Global bifurcation of capillary-gravity water waves with overhanging profiles and arbitrary vorticity

Abstract

We study two-dimensional periodic capillary-gravity water waves propagating at the free surface of water in a flow with arbitrary, prescribed vorticity over a flat bed. Using conformal mappings and a new Babenko-type reformulation of Bernoulli's equation, the problem is equivalently cast into the form "identity plus compact", which is amenable to Rabinowitz' global bifurcation theorem, while no restrictions on the geometry of the surface profile and no assumptions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local and global solution curves, bifurcating from laminar flows with a flat surface, are constructed.

Large-amplitude steady gravity water waves with general vorticity and critical layers

“…(b) It is important to point out that we could manage to lower the norm for blow-up of w in alternative (i)(b) to C 0,δ , 12/13 < δ ≤ 1. This seems to be the first time (except for the paper [51] of the present authors dealing with the capillary-gravity case) that this norm could be lowered that much (without restricting to a special case for γ in the first place).…”

“…Transversality condition As for the transversality condition, we state here a theorem of [51], which was proved in the case of nonzero surface tension, but, together with its proof, is also valid in the case of zero surface tension.…”

“…We also mention [4] where a similar strategy is used for vortex sheets in the irrotational context, allowing for overhanging waves and using a reformulation as a nonlinear singular integral equation in terms of the tangent angle. Then, the paper [51] first allowed for arbitrary vorticity, stagnation points, critical layers, and overhanging profiles, also crucially making use of nonzero surface tension to derive an "identity plus compact" operator equation via a Babenko-type reformulation of Bernoulli's equation. The main goal of the present paper is to extend this strategy and the results to the pure gravity case.…”

Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl

“…18, this means that the bifurcation conditions are much less explicit and that we require qualitative methods. The second aim is to construct large-amplitude solutions using global bifurcation theory and a reformulation of the problem inspired by the recent paper 5 on the two-dimensional gravity-capillary water wave problem with vorticity.…”

Bifurcation analysis for axisymmetric capillary water waves with vorticity and swirl

“…In contrast to [23], this means that the bifurcation conditions are much less explicit and that we require qualitative methods. The second aim is to construct large-amplitude solutions using global bifurcation theory and a reformulation of the problem inspired by the recent paper [25] on the 2D gravity-capillary water wave problem with vorticity.…”