High Mach number limit of one-dimensional piston problem for non-isentropic compressible Euler equations: Polytropic gas

Abstract

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.

Hypersonic limit of two‐dimensional steady compressible Euler flows passing a straight wedge

“…However, in this work, we found that hypersonic limit is not the vanishing pressure limit, and the singularity is not a δ-shock. This is also true for the higher Mach number limit for the one-dimensional piston problem [14].…”

“…This is also true for the high-Mach-number-limit for the one-dimensional piston problem. [14] We remark by passing that readers shall not be confused by the concept of measure solutions called in this paper (or measurevalued solutions called in [4]) with that of measure-valued solutions proposed by DiPerna. [7] The latter has been intensively studied recently (see, for example, [2,3] and references therein).…”

Measure solutions of one-dimensional piston problem for compressible Euler equations of Chaplygin gas

“…The hypersonic-limit problem of polytropic gas passing a two-dimensional wedge was studied in [7], where the authors also proposed a general concept of measure solutions to the two-dimensional steady non-isentropic compressible Euler equations. In all these works, the basic idea is to relax the nonlinearity in the Euler equations by considering all mass, momentum and pressure to be Radon measures on the physical Euclidean spaces, and then requiring that momentum etc.…”

On Two-Dimensional Steady Hypersonic-Limit Euler Flows Passing Ramps and Radon Measure Solutions of Compressible Euler Equations

“…See [1,2,16,22] for the background and physical theory of hypersonic flows, especially [16,Chapter 3] for a detailed introduction to the Newton's theory of hypersonic flow. In [25,23], the authors also studied the related problems of measure solutions of high Mach number limits of piston problems for polytropic gases and Chaplygin gas. These papers demonstrate that the concept of measure solutions we proposed works well for these fundamental physical problems.…”

“…It is well-known that to study weak solutions, one shall choose the correct representations of the Euler equations and the state function. Previous works [24,25,23] It is a classic problem in gas dynamics, and has been studied extensively (see, for example, [9,17,18] and references therein). Actually it's Hu and Zhang's work [17,18] that motivates us to study the hypersonic-limit flow.…”