Rarefaction wave interaction of pressure-gradient system

by Bang, Seunghoon.

iii The pressure-gradient system is a relatively new system of conservation laws. This system is a reduction from the two-dimensional(2-D) compressible Euler equations. Separating the pressure from the inertia in the flux of the Euler equations, we obtain the transport(or convective) system without pressure terms and the pressure-gradient system with pressure terms. Since the Cauchy problem for this system is still very difficult, We consider the Riemann problem, but it is also a complex open problem. According to the initial data, we have 12 main configurations that explain the interaction between various waves such as rarefaction waves, shocks and contact discontinuities. Here, we are interested in demonstrating the four rarefaction waves analytically. This problem is a mixed type nonlinear equation(elliptic, parabolic and hyperbolic). Yuxi Zheng proved that there exists a weak solution in the elliptic region, and we show that there exists a continuous and piecewise smooth solution in the hyperbolic region up to the domain of determinacy.