Traveling Quasi-periodic Water Waves with Constant Vorticity

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.

Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations

“…In the next lemma we collect some useful classical results dealing with various operations in weighted Sobolev spaces. The proofs are very close to those in [12,13,14], so we omit them. Lemma 4.1.…”

“…However in the quasi-linear case where the nonlinearity is unbounded and has the same order as the linear part the situation turns to be much more tricky. This is the case for instance in the water-wave equations where several results has been obtained in the past few years on the periodic or quasi-periodic, standing or traveling settings, see [1,4,12,13,14,44,50].…”

“…We shall focus in this section on some useful norms related to suitable operators class. These notions were used before in [4,12,13,14]. We consider a smooth family of bounded operators on Sobolev spaces…”

“…This procedure will be discussed in Proposition 6.2 and Proposition 6.3. We basically use KAM techniques as in [12,26] in order to conjugate the operator L εr , through a suitable quasi-periodic symplectic change of coordinates B, to a transport operator with constant coefficients. Indeed, we may construct an invertible transformation…”

On the analyticity of the Dirichlet-Neumann operator and Stokes waves

“…Quasi-periodic traveling waves. More general 1d time quasi-periodic traveling Stokes waves have been recently obtained in Berti-Franzoi-Maspero [11,12], with or without surface tension, and Feola-Giuliani [21], by means of a Nash-Moser implicit function iterative scheme. We remark that these solutions are not steady in any moving frame.…”