On the Ultra Relativistic Euler Equations

We consider the relativistic Euler equations in isentropic fluids with the equation of state , which is the ultra-relativistic limit. We analyze the single shocks. We study the shock interaction, and give explicit example for the non-backward uniqueness. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary

We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has played an important role in prior work on the non-relativistic compressible Euler equations. Our main result is the derivation, relative to Lagrangian (also known as co-moving) coordinates, of local-in-time a priori estimates for the solution. The solution features a fluid-vacuum boundary, transported by the fluid four-velocity, along which the hyperbolicity of the equations degenerates. In this context, the relativistic Euler equations are equivalent to a degenerate quasilinear hyperbolic wave-map-like system that cannot be treated using standard energy methods.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.     

Interaction of three and four rarefaction waves of the pressure-gradient system

We study the two-dimensional pressure-gradient system, a subsystem of the two-dimensional compressible Euler system. We consider the problem of interaction of four rarefaction waves which is one case of two-dimensional Riemann problems. It is known that, when two planar waves interact, there exists a smooth solution in the interaction region. In this paper, we establish the existence of a smooth solution in the hyperbolic domain of determinacy, in which we encounter the interaction of simple and planar waves and shock prevention in simple waves.     

Global entropy solutions to the relativistic Euler equations for a class of large initial data

We are concerned with global entropy solutions to the relativistic Euler equations for a class of large initial data which involve the interaction of shock waves and rarefaction waves. We first carefully analyze the global behavior of the shock curves, the rarefaction wave curves, and their corresponding inverse curves in the phase plane. Based on these analyses, we use the Glimm scheme to construct global entropy solutions to the relativistic Euler equations for the class of large discontinuous initial data.Received: May 23, 2004     

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

STABILIZATION EFFECT OF FRICTIONS FOR TRANSONIC SHOCKS IN STEADY COMPRESSIBLE EULER FLOWS PASSING THREE-DIMENSIONAL DUCTS

Transonic shocks play a pivotal role in designation of supersonic inlets and ramjets. For the three-dimensional steady non-isentropic compressible Euler system with frictions, we constructe a family of transonic shock solutions in rectilinear ducts with square cross-sections. In this article, we are devoted to proving rigorously that a large class of these transonic shock solutions are stable, under multidimensional small perturbations of the upcoming supersonic flows and back pressures at the exits of ducts in suitable function spaces.This manifests that frictions have a stabilization effect on transonic shocks in ducts, in consideration of previous works which shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, because frictions lead to a stronger coupling between the elliptic and hyperbolic parts of the three-dimensional steady subsonic Euler system, we develop the framework established in previous works to study more complex and interesting Venttsel problems of nonlocal elliptic equations.     

Geometrical optics and post shock behavior for nonlinear conservation laws

In this paper we present an approach to the study of nonlinear waves and shocks associated with signaling problems for hyperbolic systems of conservation laws. Our approach employs a nonlinear phase variable, treats problems for which the flux function (and hence the matrix coefficients in the quasilinear system of partial differential equations) has explicit spatial dependence, and provides the post shock representation for the solution.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.     

The generation,propagation, and extinction of multiphases in the KdV zero‐dispersion limit

We study the multiphases in the KdV zero‐dispersion limit. These phases are governed by the Whitham equations, which are 2g + 1 quasi‐linear hyperbolic equations where g is the number of phases. We are interested in both the interaction of two single phases and the breaking of a single phase for general initial data. We analyze in detail how a double phase is generated from the interaction or breaking, how it propagates in space‐time, and how it collapses to a single phase in a finite time. The Whitham equations are known to be integrable via a hodograph transform. The crucial step in our approach is to formulate the hodograph transform in terms of the Euler‐Poisson‐Darboux solutions. Under our scheme, the zeros of the Jacobian of the transform are given by the zeros of the Euler‐Poisson‐Darboux solution. Hence, the problem of inverting the hodograph transform to give the Whitham solution reduces to that of counting the zeros of the Euler‐Poisson‐Darboux solution. © 2002 Wiley Periodicals, Inc.     

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.     

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 a Mach configuration

We prove the stability of a Mach configuration, which occurs in shock reflection off an obstacle or shock interaction in compressible flow. The compressible flow is described by a full, steady Euler system of gas dynamics. The unperturbed Mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity. Among four regions divided by these four lines in the neighborhood of the intersection, two are supersonic regions, and other two are subsonic regions. We prove that if the constant states in the supersonic regions are slightly perturbed, then the structure of the whole configuration holds, while the other two shock fronts and the slip line, as well as the flow field in the subsonic regions, are also slightly perturbed. Such a conclusion asserts the existence and stability of the general Mach configuration in shock theory. In order to prove the result, we reduce the problem to a free boundary value problem, where two unknown shock fronts are free boundaries, while the slip line is transformed to a fixed line by a Lagrange transformation. In the region where the solution is to be determined, we have to deal with an elliptic‐hyperbolic composed system. By decoupling this system and combining the technique for both hyperbolic equations and elliptic equations, we establish the required estimates, which are crucial in the proof of the existence of a solution to the free boundary value problem. © 2005 Wiley Periodicals, Inc.     

Transonic shocks for steady Euler flows with cylindrical symmetry

In this paper, we prove the existence of transonic shocks adjacent to a uniform one for the full Euler system for steady compressible fluids with cylindrical symmetry in a cylinder, and consequently show the stability of such uniform transonic shocks. Mathematically we solve a free boundary problem for a quasi-linear elliptic–hyperbolic composite system. This reveals that the boundary conditions and equations interact in a subtle way. The key point is to “separate” in a suitable way the elliptic and hyperbolic parts of the system. The approach developed here can be applied to deal with certain multidimensional problems concerning stability of transonic shocks for the full Euler system.     

Introduction to nonclassical shock solutions of the modified Burgers‐Korteweg‐de Vries equation

A general property of nonlinear hyperbolic equations is the eventual formation of discontinuities in the propagating signal. These discontinuities are not uniquely defined by the initial data for the problem and a central issue is the identification of acceptable weak solutions. Particular difficulties arise when the hyperbolic system ceases to be genuinely nonlinear in some of its characteristic fields. This equates in the case of a scalar law to the lack of convexity in the flux function. Here a representative example is provided by the modified Korteweg‐de Vries‐Burgers equation which exhibits a quadratic as well as a cubic nonlinear term and arises in a variety of engineering applications including weakly nonlinear waves in fluidized beds and two‐layer fluid flows. Its solutions have the distinguishing feature to generate undercompressive or nonclassical shocks in the hyperbolic limit with dispersion and dissipation balanced. The resulting rich variety of wave phenomena: shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations, which have no counterpart in classical shock theories is discussed.     

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.     

Extension of Euler’s method to parabolic equations

Euler generalized d’Alembert’s solution to a wide class of linear hyperbolic equations with two independent variables. He introduced in 1769 the quantities that were rediscovered by Laplace in 1773 and became known as the Laplace invariants. The present paper is devoted to an extension of Euler’s method to linear parabolic equations with two independent variables. The new method allows one to derive an explicit formula for the general solution of a wide class of parabolic equations. In particular, the general solution of the Black–Scholes equation is obtained.