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On some properties of traveling water waves with vorticity
We prove that for a large class of vorticity functions the crests of any corresponding traveling gravity water wave of finite depth are necessarily points of maximal horizontal velocity. We also show that for waves with nonpositive vorticity the pressure everywhere in the fluid is larger than the atmospheric pressure. A related a priori estimate for waves with nonnegative vorticity is also given
On the existence of steady periodic capillary-gravity stratified water waves
We prove the existence of small steady periodic capillary-gravity water waves
for general stratified flows, where we allow for stagnation points in the flow.
We establish the existence of both laminar and non-laminar flow solutions for
the governing equations. This is achieved by using bifurcation theory and
estimates based on the ellipticity of the system, where we regard, in turn, the
mass-flux and surface tension as bifurcation parameters.Comment: 17 pages, to appear in Ann. Scuola Norm. Sup. Pisa Cl. Sc
Stokes waves with vorticity
The existence of periodic waves propagating downstream on the surface of a
two-dimensional infinitely deep water under gravity is established for a
general class of vorticities. When reformulated as an elliptic boundary value
problem in a fixed semi-infinite strip with a parameter, the operator
describing the problem is nonlinear and non-Fredholm. A global connected set of
nontrivial solutions is obtained via singular theory of bifurcation. Each
solution on the continuum has a symmetric and monotone wave profile. The proof
uses a generalized degree theory, global bifurcation theory and Wyburn's lemma
in topology, combined with the Schauder theory for elliptic problems and the
maximum principle