Stationary free surface viscous flows without surface tension in three dimensions. (Q1945761)

Suppose \(\Omega ^{\gamma }_{\rho }\) is the unbounded three-dimensional channel \(\Omega ^{\gamma }_{\rho }=\{ (x,y,z)\in \mathbb {R}^3\: \gamma (x,y)<z<z_0+\rho (x,y)\}\) with the fixed bottom \(\Gamma _{\gamma }=\{ (x,y,z)\in \mathbb {R}^3\: z=\gamma (x,y)\}\) and the free surface \(\Sigma _{\rho }=\{(x,y,z)\in \mathbb {R}^3\: z=z_0+\rho (x,y)\}\), where \(z_0\) is the height of the free surface when \(x,y\) goes to \(\infty \) . The gravitational field \(f\) has coordinates \((g \sin \alpha , 0, g \cos \alpha )\). The authors consider the flow of incompressible, viscous fluid in inclined channel \(\Omega ^{\gamma }_{\rho }\) with free boundary. They derive conditions on \(\alpha \) and \(\rho \) that guarantee the existence of a unique stationary solution in weighted Sobolev spaces.