Global solutions to rotating motion of isentropic flows with cylindrical symmetry

Abstract

We are concerned with global weak solutions to the isentropic compressible Euler equations with cylindrically symmetric rotating structure, in which the origin is included. Due to the presence of the singularity at the origin, only the case excluding the origin $|\vec{x}|\geq1$ has been considered by Chen-Glimm \cite{Chen3}. The convergence and consistency of the approximate solutions are proved by using $L^{\infty}$ compensated compactness framework and vanishing viscosity method. We observe that if the blast wave initially moves outwards and if initial density and velocity decay to zero at certain algebraic rate near the origin, then the density and velocity decay at the same rate for any positive time. In particular, the initial normal velocity is assumed to be non-negative, and there is no restriction on the sign of initial angular velocity.