2,488 research outputs found
An analytical and numerical study of steady patches in the disc
In this paper, we prove the existence of -fold rotating patches for the
Euler equations in the disc, for both simply-connected and doubly-connected
cases. Compared to the planar case, the rigid boundary introduces rich dynamics
for the lowest symmetries and . We also discuss some numerical
experiments highlighting the interaction between the boundary of the patch and
the rigid one.Comment: 56 page
Doubly connected V-states for the generalized surface quasi-geostrophic equations
In this paper, we prove the existence of doubly connected V-states for the
generalized SQG equations with They can be described by
countable branches bifurcating from the annulus at some explicit "eigenvalues"
related to Bessel functions of the first kind. Contrary to Euler equations
\cite{H-F-M-V}, we find V-states rotating with positive and negative angular
velocities. At the end of the paper we discuss some numerical experiments
concerning the limiting V-states.Comment: 65 page
Existence of corotating and counter-rotating vortex pairs for active scalar equations
In this paper, we study the existence of corotating and counter-rotating
pairs of simply connected patches for Euler equations and the
equations with From the numerical
experiments implemented for Euler equations in \cite{DZ, humbert, S-Z} it is
conjectured the existence of a curve of steady vortex pairs passing through the
point vortex pairs. There are some analytical proofs based on variational
principle \cite{keady, Tur}, however they do not give enough information about
the pairs such as the uniqueness or the topological structure of each single
vortex. We intend in this paper to give direct proofs confirming the numerical
experiments and extend these results for the equation
when . The proofs rely on the contour dynamics equations
combined with a desingularization of the point vortex pairs and the application
of the implicit function theorem.Comment: 39 pages, we unified some section
Bifurcation of rotating patches from Kirchhoff vortices
In this paper we prove the existence of countable branches of rotating
patches bifurcating from the ellipses at some implicit angular velocities.Comment: 21 page
Relative equilibria with holes for the surface quasi-geostrophic equations
We study the existence of doubly connected rotating patches for the inviscid
surface quasi- geostrophic equation left open in \cite{HHH}. By using the
approach proposed by \cite{CCGS} we also prove that close to the annulus the
boundaries are actually analytic curves
A generalisation of Schramm's formula for SLE(2)
The scaling limit of planar loop-erased random walks is described by a
stochastic Loewner evolution with parameter kappa=2. In this note SLE(2) in the
upper half-plane H minus a simply-connected compact subset K of H is studied.
As a main result, the left-passage probability with respect to K is explicitly
determined.Comment: 16 pages, 3 figures, Tik
Degenerate bifurcation of the rotating patches
In this paper we study the existence of doubly-connected rotating patches for
Euler equations when the classical non-degeneracy conditions are not satisfied.
We prove the bifurcation of the V-states with two-fold symmetry, however for
higher fold symmetry with the bifurcation does not occur. This
answers to a problem left open in \cite{H-F-M-V}. Note that, contrary to the
known results for simply-connected and doubly-connected cases where the
bifurcation is pitchfork, we show that the degenerate bifurcation is actually
transcritical. These results are in agreement with the numerical observations
recently discussed in \cite{H-F-M-V}. The proofs stem from the local structure
of the quadratic form associated to the reduced bifurcation equation.Comment: 39 page
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