SHOCK DIFFRACTION PROBLEM BY CONVEX CORNERED WEDGES FOR ISOTHERMAL GAS

doi:10.1007/s10473-021-0407-7

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (4): 1130-1140.doi: 10.1007/s10473-021-0407-7

SHOCK DIFFRACTION PROBLEM BY CONVEX CORNERED WEDGES FOR ISOTHERMAL GAS

王钦¹, Kyungwoo SONG²

1. Department of Mathematics, Yunnan University, Kunming 650091, China;
2. Department of Mathematics, Kyung Hee University, Seoul 02447, Korea

收稿日期:2020-02-13 修回日期:2020-05-14 出版日期:2021-08-25 发布日期:2021-09-01
通讯作者: Kyungwoo SONG E-mail:kyusong@khu.ac.kr
作者简介:Qin WANG,E-mail:mathwq@ynu.edu.cn;Kyungwoo SONG,E-mail:kyusong@khu.ac.kr
基金资助:
The research of Qin Wang is supported by National Natural Science Foundation of China (11761077), NSF of Yunnan province (2019FY003007) and Project for Innovation Team (Cultivation) of Yunnan Province, (202005AE160006); the research of Kyungwoo Song is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (NRF-2019R1F1A1057766).

SHOCK DIFFRACTION PROBLEM BY CONVEX CORNERED WEDGES FOR ISOTHERMAL GAS

Qin WANG¹, Kyungwoo SONG²

1. Department of Mathematics, Yunnan University, Kunming 650091, China;
2. Department of Mathematics, Kyung Hee University, Seoul 02447, Korea

Received:2020-02-13 Revised:2020-05-14 Online:2021-08-25 Published:2021-09-01
Contact: Kyungwoo SONG E-mail:kyusong@khu.ac.kr
Supported by:
The research of Qin Wang is supported by National Natural Science Foundation of China (11761077), NSF of Yunnan province (2019FY003007) and Project for Innovation Team (Cultivation) of Yunnan Province, (202005AE160006); the research of Kyungwoo Song is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (NRF-2019R1F1A1057766).

摘要/Abstract

摘要： We are concerned with the shock diffraction configuration for isothermal gas modeled by the conservation laws of nonlinear wave system. We reformulate the shock diffraction problem into a linear degenerate elliptic equation in a fixed bounded domain. The degeneracy is of Keldysh type-the derivative of a solution blows up at the boundary. We establish the global existence of solutions and prove the $C^{0,\frac{1}{2}}$-regularity of solutions near the degenerate boundary. We also compare the difference of solutions between the isothermal gas and the polytropic gas.

关键词: Nonlinear wave system, isothermal gas, shock diffraction, degenerate elliptic equation, Riemann problem, regularity

Abstract: We are concerned with the shock diffraction configuration for isothermal gas modeled by the conservation laws of nonlinear wave system. We reformulate the shock diffraction problem into a linear degenerate elliptic equation in a fixed bounded domain. The degeneracy is of Keldysh type-the derivative of a solution blows up at the boundary. We establish the global existence of solutions and prove the $C^{0,\frac{1}{2}}$-regularity of solutions near the degenerate boundary. We also compare the difference of solutions between the isothermal gas and the polytropic gas.

Key words: Nonlinear wave system, isothermal gas, shock diffraction, degenerate elliptic equation, Riemann problem, regularity

中图分类号:

35M10

王钦, Kyungwoo SONG. SHOCK DIFFRACTION PROBLEM BY CONVEX CORNERED WEDGES FOR ISOTHERMAL GAS[J]. 数学物理学报(英文版), 2021, 41(4): 1130-1140.

Qin WANG, Kyungwoo SONG. SHOCK DIFFRACTION PROBLEM BY CONVEX CORNERED WEDGES FOR ISOTHERMAL GAS[J]. Acta mathematica scientia,Series B, 2021, 41(4): 1130-1140.

