Study Of Global Smooth Solutions For Two-Dimensional Generalized Pressure Gradient Equations

The system of nonlinear hyperbolic conservation law is a major branch in the field of partial differential equations.The study of the expansion of gas to vacuum is of great significance in both theory and practice.This paper focuses on the existence of smooth solutions of two-dimensional generalized pressure gradient system for degenerate hyperbolics in gas dynamics.The main contents are shown as follows:The characteristic decomposition of the pressure is obtained in the self-similarity plane,and its simple waves are discussed.The characteristic decomposition of pressure is directly established in polar coordinates by using the second-order equation form of pressure.Then,using the existence of local smooth solutions,a priori estimates inC~2 established by the characteristic decomposition and the continuation method,we prove the existence of smooth solutions for the degenerate Goursat problem with gas to the vacuum boundary.Finally,using pressure decomposition and continuity induction,we show that the vacuum region degenerates into vacuum single-point.