Two-dimensional cavity flow in an infinitely long channel with non-zero vorticity

The main object of this paper is to investigate the well-posedness theory of the incompressible inviscid cavity flow in an infinitely long channel. The flow is governed by two-dimensional incompressible, steady Euler system. The main results read that given a mass flux and a constant vorticity in the inlet of the channel, firstly, we establish the existence and the uniqueness of the incompressible cavity flow in an infinitely long symmetric channel, which contains a smooth free surface detaching at the boundary point of the obstacle. Secondly, some fundamental properties, such as the asymptotic behaviors of the cavity flow and the free boundary in the upstream and downstream, and the positivity of the horizontal velocity, are also obtained. Finally, we show that there does not exist a finite cavity or a cusped cavity in the infinitely long nozzle, which gives a positive answer to the conjecture by H. Villat in 1913 on the non-existence of a symmetric finite cusped cavity behind an obstacle.