On the stability of travelling waves with vorticity obtained by minimization

Boris Buffoni, Geoffrey R Burton

Research output: Contribution to journalArticle

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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.
Original languageEnglish
Pages (from-to)1597-1629
Number of pages33
JournalNoDEA-Nonlinear Differential Equations and Applications
Volume20
Issue number5
Early online date19 Mar 2013
DOIs
Publication statusPublished - Oct 2013

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Stream Function
Vorticity
Traveling Wave
Periodic Traveling Waves
Conditional Stability
Water Waves
Surface Waves
Lagrange multipliers
Free Boundary
Minimizer
Smooth function
Impulse
Function Space
Velocity Field
Stability Analysis
Vanish
Parallel flow
Horizontal
Water waves
Perturbation

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On the stability of travelling waves with vorticity obtained by minimization. / Buffoni, Boris; Burton, Geoffrey R.

In: NoDEA-Nonlinear Differential Equations and Applications, Vol. 20, No. 5, 10.2013, p. 1597-1629.

Research output: Contribution to journalArticle

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