Existence and Uniqueness of the Global L1 Solution of the Euler Equations for Chaplygin Gas

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 Lloc1. 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.

Automated summary

This paper establishes 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. The method used is Lagrangian representation with characteristic analysis, and the assumptions on the initial data are believed to 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.