Communications in Mathematical Sciences

Volume 20 (2022)

Number 5

On two-dimensional steady hypersonic-limit Euler flows passing ramps and radon measure solutions of compressible Euler equations

Pages: 1331 – 1361



Yunjuan Jin (Center for Partial Differential Equations, School of Mathematical Sciences, East China Normal University, Shanghai, China)

Aifang Qu (Department of Mathematics, Shanghai Normal University, Shanghai, China)

Hairong Yuan (School of Mathematical Sciences and Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai, China)


We proposed rigorous definitions of Radon measure solutions for boundary value problems of steady compressible Euler equations which model hypersonic-limit inviscid flows passing two-dimensional ramps, and their interactions with still gas and pressureless jets. We proved the Newton–Busemann pressure law of drags on a body in hypersonic flow, and constructed various physically interesting measure solutions with density containing Dirac measures supported on curves, also exhibited examples of blow up of certain measure solutions. This established a mathematical foundation for applications in engineering and further studies of measure solutions of compressible Euler equations.


compressible Euler equations, hypersonic, Newton–Busemann pressure law, shock layer, free layer, Dirac measure, measure solution, vacuum, singular Riemann problem

2010 Mathematics Subject Classification

35L65, 35L67, 35Q31, 35R06, 35R35, 76K05

The research of Aifang Qu is supported by National Natural Science Foundation of China (NNSFC) under Grants No.11571357, No.11871218, and No.12071298.

Hairong Yuan is supported by NNSFC under Grants No.11871218, No.12071298, and by Science and Technology Commission of Shanghai Municipality (STCSM) under Grant No. 18dz2271000.

Received 20 August 2021

Accepted 30 November 2021

Published 26 May 2022