We study the 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 the 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.

In this paper, we establish the global existence of supersonic entropy solutions with a strong contact discontinuity over a Lipschitz wall governed by the two-dimensional steady exothermically reacting Euler equations, when the total variation of both the initial data and slope of the Lipschitz wall is sufficiently small. Local and global estimates are developed and a modified Glimm-type functional is carefully designed. Next the validation of the quasi-one-dimensional approximation in the domain bounded by the wall and the strong contact discontinuity is rigorous justified by proving that the difference between the average of weak solution and the solution of quasi-one-dimensional system can be bounded by the square of the total variation of both the initial data and slope of the Lipschitz wall.