Communications in Mathematical Sciences

Volume 18 (2020)

Number 8

Nonlinear stability of the boundary layer and rarefaction wave for the inflow problem governed by the heat-conductive ideal gas without viscosity

Pages: 2235 – 2261



Meichen Hou (School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China)

Lili Fan (School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan, China)


This paper is devoted to studying the inflow problem for an ideal polytropic model with non-viscous gas in one-dimensional half space. We show the existence of the boundary layer in different areas. By employing the energy method, we also prove the unique global-in-time existence of the solution and the asymptotic stability of both the boundary layers, the $3$‑rarefaction wave and their superposition wave under some smallness conditions. Series of simple but tricky operations on boundary need to be carefully done by taking good advantage of construction on the system and domain properties.


non-viscous, inflow problem, boundary layer, rarefaction wave

2010 Mathematics Subject Classification

35B35, 35B40, 35M33, 35Q35, 76N10, 76N15

This work was supported by the Fundamental Research grants from the Science Foundation of Hubei Province under the contract 2018CFB693. The research of L.L. Fan was supported by the Natural Science Foundation of China #11871388 and and in part by the Natural Science Foundation of China #11701439.

Received 8 March 2020

Accepted 17 June 2020

Published 22 December 2020