94 research outputs found
On the Yudovich solutions for the ideal MHD equations
In this paper, we address the problem of weak solutions of Yudovich type for
the inviscid MHD equations in two dimensions. The local-in-time existence and
uniqueness of these solutions sound to be hard to achieve due to some terms
involving Riesz transforms in the vorticity-current formulation. We shall prove
that the vortex patches with smooth boundary offer a suitable class of initial
data for which the problem can be solved. However this is only done under a
geometric constraint by assuming the boundary of the initial vorticity to be
frozen in a magnetic field line.
We shall also discuss the stationary patches for the incompressible Euler
system and the MHD system. For example, we prove that a stationary simply
connected patch with rectifiable boundary for the system is necessarily
the characteristic function of a disc.Comment: 40 page
On the trivial solutions for the rotating patch model
In this paper we study the clockwise simply connected rotating patches for
Euler equations. By using the moving plane method we prove that Rankine
vortices are the only solutions to this problem in the class of {\it slightly}
convex domains. We discuss in the second part of the paper the case where the
angular velocity and we show without any geometric condition
that the set of the V-states is trivial and reduced to the Rankine vortices.Comment: 14 page
On the inviscid Boussinesq system with rough initial data
We deal with the local well-posedness theory for the two-dimensional inviscid
Boussinesq system with rough initial data of Yudovich type. The problem is in
some sense critical due to some terms involving Riesz transforms in the
vorticity-density formulation. We give a positive answer for a special
sub-class of Yudovich data including smooth and singular vortex patches. For
the latter case we assume in addition that the initial density is constant
around the singular part of the patch boundary.Comment: 26 page
On the V-states for the generalized quasi-geostrophic equations
We prove the existence of the V-states for the generalized inviscid SQG
equations with These structures are special rotating simply
connected patches with fold symmetry bifurcating from the trivial solution
at some explicit values of the angular velocity. This produces, inter alia, an
infinite family of non stationary global solutions with uniqueness.Comment: 54 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
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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