Global bifurcation for steady viscous roll waves on an incline

We construct a branch of travelling periodic `roll wave’ solutions to the free-boundary incompressible Navier–Stokes equations on an inclined plane in two dimensions. These solutions bifurcate from a parallel shear flow, under natural assumptions on the related Orr–Sommerfeld equation. Using techniques from analytic global bifurcation theory, we extend the local branch to a global curve of solutions. A key step of the proof is reformulating the problem, including the unknown free boundary, as an elliptic system in the sense of Agmon–Douglis–Nirenberg. Finally, we verify the hypotheses on the Orr–Sommerfeld equation for two regimes: small wavenumber and low Reynolds number.