Steady waves in flows over periodic bottoms

Abstract

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 a steady solution from the trivial solution for a flat bottom, with the exception of a sequence of velocities $c_{k}$. The main contribution is the proof that at least two steady solutions for a near-flat bottom persist close to a non-degenerate $S^{1}$-orbit of non-constant steady waves for a flat bottom. As a consequence, we obtain the persistence of at least two steady waves close to a non-degenerate $S^{1}$-orbit of Stokes waves arising from the velocities $c_{k}$ for a flat bottom