Simple waves for two-dimensional compressible pseudo-steady Euler system

A simple wave is defined as a flow in a region whose image is a curve in the phase space. It is well known that ＂the theory of simple waves is fundamental in building up the solutions of flow problems out of elementary flow patterns＂ see Courant and Friedrichs＇s chassical book ＂Supersonic Flow and Shock Waves＂. This paper mainly concerned with the geometric construction of simple waves for the 2D pseudo-steady compressible Euler system. Based on the geometric interpretation, the expansion or compression simple wave flow construction around a pseudo-stream line with a bend part are constructed. It is a building block that appears in the global solution to four contact discontinuities Riemann problems.

SIMPLE WAVES OF THE TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS

In this paper, we establish the existence of four families of simple wave solu- tion for two dimensional compressible full Euler system in the self-similar plane. For the 2 × 2 quasilinear non-reducible hyperbolic system, there not necessarily exists any simple wave solution. We prove the result that there are simple wave solutions for this 4× 4 non- reducible hyperbolic system, its simple wave flow is covered by four straight characteristics λ0 =λ1,λA2, λ3 and the solutions keep constants along these lines. We also investigate the existence of simple wave solution for the isentropic relativistic hydrodynamic system in the self-similar plane.

ANALYTIC SOLUTIONS OF SIMPLE WAVES

B. Riemann furnished the general solution of simple waves in 1860[1] But it is difficult to find out the exact forms of the arbitrary function contained in the general solution which must satisfy boundary or initial conditions. For this reason it is inconvenient to probe into the characteristics of concrete problems. In this paper the analytic solutions of simple waves are afforded according to the geometric theory of quasi-linear partial differential equation, and they are determined with boundary or initial conditions. By using these solutions the specific properties of certain flows are discussed and novel results are obtained.

NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL

In this article,we study the exhaustive analysis of nonlinear wave interactions for a 2×2 homogeneous system of quasilinear hyperbolic partial differential equations(PDEs)governing the macroscopic production.We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations.Furthermore,we study the interaction between simple waves in detail through exact solution of general initial value problem.Finally,we discuss the all possible interaction of elementary waves using the solution of Riemann problem.

Centered Waves for the Two-dimensional Pseudo-Steady van der Waals Gas Satisfied Maxwell's Law Around a Sharp Corner

In this paper,the authors study the centered waves for the two-dimensional(2D for short)pseudo-steady supersonic flow with van der Waals gas satisfied Maxwell's law around a sharp corner.In view of the initial value of the specific volume and the properties of van der Waals gas,the centered waves at the sharp corner are constructed by classification.It is shown that the supersonic incoming flow turns the sharp corner by a centered simple wave or a centered simple wave with right-contact discontinuity or a composite wave(jump-fan,fan-jump or fan-jump-fan),or a combination of waves and constant state.Moreover,the critical angle of the sharp corner corresponding to the appearance of the vacuum phenomenon is obtained.

The Configuration of Shock Wave Reflection for the TSD Equation

In this paper, we mainly study the nonlinear wave configuration caused by shock wave reflection for the TSD （Transonic Small Disturbance） equation and specify the existence and nonexistence of various nonlinear wave configurations. We give a condition under which the solution of the RR （Regular reflection） for the TSD equation exists. We also prove that there exists no wave configuration of shock wave reflection for the TSD equation which consists of three or four shock waves. In phase space, we prove that the TSD equation has an IR （Irregular reflection） configuration containing a centered simple wave. Furthermore, we also prove the stability of RR configuration and the wave configuration containing a centered simple wave by solving a free boundary value problem of the TSD equation.

Global Exact Boundary Controllability for General First-Order Quasilinear Hyperbolic Systems

For general first-order quasilinear hyperbolic systems,based on the analysis of simple wave solutions along characteristic trajectories,the global two-sided exact boundary controllability is achieved in a relatively short controlling time.