Special relativistic effects revealed in the Riemann problem for three-dimensional relativistic Euler equations

We consider the Riemann problem of three-dimensional relativistic Euler equations with two discontinuous initial states separated by a planar hypersurface. Based on the detailed analysis on the Riemann solutions, special relativistic effects are revealed, which are the variations of limiting relative normal velocities and intermediate states and thus the smooth transition of wave patterns when the tangential velocities in the initial states are suitably varied. While in the corresponding non-relativistic fluid, these special relativistic effects will not occur.     

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.     

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.     

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.     

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.     

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.     

Photon radiation of an extended relativistic object

The operator of the electromagnetic interaction of an extended relativistic object is constructed. It is shown that the main difficulty, which arises due to noncommutativity of the coordinates and velocities of the charges in relativistic quantum mechanics, may be overcome by a special ordering of the operators. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 2, pp. 283–294, August, 1997.     

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     

On global symmetric solutions to the relativistic Vlasov–Poisson equation in three space dimensions

A collisionless plasma is modelled by the Vlasov–Maxwell system. In the presence of very large velocities, relativistic corrections are meaningful. When magnetic effects are ignored this formally becomes the relativistic Vlasov–Poisson equation. The initial datum for the phase space density ƒ₀(x, v) is assumed to be sufficiently smooth, non‐negative and cylindrically symmetric. If the (two‐dimensional) angular momentum is bounded away from zero on the support of ƒ₀(x, v), it is shown that a smooth solution to the Cauchy problem exists for all times. Copyright © 2001 John Wiley & Sons, Ltd.     

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.     

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.     

The ultra‐relativistic Euler equations

We study the ultra‐relativistic Euler equations for an ideal gas, which is a system of nonlinear hyperbolic conservation laws. We first analyze the single shocks and rarefaction waves and solve the Riemann problem in a constructive way. Especially, we develop an own parametrization for single shocks, which will be used to derive a new explicit shock interaction formula. This shock interaction formula plays an important role in the study of the ultra‐relativistic Euler equations. One application will be presented in this paper, namely, the construction of explicit solutions including shock fronts, which gives an interesting example for the non‐backward uniqueness of our hyperbolic system. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

On spherically symmetric solutions of the relativistic Euler equation

This work investigates the spherical symmetric solutions of more realistic equation of states. We generalize the method of Hsu et al. (Methods Appl. Anal. 8 (2001) 159) to show the existence of spherical symmetric weak solution of the relativistic Euler equation with initial data containing the vacuum state.     

A common characterization of classical and relativistic addition

A common system of axioms is presented which, applied to the relativistic or to the classical situation, characterizes the addition formula for velocities in both cases. Received 19 September 2001.     

相变演化Suliciu模型中波的相互作用

描述相变演化的Suliciu模型，其基本波可由行波分析得到．对于任何给定分两段常值的初始状态，相应的Riemann解是某些基本波的组合．对分三段常值的初始状态，解由上述Piemann解构成，其中相邻两状态间以基本波连接．当基本波发生碰撞时，新的Riemann问题形成．通过研究这些Riemann。问题，可以在适当的参数空间中对基本波之间复杂的相互作用加以分类．     

Steger–Warming flux vector splitting method for special relativistic hydrodynamics

This paper discusses the properties of the rotational invariance and hyperbolicity in time of the governing equations of the ideal special relativistic hydrodynamics and proves for the first time that the ideal relativistic hydrodynamical equations satisfy the homogeneity property, which is the footstone of the Steger–Warming flux vector splitting method [J. L. Steger and R. F. Warming, J. Comput. Phys., 40(1981), 263–293]. On the basis of this remarkable property, the Steger–Warming flux vector splitting (SW‐FVS) is given. Two high‐resolution SW‐FVS schemes are also given on the basis of the initial reconstructions of the solutions and the fluxes, respectively. Several numerical experiments are conducted to validate the performance of the SW‐FVS method. Copyright © 2013 John Wiley & Sons, Ltd.     

Initial—boundary-value problem for the relativistic string with massive ends

It is shown that the initial-boundary-value problem for a relativistic string with masses at the ends can be solved for the most general form of specification of the initial position and initial velocities of the points of the string. An investigation is made of the connection between the freedom in the parametrization of the initial curve and the reparametrization invariance preserving linearity of the equations of motion of the string. The posed problem is solved by extending the solution determined by the initial conditions from the restricted initial region to the entire world surface of the string by means of boundary conditions of various types: the mass at a given end is equal to zero, is infinitely large, or is finite. In the last case it is shown that the problem of the extension reduces to the solution of a normal system of ordinary differential equations. Specific examples are considered.Joint Institute for Nuclear Research, Dubna; State University, Tver. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 253–271, November, 1994.     

Global weak solutions to the relativistic BGK equation

Abstract

In this paper, the global existence of weak solutions to the relativistic BGK model for the relativistic Boltzmann equation is analyzed. The proof relies on the strong compactness of the density, velocity, and temperature under minimal assumptions on the control of some moments of the initial condition together with the initial entropy.