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On the stability of travelling waves with vorticity obtained by minimization


Reference:

Buffoni, B. and Burton, G. R., 2013. On the stability of travelling waves with vorticity obtained by minimization. NoDEA-Nonlinear Differential Equations and Applications, 20 (5), pp. 1597-1629.

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    Official URL:

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

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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.

    Details

    Item Type Articles
    CreatorsBuffoni, B.and Burton, G. R.
    DOI10.1007/s00030-013-0223-4
    Related URLs
    URLURL Type
    http://arxiv.org/abs/1207.7198Free Full-text
    DepartmentsFaculty of Science > Mathematical Sciences
    Publisher Statementbuffoniburton.pdf: The original publication is available at www.springerlink.com
    RefereedYes
    StatusPublished
    ID Code32406

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