Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (3): 941-958.doi: 10.1007/s10473-021-0319-6

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Tingting CHEN1, Aifang QU2, Zhen WANG3   

  1. 1. School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China;
    2. Department of Mathematics, Shanghai Normal University, Shanghai 200234, China;
    3. School of Science, Wuhan University of Technology, Wuhan 430070, China
  • Received:2020-03-10 Revised:2020-10-27 Online:2021-06-25 Published:2021-06-07
  • Contact: Aifang QU
  • About author:Tingting CHEN,,;Zhen WANG,,
  • Supported by:
    The research of Tingting Chen is supported by the Central Universities, China University of Geosciences (Wuhan) (CUGL180827). The research of Aifang Qu is supported by the National Natural Science Foundation of China (11871218, 12071298). The research of Zhen Wang is supported by the National Natural Science Foundation of China (11771442).

Abstract: In this paper, we establish the global existence and uniqueness of the solution of the Cauchy problem of a one-dimensional compressible isentropic Euler system for a Chaplygin gas with large initial data in the space $L_{\rm loc}^1$. The hypotheses on the initial data may be the least requirement to ensure the existence of a weak solution in the Lebesgue measurable sense. The novelty and also the essence of the difficulty of the problem lie in the fact that we have neither the requirement on the local boundedness of the density nor that which is bounded away from vacuum. We develop the previous results on this degenerate system. The method used is Lagrangian representation, the essence of which is characteristic analysis. The key point is to prove the existence of the Lagrangian representation and the absolute continuity of the potentials constructed with respect to the space and the time variables. We achieve this by finding a property of the fundamental theorem of calculus for Lebesgue integration, which is a sufficient and necessary condition for judging whether a monotone continuous function is absolutely continuous. The assumptions on the initial data in this paper are believed to also be necessary for ruling out the formation of Dirac singularity of density. The ideas and techniques developed here may be useful for other nonlinear problems involving similar difficulties.

Key words: Compressible Euler equations, linearly degenerate fields, initial data in $L_{\rm loc}^1$ space without uniform bounds, global well-posedness, regularity

CLC Number: 

  • 35L65