Two-dimensional stationary fluids with gravity and nonflat bottom

We establish well-posedness theory for the two-dimensional stationary fluids with gravity and a nonflat bottom. The major purpose in this paper is to study free streamline theory in hydrodynamics of the fluids with gravity in rotational flows. More precisely, given any positive horizontal velocity of incoming flow in the inlet of the nozzle, there exists a smooth fluids with gravity. In addition, there exists a smooth free surface, which goes to infinity. Furthermore, we obtain the asymptotic behavior of the fluids with gravity in upstream and downstream by means of a blow-up argument. As byproducts, the monotone relationship between the location of the free boundary and the asymptotic height in the inlet are also established.