Gravity water waves over constant vorticity flows: from laminar flows to touching waves

Abstract: In a recent paper, Hur and Wheeler proved the existence of periodic steady water waves over an infinitely deep, two-dimensional and constant vorticity flow and subject to gravity whose profile overhangs, among which, waves whose surface touches at a point, enclosing a bubble of air. We take this further, proving the existence of a continuous curve of water waves from a laminar flow up to a touching wave for fixed non-zero gravity. This implies the existence of a wave profile that is vertical at a point but not overhanging, which is referred to as a breaking wave. This allows us to study the behaviour of critical layers, which are points where the horizontal velocity vanishes, at locations where the wave profile is vertical. This applies to both overhanging and breaking waves. We also extend our results regarding the continuous curve of water waves from a laminar flow up to a touching wave to finite but very large depth. We formulate our problem as a modified version of the Babenko equation. We then use methods from local bifurcation theory to construct solutions near the laminar flow and use a compactness argument to ensure the maps obtained from the different Implicit Function Theorems coincide. In the last Section, we extend our results to the finite depth case. To do this, we formulate the problem utilising the periodic Hilbert transform on a finite strip. Properties of this operator discovered by Constantin, Strauss and Varvaruca turn out instrumental for our purposes.