Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (3): 887-902.doi: 10.1007/s10473-022-0305-7

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Ning-An LAI1,2, Wei XIANG3, Yi ZHOU4   

  1. 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 Online:2022-06-26 Published:2022-06-24
  • Contact: Ning-An LAI,
  • 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).

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

CLC Number: 

  • 35L60