarefaction waves tends to a vacuum state. Finally, some numerical results exhibiting the formation of δ-shocks and vacuum states are presented as the pressure decreases.     

Interaction of Shallow Water Waves

In this paper, we consider the Riemann problem and interaction of elementary waves for a nonlinear hyperbolic system of conservation laws that arises in shallow water theory. This class of equations includes as a special case the equations of classical shallow water equations. We study the bore and dilatation waves and their properties, and show the existence and uniqueness of the solution to the Riemann problem. Towards the end, we discuss numerical results for different initial data along with all possible interactions of elementary waves. It is noticed that in contrast to the p -system, the Riemann problem is solvable for arbitrary initial data, and its solution does not contain vacuum state.     

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.     

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.     

Dynamical behaviors for 1D compressible Navier-Stokes equations with density-dependent viscosity

The dynamical behaviors of vacuum states for one-dimensional compressible Navier-Stokes equations with density-dependent viscosity coefficient are considered. It is first shown that a unique strong solution to the free boundary value problem exists globally in time, the free boundary expands outwards at an algebraic rate in time, and the density is strictly positive in any finite time but decays pointwise to zero time-asymptotically. Then, it is proved that there exists a unique global weak solution to the initial boundary value problem when the initial data contains discontinuously a piece of continuous vacuum and is regular away from the vacuum. The solution is piecewise regular and contains a piece of continuous vacuum before the time T_∗>0, which is compressed at an algebraic rate and vanishes at the time T_∗, meanwhile the weak solution becomes either a strong solution or a piecewise strong one and tends to the equilibrium state exponentially.     

Stability of Riemann solutions with large oscillation for the relativistic Euler equations

We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L^∞∩BV_loc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL¹∩L^∞∩BV_loc perturbation of the Riemann initial data, as long as the corresponding solutions are in L^∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L^∞ with arbitrarily large oscillation.     

A generalized Riemann problem for quasi-one-dimensional gas flows

A generalization of the Riemann problem for gas dynamical flows influenced by curved geometry, such as flows in a variable-area duct, is solved. For this generalized Riemann problem the initial data consist of a pair of steady-state solutions separated by a jump discontinuity. The solution of the generalized Riemann problem is used as a basis for a random choice method in which steady-state solutions are used as an Ansatz to approximate the spatial variation of the solution between grid points. For nearly steady flow in a Laval nozzle, where this Ansatz is appropriate, this generalized random choice method gives greatly improved results.     

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.     

Oil displacement by water and gas in a porous medium: the Riemann problem

In this work we present the construction of the Riemann solution for a system of two conservation laws representing displacement in immiscible three-phase flow. The porousmedium is initially filled with oil and small amounts of water and gas; then a fixed proportion of water and gas is injected. We use the wave curve method to determine the wave sequences in the Riemann solution for arbitrary initial and injection data in the above mentioned class. We show the L_Loc¹-stability of the Riemann solution with variation of data. We do not verify uniqueness of the Riemann solution, but we believe that it is valid.     

Global solutions with shock waves to the generalized Riemann problem for a system of hyperbolic conservation laws with linear damping

It is proven that a class of the generalized Riemann problem for quasilinear hyperbolic systems of conservation laws with the uniform damping term admits a unique global piecewise C¹ solution u=u(t,x) containing only n shock waves with small amplitude on t?0 and this solution possesses a global structure similar to that of the similarity solution of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data. We also give an example to show that the uniform damping mechanism is not strong enough to prevent the formation of shock waves.     

The Riemann problem for general systems of conservation laws

An extended entropy condition (E) has previously been proposed, by which we have been able to prove uniqueness and existence theorems for the Riemann problem for general 2-conservation laws. In this paper we consider the Riemann problem for general n-conservation laws. We first show how the shock are related to the characteristic speeds. A uniqueness theorem is proved subject to condition (E), which is equivalent to Lax's shock inequalities when the system is “genuinely nonlinear.” These general observations are then applied to the equations of gas dynamics without the convexity condition P_vv(v, s) > 0. Using condition (E), we prove the uniqueness theorem for the Riemann problem of the gas dynamics equations. This answers a question of Bethe. Next, we establish the relation between the shock speed σ and the entropy S along any shock curve. That the entropy S increases across any shock, first proved by Weyl for the convex case, is established for the nonconvex case by a different method. Wendroff also considered the gas dynamics equations without convexity conditions and constructed a solution to the Riemann problem. Notice that his solution does satisfy our condition (E).     

The analytic continuation of solutions to elliptic boundary value problems in two independent variables

An explicit representation is derived for the continuation across an analytic boundary of the solution to a boundary value problem for an analytic elliptic equation of second order in two independent variables. The representation is in terms of Cauchy data on the boundary and the complex Riemann function. This is equivalent to a representation for the solution to Cauchy's problem given by Henrici in 1957. It is confirmed that the method of complex characteristics is satisfactory for locating real singularities in the solution provided that the Riemann function is entire in its four arguments. Applications to Laplace's and Helmholtz's equations are discussed. By inserting known, simple solutions to the latter equation into the representation formula, several nontrivial integral relations involving the Bessel function J₀, and a possibly new series expansion for J_μ(x), are found.     

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.     

Wave Interactions and Stability of Riemann Solutions of the Aw–Rascle Model for Generalized Chaplygin Gas

In this paper, we mainly study the interactions of the elementary waves of the Aw–Rascle model for the generalized Chaplygin gas. With the characteristic analysis method, the solutions are obtained constructively when the initial data are three piecewise constant states. Moreover, by studying the limits of the perturbed Riemann solutions as the perturbed parameter \(\varepsilon\) tends to zero, we find that the Riemann solutions are stable for such perturbations on the initial data.