On the stability of travelling waves with vorticity obtained by minimisation

TL;DR

This paper develops a modified variational approach to prove the existence of periodic water waves with vorticity, enabling stability analysis by enlarging the function space and incorporating additional constraints.

Contribution

It introduces a new method for constructing traveling water waves with vorticity via energy minimization in an expanded function space, including circulation and impulse constraints.

Findings

Existence of periodic water waves with vorticity established.

The approach allows for stability analysis of these waves.

Method accounts for non-vanishing velocity components at the free boundary.

Abstract

We modify the approach of Burton and Toland [Comm. Pure Appl. Math. (2011)] to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the function space to a class of stream functions that do not correspond necessarily to travelling profiles. In particular, for smooth profiles and smooth stream functions, the normal component of the velocity field at the free boundary is not required a priori to vanish in some Galilean coordinate system. Travelling periodic waves are obtained by a direct minimisation of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimisation appears in Burton and Toland after a first maximisation). In addition, we not only use the circulation along the upper boundary as a constraint, but also the total horizontal impulse (the velocity becoming a Lagrange multiplier). This allows us to preclude parallel flows by choosing appropriately the values of these two constraints and the sign of the vorticity. By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period.