Steady incompressible axially symmetric Réthy flows

In this paper, we are concerned with a jet and a cavity issuing from a semi-infinite long nozzle around a given obstacle in the axially symmetric case, which is called the axially symmetric incompressible Rethy flow. The existence and uniqueness of the axially symmetric Rethy flow are obtained via a variational approach when the nozzle and the obstacle are y-graphs and a mass flux at the inlet of the nozzle is given. This is the first result on the existence of axially symmetric flow with two free boundaries, which can be either jet or cavity. In order to show the existence, we develop a new method to show the vanishing and non-vanishing of the free boundaries by employing carefully the variational structures and the decay rate of the asymptotic behaviours of the solutions at the downstream.