参考文献

[1] Bae M, Chen G Q, Feldman M. Regularity of solutions to regular shock reflection for potential flow. Invent Math, 2009, 175:505-543
[2] Bae M, Chen G Q, Feldman M. Prandtl-Meyer reflection configurations, transonic shocks, and free boundary problems. arXiv:1901.05916
[3] Čanić S, Keyfitz B L. An elliptic problem arising from the uns teady transonic small disturbance equation. J Differ Equ, 1996, 125:548-574
[4] Čanić S, Keyfitz B L, Kim E H. Free boundary problems for nonli near wave equations:Mach stems for interacting shocks. SIAM J Math Anal, 2006, 37:1947-1977
[5] Chang T, Hsiao L. The Riemann Problem and Interaction of Waves in Gas Dynamics. NASA STI/Recon Technical Report A, 1989
[6] Chen G Q, Deng X M, Xiang W. Shock diffraction by convex cornered wedges for the nonlinear wave system. Arch Ration Mech Anal, 2014, 211:61-112
[7] Chen G Q, Feldman M. Global solutions of shock reflection by large-angle wedges for potential flow. Ann Math, 2010, 171:1067-1182
[8] Chen G Q, Feldman M. The Mathematics of Shock reflection-diffraction and Von Neumann's Conjectures, volume 359. Princeton University Press, 2018
[9] Chen G Q, Feldman M, Hu J C, Xiang W. Loss of regularity of solutions of the Lighthill problem for shock diffraction for potential flow. SIAM J Math Anal, 2020, 52:1096-1114
[10] Chen G Q, Feldman M, Xiang W. Convexity of Self-Similar Transonic Shocks and Free Boundaries for the Euler Equations for Potential Flow. Arch Ration Mech Anal, 2020, 238:47-124
[11] Chen S X. A mixed equation of Tricomi-Keldysh type. J Hyper Differ Equ, 2012, 9:545-553
[12] Chen S X, Qu A F. Two-dimensional Riemann problems for Chaplygin gas. SIAM J Math Anal, 2012, 44:2146-2178
[13] Chen S X, Qu A F. Piston Problems of Two-Dimensional Chaplygin Gas. Chinese Annals of Mathematics, 2019, 40B:843-868
[14] Cheng H J, Yang H C. Riemann problem for the isentropic relativistic Chaplygin Euler equations. Z Angew Math Phys, 2012, 63:429-440
[15] Dafermos C. Hyperbolic Conservation Laws in Continuum Physics. Springer, 2000, 325
[16] Elling V, Liu T P. Supersonic flow onto a solid wedge. Comm Pure Appl Math, 2008, 61:1347-1448
[17] Glimm J. Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math, 1965, 18:697-715
[18] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Springer, 2015
[19] Hunter J K. Transverse diffraction of nonlinear waves and singular rays. SIAM J Appl Math, 1988, 48:1-37
[20] Keldysh M. On some cases of degenerate elliptic equations on the boundary of a domain. Doklady Acad Nauk USSR, 1951, 77:181-183
[21] Keller J B, Blank A. Diffraction and reflection of pulses by wedges and corners. Comm Pure Appl Math, 1951, 4:75-94
[22] Kim E H. A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation. J Differ Equ, 2010, 248:2906-2930
[23] Kim E H. An interaction of a rarefaction wave and a transonic shock for the self-similar two-dimensional nonlinear wave system. Comm Part Differ Equ, 2012, 37:610-646
[24] Kohn J J, Nirenberg L. Degenerate elliptic-parabolic equations of second order. Comm Pure Appl Math, 1967, 20:797-872
[25] Morawetz C S. Potential theory for regular and mach reflection of a shock at a wedge. Comm Pure Appl Math, 1994, 47:593-624
[26] Otway T H. The Dirichlet Problem for Elliptic-hyperbolic Equations of Keldysh Type. Springer, 2012
[27] Prandtl L. Allgemeine Uberlegungen über die Strömung zusammendrückbarer Flüssigkeiten. Z Angew Math Me, 1936, 16:129-142
[28] Serre D. Multidimensional shock interaction for a Chaplygin gas. Arch Ration Mech Anal, 2009, 191:539-577
[29] Song K, Zheng Y X. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete Contin Dyn Syst, 2009, 24:1365-1380
[30] Wang Q, Zheng Y X. The regularity of semi-hyperbolic patches at sonic lines for the pressure gradient equation in gas dynamics. Indiana Univ Math J, 2014, 63:385-402
[31] Wang Q, Zhang J Q, Yang H C. Two dimensional Riemann-type problem and shock diffraction for the Chaplygin gas. Appl Math Lett, 2019:106046
[32] Yang H C, Zhang M M, Wang Q. Global solutions of shock reflection problem for the pressure gradient system. Comm Pure Appl Anal, 2020, 19:3387-3428
[33] Zheng Y X. Existence of solutions to the transonic pressure-gradient equations of the compressible Euler equations in elliptic regions. Comm Part Differ Equ, 1997, 22:1849-1868
[34] Zheng Y X. A global solution to a two-dimensional Riemann problem involving shocks as free boundaries. Acta Mathematicae Applicatae Sinica, English Series, 2003, 19:559-572
[35] Zheng Y X. Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Mathematicae Applicatae Sinica, English Series, 2006, 22:177-210

SHOCK DIFFRACTION PROBLEM BY CONVEX CORNERED WEDGES FOR ISOTHERMAL GAS

SHOCK DIFFRACTION PROBLEM BY CONVEX CORNERED WEDGES FOR ISOTHERMAL GAS

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