126 research outputs found
Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation
We construct families of smooth travelling-wave solutions to the inviscid
surface quasi-geostrophic equation (SQG). These solutions can be viewed as the
equivalents for this equation of the vortex anti-vortex pairs in the context of
the incompressible Euler equation. Our argument relies on the stream function
formulation and eventually amounts to solving a fractional nonlinear elliptic
equation by variational methods
On the linear stability of vortex columns in the energy space
We investigate the linear stability of inviscid columnar vortices with
respect to finite energy perturbations. For a large class of vortex profiles,
we show that the linearized evolution group has a sub-exponential growth in
time, which means that the associated growth bound is equal to zero. This
implies in particular that the spectrum of the linearized operator is entirely
contained in the imaginary axis. This contribution complements the results of a
previous work, where spectral stability was established for the linearized
operator in the enstrophy space.Comment: Major revision, including a complete rewriting of Section
Leapfrogging vortex rings for the three dimensional Gross-Pitaevskii equation
Leapfrogging motion of vortex rings sharing the same axis of symmetry was
first predicted by Helmholtz in his famous work on the Euler equation for
incompressible fluids. Its justification in that framework remains an open
question to date. In this paper, we rigorously derive the corresponding
leapfrogging motion for the axially symmetric three-dimensional
Gross-Pitaevskii equation.Comment: 39 pages, 2 figure
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