Formation Of Singularities For The Non-isentropic Compressible Euler Equations With Generalized Chaplygin Gas

In the fields of mathematics,physics,fluid mechanics and other natural sciences,many important first-order quasilinear hyperbolic systems with two independent variables have arised.An example is the Euler equations in gas dynamics.The study of the global behavior of classical solutions to the Euler equations and numerical computations,are based on the premise of the global existence of classical solutions.Therefore,the study of the classical solution of compressible Euler equations is of great significance in theory and practice.This paper mainly studies the singularity formation problem of classical solutions of non-isentropic compressible Euler equations under generalized Chaplygin gases.The main contents are: Firstly,by using the characteristic form of the compressible Euler equations,a characteristic decomposition in terms of specific volume is established.Through this decomposition,an appropriate pressure gradient variable is identified and a corresponding coupled Riccati equations is derived.Secondly,Riemann invariants are introduced to establish the corresponding differential relations,under certain conditions,smooth solutions can be estimated with maximum modulus.Finally,by decoupling the solutions of the Riccati equations for pressure gradient variables,a refined lower bound for density is obtained,which is then used to prove the blowup of gradient under certain conditions.