GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW

doi:10.1007/s10473-022-0305-7

数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (3): 887-902.doi: 10.1007/s10473-022-0305-7

GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW

赖宁安^1,2, 向伟³, 周忆⁴

1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China;
2. Department of Mathematics, Lishui University, Lishui, 323000, China;
3. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, 999077, China;
4. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

收稿日期:2020-07-04 发布日期:2022-06-24
通讯作者: Ning-An LAI,E-mail:ninganlai@lsu.edu.cn E-mail:ninganlai@lsu.edu.cn
基金资助:
N. A. Lai and Y. Zhou were supported by NSFC (12171097). W. Xiang was supported in part by the Research Grants Council of the HKSAR, China (Project No.CityU 11303518, Project CityU 11304820 and Project CityU 11300021).

GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW

Ning-An LAI^1,2, Wei XIANG³, Yi ZHOU⁴

1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China;
2. Department of Mathematics, Lishui University, Lishui, 323000, China;
3. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, 999077, China;
4. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

Received:2020-07-04 Published:2022-06-24
Contact: Ning-An LAI,E-mail:ninganlai@lsu.edu.cn E-mail:ninganlai@lsu.edu.cn
Supported by:
N. A. Lai and Y. Zhou were supported by NSFC (12171097). W. Xiang was supported in part by the Research Grants Council of the HKSAR, China (Project No.CityU 11303518, Project CityU 11304820 and Project CityU 11300021).

摘要/Abstract

摘要： In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. A non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity which is a perturbation of plane waves. Therefore, unlike in the one-dimensional case, the multi-dimensional plane shocks are not stable globally. Moreover, a sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.

关键词: Blow-up, global solution, instability, shock, contact disctinuity, Euler equations, isothermal, generalized Riemann problem, nonlinear wave equations

Abstract: In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. A non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity which is a perturbation of plane waves. Therefore, unlike in the one-dimensional case, the multi-dimensional plane shocks are not stable globally. Moreover, a sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.

Key words: Blow-up, global solution, instability, shock, contact disctinuity, Euler equations, isothermal, generalized Riemann problem, nonlinear wave equations

中图分类号:

35L60

赖宁安, 向伟, 周忆. GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW[J]. 数学物理学报(英文版), 2022, 42(3): 887-902.

Ning-An LAI, Wei XIANG, Yi ZHOU. GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW[J]. Acta mathematica scientia,Series B, 2022, 42(3): 887-902.

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GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW

GLOBAL INSTABILITY OF MULTI-DIMENSIONAL PLANE SHOCKS FOR ISOTHERMAL FLOW

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