On the existence of extreme waves and the Stokes conjecture with vorticity

Abstract

This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120 • or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120 • if the vorticity is nonnegative near the free surface.

Steady water waves

“…Location of the stagnation points. If the vorticity function γ(·) is nonpositive, in the context of Theorem 4.1, it is known [80] that P (x, y) ≥ P atm everywhere and that the horizontal velocity u(x, y) is maximized at the crest. Furthermore, if Q is bounded for waves in C, then there exists an "extreme" wave such that max u = c, which is obtained as a weak limit of nonextreme waves [81].…”

Steady Water Waves with Multiple Critical Layers

“…This opened the way, not only for a new branch of research, but also for the existence of new types of waves. In many aspects, the features of rotational waves resemble those of irrotational waves: results on symmetry [4,5,15], regularity [6], and waves of greatest height [23] have now been extended from irrotational waves to waves with a general vorticity distribution. In most of those cases the differences have manifested themselves as technical difficulties, not different properties per se.…”

Solitary Water Waves of Large Amplitude Generated by Surface Pressure

“…With R ≡ 0, this was proven by Varvaruca [28] and also repeated in [31]. The modifications for R nonzero are straightforward and hence omitted.…”