2,183 research outputs found
Regularization of point vortices for the Euler equation in dimension two
In this paper, we construct stationary classical solutions of the
incompressible Euler equation approximating singular stationary solutions of
this equation.
This procedure is carried out by constructing solutions to the following
elliptic problem [ -\ep^2 \Delta
u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p, \quad & x\in\Omega, u=0, \quad
& x\in\partial\Omega, ] where , is a bounded
domain, is a harmonic function.
We showed that if is simply-connected smooth domain, then for any
given non-degenerate critical point of Kirchhoff-Routh function
with the same strength , there is a
stationary classical solution approximating stationary points vortex
solution of incompressible Euler equations with vorticity .
Existence and asymptotic behavior of single point non-vanishing vortex
solutions were studied by D. Smets and J. Van Schaftingen (2010).Comment: 32page
Vortex motion for the lake equations
The lake equations model the vertically
averaged horizontal velocity in an inviscid incompressible flow of a fluid in a
basin whose variable depth is small in comparison
with the size of its two-dimensional projection . When
the depth is positive everywhere in and constant on the boundary, we
prove that the vorticity of solutions of the lake equations whose initial
vorticity concentrates at an interior point is asympotically a multiple of a
Dirac mass whose motion is governed by the depth function .Comment: Minor revision, 43 page
On non-topological solutions for planar Liouville Systems of Toda-type
Motivated by the study of non abelian Chern Simons vortices of non
topological type in Gauge Field Theory, we analyse the solvability of planar
Liouville systems of Toda type in presence of singular sources. We identify
necessary and sufficient conditions on the "flux" pair which ensure the radial
solvability of the system. Since the given system includes the (integrable) 2 X
2 Toda system as a particular case, thus we recover the existence result
available in this case. Our method relies on a blow-up analysis, which even in
the radial setting, takes new turns compared with the single equation case
The Kosterlitz-Thouless Phenomenon on a Fluid Random Surface
The problem of a periodic scalar field on a two-dimensional dynamical random
lattice is studied with the inclusion of vortices in the action. Using a random
matrix formulation, in the continuum limit for genus zero surfaces the
partition function is found exactly, as a function of the chemical potential
for vortices of unit winding number, at a specific radius in the plasma phase.
This solution is used to describe the Kosterlitz- Thouless phenomenon in the
presence of 2D quantum gravity as one passes from the ultra-violet to the
infra-red.Comment: 15 pages. This version to appear in Nucl.Phys.B contains less
introductory material (revised
Multidimensional Conservation Laws: Overview, Problems, and Perspective
Some of recent important developments are overviewed, several longstanding
open problems are discussed, and a perspective is presented for the
mathematical theory of multidimensional conservation laws. Some basic features
and phenomena of multidimensional hyperbolic conservation laws are revealed,
and some samples of multidimensional systems/models and related important
problems are presented and analyzed with emphasis on the prototypes that have
been solved or may be expected to be solved rigorously at least for some cases.
In particular, multidimensional steady supersonic problems and transonic
problems, shock reflection-diffraction problems, and related effective
nonlinear approaches are analyzed. A theory of divergence-measure vector fields
and related analytical frameworks for the analysis of entropy solutions are
discussed.Comment: 43 pages, 3 figure
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