587 research outputs found
Restrictions on the geometry of the periodic vorticity equation
We prove that several evolution equations arising as mathematical models for
fluid motion cannot be realized as metric Euler equations on the Lie group of
all smooth and orientation-preserving diffeomorphisms on the circle. These
include the quasi-geostrophic model equation, the axisymmetric Euler flow in
higher space dimensions, and De Gregorio's vorticity model equation.Comment: 14 pages, 1 tabl
Steady water waves with multiple critical layers
We construct small-amplitude periodic water waves with multiple critical
layers. In addition to waves with arbitrarily many critical layers and a single
crest in each period, two-dimensional sets of waves with several crests and
troughs in each period are found. The setting is that of steady two-dimensional
finite-depth gravity water waves with vorticity.Comment: 16 pages, 2 figures. As accepted for publication in SIAM J. Math.
Ana
The periodic b-equation and Euler equations on the circle
In this note we show that the periodic b-equation can only be realized as an
Euler equation on the Lie group Diff(S^1) of all smooth and orientiation
preserving diffeomorphisms on the cirlce if b=2, i.e. for the Camassa-Holm
equation. In this case the inertia operator generating the metric on Diff(S^1)
is given by A=1-d^2/dx^2. In contrast, the Degasperis-Procesi equation, for
which b=3, is not an Euler equation on Diff(S^1) for any inertia operator. Our
result generalizes a recent result of B. Kolev.Comment: 8 page
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