Methods and Applications of Analysis

Volume 28 (2021)

Number 4

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part III

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Study of boundary layers in compressible non-isentropic flows

Pages: 453 – 466

DOI: https://dx.doi.org/10.4310/MAA.2021.v28.n4.a3

Authors

Cheng-Jie Liu (Institute of Natural Sciences, School of Mathematical Sciences, Center of Applied Mathematics, Shanghai Jiao Tong University, Shanghai, China)

Ya-Guang Wang (School of Mathematical Sciences, Center of Applied Mathematics, Shanghai Jiao Tong University, Shanghai, China)

Tong Yang (Department of Mathematics, City University of Hong Kong)

Abstract

In this note, we review our recent study on boundary layer problems in compressible non-isentropic flows with non-slip boundary condition for the velocity, in the small viscosity and heat conductivity limit. By multi-scale analysis, we derive the problems of viscous layer profiles and thermal layer profiles for different scales of viscosity and heat conductivity, from which we obtain the interaction mechanism of viscous layers and thermal layers. Then, when the viscosity goes to zero slower than or at the same rate as the heat conductivity, we give a well-posedness result of the two-dimensional viscous layer problem, which is the Prandtl type equations coupled with a degenerated parabolic equation, in the class of tangential velocity being strictly monotonic in the normal variable. Last, when the viscosity goes to zero faster than the heat conductivity, we study the stability of the thermal layer problem at a shear flow in two or three space variables, which is an inviscid Prandtl type equations coupled with a degenerated parabolic equation.

Keywords

compressible non-isentropic flows, small viscosity and heat conductivity limit, viscous layers and thermal layers, well-posedness

2010 Mathematics Subject Classification

35M13, 35Q35, 76D03, 76D10, 76N20

Full Text (PDF format)

C. J. Liu’s research was supported by National Key R&D Program of China under Grant No. 2020YFA0712000, National Natural Science Foundation of China under Grant No. 11801364, and the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDA25010401.

Y. G. Wang’s research was supported by National Key R&D Program of China under Grant No. 2020YFA0712000, National Natural Science Foundation of China under Grant No. 12171317, Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDA25010402, and Shanghai Municipal Education Commission under Grant No. 2021-01-07-00-02-E00087.

T. Yang’s research was supported by the General Research Fund of Hong Kong City University No. 11302020.

Received 19 October 2020

Accepted 28 December 2020

Published 10 June 2022