Steady waves in flows over periodic bottoms.

We study the formation of steady waves in two dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator we establish the unique continuation of steady solution from the trivial solution for a flat bottom, with the exception of a sequence of resonant velocities $c_{k}$. Our main contribution is the proof that at least two steady solutions persist from a non-degenerate $S^{1}$-orbit of steady waves for a flat bottom. As a consequence, for near flat bottoms, we obtain the persistence of at least two steady waves close to the $S^{1}$-orbit of Stokes waves bifurcating from the velocities $c_{k}$.