Pulsating flows of the 2D Euler–Poisson equations

Abstract

It is well known that solutions of gaseous or fluid dynamical systems can easily blowup or develop shock waves in finite time.In this paper we show the existence of a class of “radially symmetric” rotational solutions to the two-dimensional pressureless Euler–Poisson equations. The flows are global (i.e., exist for all t>0), have compact support at all times, and pulsate periodically. The method of construction is novel. It comprises the piecing together of suitable shells of moving particles in a delicate manner with careful choice of initial data. Each shell is the solution of a member of a continuum of ordinary differential equations. Detailed analysis of the equations is carried out to ensure that neighboring shells can be chosen to pulsate with the same period. We also show that any such solution can be extended to a larger flow by adding annuli of pulsating flows. Our result exhibits another example of rotation preventing the blowup of solutions.