Global entropy solutions to a variant of the compressible Euler equations

DiPerna [R.J. DiPerna, Global solutions to a class of nonlinear hyperbolic systems of equations, Comm. Rat. Pure Appl. Math. 26 (1973) 1–28] use the Glimm’s scheme method to obtain a global weak solution to the Euler equations of one-dimensional, compressible fluid flow with 1<γ<31<γ<3, while in this work, we use the compensated compactness method coupled with some basic ideas of the kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor [P.L. Lions, B. Perthame, P.E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996) 599–638; P.L. Lions, B. Perthame, E. Tadmor, Kinetic formulation of the isentropic gas dynamics and pp-system, Comm. Math. Phys. 163 (1994) 415–431] to obtain the existence of global entropy solutions to the system with a uniform amplitude bound.