We formulate a problem on hypersonic limit of two-dimensional steady non-isentropic compressible Euler flows passing a straight wedge. It turns out that the Mach number of the upcoming uniform supersonic flow increases to infinity may be taken as that the adiabatic exponent of the polytropic gas decreases to 1. We propose a form of the Euler equations which is valid if the unknowns are Radon measures and construct a measure solution containing Dirac measures supported on the surface of the wedge. It is proved that as → 1, the sequence of solutions of the compressible Euler equations that contains a shock ahead of the wedge converges vaguely as measures to the measure solution constructed. This justifies the Newton theory of hypersonic flow passing obstacles in the case of two-dimensional straight wedges. The result also demonstrates the necessity of considering general measure solutions in the study of boundary-value problems of systems of hyperbolic conservation laws. K E Y W O R D Scompressible Euler equations, Dirac measure, hypersonic, measure solution, shock wave, wedge

We study high Mach number limit of the one dimensional piston problem for the full compressible Euler equations of polytropic gas, for both cases that the piston rushes into or recedes from the uniform still gas, at a constant speed. There are two different situations, and one needs to consider measure solutions of the Euler equations to deal with concentration of mass on the piston, or formation of vacuum. We formulate the piston problem in the framework of Radon measure solutions, and show its consistency by proving that the integral weak solutions of the piston problems converge weakly in the sense of measures to (singular) measure solutions of the limiting problems, as the Mach number of the piston increases to infinity.

This paper studies the large time behavior of solution for a class of nonlinear massless Dirac equations in R 1+1 . It is shown that the solution will tend to travelling wave solution when time tends to infinity.2010 Mathematics Subject Classification. 35Q41, 35Q40, 35L60.

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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 had constructed a family of transonic shock solutions in rectilinear ducts with square cross-sections, and this paper is 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 friction has a stabilization effect on transonic shocks in ducts, in consideration of previous works have shown that transonic shocks in purely steady Euler flows are not stable in such ducts. Except its implications to applications, since 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.