On a regularization of the compressible Euler equations for an isothermal gas

We consider a Leray-type regularization of the compressible Euler equations for an isothermal gas. The regularized system depends on a small parameter α>0. Using Riemann invariants, we prove the existence of smooth solutions for the regularized system for every α>0. The regularization mechanism is a non-linear bending of characteristics that prevents their finite-time crossing. We prove that, in the α→0 limit, the regularized solutions converge strongly. However, based on our analysis and numerical simulations, the limit is not the unique entropy solution of the Euler equations. The numerical method used to support this claim is derived from the Riemann invariants for the regularized system. This method is guaranteed to preserve the monotonicity of characteristics.     

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.     

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 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.     

Limit relations for three simple hyperbolic systems of conservation laws

This paper is concerned with the limit relations from the Euler equations of one‐dimensional compressible fluid flow and the magnetohydrodynamics equations to the simplified transport equations, where the δ‐shock waves occur in their Riemann solutions of the latter two equations. The objective is to prove that the Riemann solutions of the perturbed equations coming from the one‐dimensional simplified Euler equations and the magnetohydrodynamics equations converge to the corresponding Riemann solutions of the simplified transport equations as the perturbation parameterx ε tends to zero. Furthermore, the result can also be generalized to more general situations. Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Relaxation approximation of the Euler equations

The aim of this paper is to show how solutions to the one-dimensional compressible Euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term. The enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate Riemann solver. From a theoretical point of view, the convergence of solutions to the enlarged system towards solutions to the Euler equations is proved for local in time smooth solutions. We also show that arbitrarily large shock waves for the Euler equations admit smooth shock profiles for the enlarged relaxation system. In the end, we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system. We test it on various cases.     

Riemann problem for the relativistic Chaplygin Euler equations

The relativistic Euler equations for a Chaplygin gas are studied. The Riemann problem is solved constructively. There are five kinds of Riemann solutions, in which four only contain different contact discontinuities and the other involves delta shock waves. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta-shock solutions are established.     

Blowup phenomena of the compressible Euler equations

The blowup phenomena of solutions of the compressible Euler equations is investigated. The approach is to construct the special solutions and use phase plane analysis. In particular, the special explicit solutions with velocity of the form c(t)x are constructed to show the blowup and expanding properties.     

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.     

From gas dynamics to pressureless gas dynamics

This paper is devoted to the convergence of solutions of the compressible Euler equations towards solutions of the pressureless gas dynamics system, when the pressure tends to 0. The goal is to prove accurate uniform bounds for particular solutions of the Euler equations.

     

Note for “Some Exact Blowup Solutions to the pressureless Euler equations in R” [Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 2993-2998]

In this supplementary note, we can generalize the exact solutions for the pressureless Euler equations in [Yuen MW. Some exact blowup solutions to the pressureless Euler equations in R^N, Commun. Nonlinear Sci. Numer. Simulat. 2011;16:2993-8]. Here, the solutions are constructed in implicit or explicit forms.     

Numerical study of the interaction between shocks and rarefaction waves in an ideal gas

The interaction between shock waves and rarefaction waves is numerically studied using the one-dimensional Euler equations for an ideal gas. A specific form of solutions, which are called contact regions, is detected. They represent extended zones with continuously varying density and temperature at constant pressure and velocity. It is shown that, at long times, the solutions to the interaction problem tend to those to the Riemann problems with the contact discontinuity replaced by a contact region.     

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.     

Bidimensional shallow water model with polynomial dependence on depth through vorticity

In this paper, we obtain a bidimensional shallow water model with polynomial dependence on depth. With this aim, we introduce a small non-dimensional parameter ε and we study three-dimensional Euler equations in a domain depending on ε (in such a way that, when ε becomes small, the domain has small depth). Then, we use asymptotic analysis to study what happens when ε approaches to zero. Asymptotic analysis allows us to obtain a new bidimensional shallow water model that not only computes the average velocity (as the classical model does) but also provides the horizontal velocity at different depths. This represents a significant improvement over the classical model. We must also remark that we obtain the model without making assumptions about velocity or pressure behavior (only the usual ansatz in asymptotic analysis). Finally, we present some numerical results showing that the new model is able to approximate the non-constant in depth solutions to Euler equations, whereas the classical model can only obtain the average velocity.     

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 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.     

Blowup phenomena of solutions to the Euler equations for compressible fluid flow

The blowup phenomena of solutions is investigated for the Euler equations of compressible fluid flow. The approach is to construct special explicit solutions with spherical symmetry to study certain blowup behavior of multi-dimensional solutions. In particular, the special solutions with velocity of the form c(t)x are constructed to show the expanding and blowup properties. The solution with velocity of the form for γ?1 and for any space dimensions is obtained as a corollary. Another conclusion is that there is only trivial solution with velocity of the form c(t)|x|^α-1x for α≠1 and multi-space dimensions.     

Convergence of compressible Euler-Poisson system to incompressible Euler equations

In this paper, we study the quasi-neutral limit of compressible Euler-Poisson equations in plasma physics in the torus T^d. For well prepared initial data the convergence of solutions of compressible Euler-Poisson equations to the solutions of incompressible Euler equations is justified rigorously by an elaborate energy methods based on studies on an λ-weighted Lyapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to λ is to use the curl-div decomposition of the gradient.     

Blowup of solutions for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations

The blowup phenomenon of solutions is investigated for the initial boundary value problem of the 3‐dimensional compressible damped Euler equations. It is shown that if the initial functional associated with a general test function is large enough, the solutions of the damped Euler equations will blow up on finite time. Hence, a class of blowup conditions is established. Moreover, the blowup time could be estimated.