Abstract
We prove the first bifurcation result of time quasi-periodic traveling wave solutions for space periodic water waves with vorticity. In particular, we prove the existence of small amplitude time quasi-periodic solutions of the gravity-capillary water waves equations with constant vorticity, for a bidimensional fluid over a flat bottom delimited by a space-periodic free interface. These quasi-periodic solutions exist for all the values of depth, gravity and vorticity, and restrict the surface tension to a Borel set of asymptotically full Lebesgue measure.
Highlights
Introduction and Main ResultThe search for traveling surface waves in inviscid fluids is a very important problem in fluid mechanics, widely studied since the pioneering work of Stokes [38] in 1847
Related difficulties appear in the search of time periodic standing waves which have been constructed in the last few years in a series of papers by Iooss, Plotnikov, Toland [22,23,25,34] for pure gravity waves, by Alazard-Baldi [1] in presence of surface tension and subsequently extended to time quasi-periodic standing waves solutions by BertiMontalto [6] and Baldi-Berti-Haus-Montalto [2]
In this paper we prove the first existence result of time quasi-periodic traveling wave solutions for the gravity-capillary water waves equations with constant vorticity for bidimensional fluids
Summary
The search for traveling surface waves in inviscid fluids is a very important problem in fluid mechanics, widely studied since the pioneering work of Stokes [38] in 1847. In this paper we prove the first existence result of time quasi-periodic traveling wave solutions for the gravity-capillary water waves equations with constant vorticity for bidimensional fluids. 1) Theorem 1.5 holds for any value of the vorticity γ , so in particular it guarantees existence of quasi-periodic traveling waves for irrotational fluids, that is γ = 0 In this case the solutions (1.19) do not reduce to those in [6], which are standing, that is even in x. There are significant differences with respect to [6], which proves the existence of quasi-periodic standing waves for irrotational fluids, in the result – the solutions of Theorem 1.5 are traveling waves of fluids with constant vorticity – and in the techniques. The analysis in [45] is complementary to Theorem 1.5; the solutions (1.19) are time-quasi-periodic traveling waves on a spatially periodic domain, whereas [45] concerns pure traveling waves with multiple spatial periods
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