Local bifurcation and regularity for steady periodic capillary–gravity water waves with constant vorticity

Existence of Wilton Ripples for Water Waves with Constant Vorticity and Capillary Effects

“…We prove now that the abstract setting presented above can be used to show that secondary bifurcation is a particular feature of the water wave problem when capillary eects are taken into account and when the ow beneath the wave has constant vorticity. To this end we shall use the new formulation for the water wave problem which has been derived in [6] in the context of gravity water waves with constant vorticity and adapted later in [23,22] to a more general setting which includes also capillary forces. More precisely, in the regime when surface tension is not negligible, two-dimensional 2π−periodic water waves traveling at constant speed over a at horizontal bottom are described, when the vorticity of the ow is assumed constant, by some of the solutions of the equation It is shown in [23,22] (see also [6]) that if w ∈ C 2+α (R) is a 2π−periodic solution of the equation (212) which satises additionally the following conditions…”

“…The constant λ can be identied as the speed at the surface of these waves as the stream function associated with these solutions is given by 18) the uid occupying the strip [0 ≤ y ≤ h] (see [6,23] for more details). We shall use here the parameter λ as a bifurcation parameter and γ as a perturbation parameter to prove that secondary bifurcation occurs on some of the bifurcation branches found in [23,22] provided that the constant vorticity is close to some critical values. To do so we need to introduce a functional analytic setting which allows us to rewrite the problem as a bifurcation equation and to use the abstract results presented in the Theorems 2.1-2.2.…”

“…As mentioned above, we restrict in this paper to considering water waves traveling over linearly sheared currents, so that the vorticity is constant, and we do not exclude the presence of stagnation points. In this setting the local bifurcation problem for capillary and capillary-gravity water waves has been investigated in [23,22] by using a formulation, obtained initially for waves conned merely to gravity forces [6], which allows one to consider waves with stagnation points and overhanging proles. The global bifurcation problem was addressed in [26], the author using the invertibility of the curvature operator to recast the governing equations as an operator equation for a compact perturbation of the identity.…”

On periodic geophysical water flows with discontinuous vorticity in the equatorial f -plane approximation

“…Extending in a natural and skillful way the method they used in [10], they proved the existence of waves of small and large amplitude. Thereafter, there have appeared works proving the existence of water waves allowing for additional complications, like the presence of stagnation points in flows of constant vorticity (see [11] for small-amplitude waves and [12] for waves of large amplitude), the incorporation of surface tension (as an additional restoring force) for flows with constant vorticity [13,14] and with discontinuous (piecewise constant) vorticity [15], of surface tension and unbounded vorticity [16], and of even piecewise constant vorticity and stagnation points [17]. The scenario of capillary-gravity water flows presenting a bounded vorticity was analysed in [18], where the existence of waves of small amplitude was proved.…”

Capillary–Gravity Water Waves with Discontinuous Vorticity: Existence and Regularity Results

“…The existence of exact capillary-gravity water waves was first established in the irrotational setting [23,24,25,40], the existence theory for rotational waves being developed more recently in the setting of waves with constant vorticity, stagnation points, and possibly with overhanging profiles [30] (see also [12,32]), or for waves with a general Hölder continuous vorticity distribution [44]. Many papers are also dedicated to the study of the properties of capillary-gravity water waves and of the flow beneath them, such as the regularity of the wave profile and that of the streamlines [19,20,2,33,45], or the description of the particle paths [18].…”