We consider an incompressible, viscous, finite depth fluid flowing down a three-dimensional channel. In the absence of surface tension, we prove the existence of a unique stationary solution in weighted Sobolev spaces. The result is based on a thorough study of the linearized problem, particularly the pseudodifferential operator relating the normal velocity of the fluid and the normal component of the associated stress tensor along the free surface, and requires the use of the Nash-Moser implicit function theorem.
"Stationary free surface viscous flows without surface tension in three dimensions." Differential Integral Equations 25 (9/10) 801 - 820, September/October 2012. https://doi.org/10.57262/die/1356012369