Exact steady periodic water waves with vorticity

Abstract

We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two-dimensional inviscid periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use bifurcation and degree theory to construct a global connected set of such solutions.

Analyticity of periodic traveling free surface water waves with vorticity

“…Given any γ ∈ C 1+α (R), an interplay between global bifurcation theory, degree theory, a priori estimates for nonlinear elliptic equations with nonlinear oblique boundary conditions in combination with sharp maximum principles ensures the existence of solutions of this type, representing waves of small amplitude as well as waves of large amplitude [6]. We will show that for all solutions whose existence has been rigorously established, all streamlines beneath the free surface must have maximal regularity being real analytic.…”

“…In swell, the particle speeds are very small compared to the wave speed unless we approach the breaking regime [18]. The assumption (2.7) expresses the fact that the waves move faster than the water (this indicates that the waves are not moving humps of water but pulses of energy moving through water) and allows us (see the discussion in [6]) to specify the vorticity ω of the flow as a function of the streamline,…”

“…Vorticity is adequate for describing currents; a current which is uniform with depth is described by zero vorticity (irrotational case) [8]. Constant nonzero vorticity is appropriate for tidal flows [19] and nonconstant vorticity is the hallmark of highly irregular currents [6].…”

“…The currently available existence theory for periodic traveling waves with vorticity is developed in the context of waves of small and large amplitude with profiles represented by Hölder continuously differentiable functions [6]. For irrotational flows a landmark theorem of Lewy [17] that generalizes the classical Schwarz reflection principle from complex function theory shows that such profiles must be real analytic (see [20]).…”

“…In the particular case of zero vorticity our approach yields an alternative short proof of the real analyticity of regular Stokes waves. 1 The approach relies on use of an appropriate hodograph change of variable that transforms the free boundary value problem (corresponding in a frame moving at the constant wave speed to the governing equations for water waves with vorticity) into a nonlinear boundary problem for a quasi-linear elliptic equation in a fixed rectangular domain [6]. Subsequently we introduce an additional parameter in the problem, and then use the implicit function theorem to exploit the analytic dependence on the parameter to obtain the analyticity of all streamlines beneath the free surface if the vorticity function is Hölder continuously differentiable.…”