Large-amplitude steady viscous surface waves on an incline

In this talk we consider periodic free-boundary incompressible Navier--Stokes flow down an inclined plane in two dimensions. By reformulating the problem as an elliptic system (in the sense of Agmon--Douglis--Nirenberg), we find a local branch of traveling solutions close to laminar flow, under assumptions on the governing Orr--Sommerfeld ODE operator. We then extend this branch to a global curve of solutions using global bifurcation theory. We verify the hypotheses for two regimes, small wavenumber and low Reynolds number. This work is joint with Miles Wheeler.