TY - JOUR

T1 - On the stability of travelling waves with vorticity obtained by minimization

AU - Buffoni, Boris

AU - Burton, Geoffrey R

PY - 2013/10

Y1 - 2013/10

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84874929797&partnerID=8YFLogxK

UR - http://arxiv.org/abs/1207.7198

UR - http://dx.doi.org/10.1007/s00030-013-0223-4

U2 - 10.1007/s00030-013-0223-4

DO - 10.1007/s00030-013-0223-4

M3 - Article

VL - 20

SP - 1597

EP - 1629

JO - NoDEA-Nonlinear Differential Equations and Applications

JF - NoDEA-Nonlinear Differential Equations and Applications

SN - 1021-9722

IS - 5

ER -