Analysis on the steady Euler flows with stagnation points in an infinitely long nozzle

A recent prominent result asserts that steady incompressible Euler flows strictly away from stagnation in a two-dimensional infinitely long strip must be shear flows. On the other hand, flows with stagnation points, very challenging in analysis, are interesting and important phenomenon in fluids. In this paper, we not only prove the uniqueness and existence of steady flows with stagnation points, but also obtain the regularity of the boundary of stagnation set, which is a class of obstacle type free boundary. First, we prove a global uniqueness theorem for steady Euler system with Poiseuille flows as upstream far field state in an infinitely long strip. Due to the appearance of stagnation points, the nonlinearity of the semilinear equation for the stream function becomes non-Lipschitz. This creates a challenging analysis problem since many classical analysis methods do not apply directly. Second, the existence of steady incompressible Euler flows, tending to Poiseuille flows in the upstream, are established in an infinitely long nozzle via variational approach. A very interesting phenomenon is the regularity of the boundary of non-stagnant region, which can be regarded as an obstacle type free boundary and is proved to be globally $C^1$. Finally, the existence of stagnation region is proved as long as the nozzle is wider than the width of the nozzle at upstream where the flows tend to Poiseuille flows.