The well-posedness of incompressible rotational jet flows with gravity

This paper is concerned with the well-posedness theory and geometric property of steady two-dimensional incompressible jet flow with vorticity and gravity. The main results show that for given incoming mass flux and atmospheric pressure at outlet, there exists a unique incompressible jet flow issuing from a semi-infinitely long nozzle, and the free boundary initiates smoothly at the separation point. Moreover, we obtain the single intersection property of the free boundary by under-over theorem. As a consequence, the monotonicity of the free boundary will be established.