70 research outputs found
Desingularization of vortex rings in 3 dimensional Euler flows
In this paper, we are concerned with nonlinear desingularization of steady
vortex rings of three-dimensional incompressible Euler fluids. We focus on the
case when the vorticity function has a simple discontinuity, which
corresponding to a jump in vorticity at the boundary of the cross-section of
the vortex ring. Using the vorticity method, we construct a family of steady
vortex rings which constitute a desingularization of the classical circular
vortex filament in several kinds of domains. The precise localization of the
asymptotic singular vortex filament is proved to depend on the circulation and
the velocity at far fields of the vortex ring. Some qualitative and asymptotic
properties are also established. Comparing with known results, our work
actually enriches and advances the study on this problem.Comment: 35 page
On desingularization of steady vortex in the lake equations
We constructed a family of steady vortex solutions for the lake equations
with general vorticity function, which constitute a desingularization of a
singular vortex. The precise localization of the asymptotic singular vortex is
shown to be the deepest position of the lake. We also study global nonlinear
stability for these solutions. Some qualitative and asymptotic properties are
also established
Desingularization of vortices for 2D steady Euler flows via the vorticity method
In this paper, we consider steady Euler flows in a planar bounded domain in
which the vorticity is sharply concentrated in a finite number of disjoint
regions of small diameter. Such flows are closely related to the point vortex
model and can be regarded as desingularization of point vortices. By an
adaption of the vorticity method, we construct a family of steady Euler flows
in which the vorticity is concentrated near a global minimum point of the Robin
function of the domain, and the corresponding stream function satisfies a
semilinear elliptic equation with a given profile function. Furthermore, for
any given isolated minimum point of the
Kirchhoff-Routh function of the domain, we prove that there exists a family of
steady Euler flows whose vorticity is supported in small regions near
, and near each the corresponding stream function
satisfies a semilinear elliptic equation with a given profile function.Comment: 28 page
Steady vortex flows of perturbation type in a planar bounded domain
In this paper, we investigate steady Euler flows in a two-dimensional bounded
domain. By an adaption of the vorticity method, we prove that for any
nonconstant harmonic function , which corresponds to a nontrivial
irrotational flow, there exists a family of steady Euler flows with small
circulation in which the vorticity is continuous and supported in a small
neighborhood of the set of maximum points of near the boundary, and the
corresponding stream function satisfies a semilinear elliptic equation with a
given profile function. Moreover, if has isolated maximum points
on the boundary, we show that there
exists a family of steady Euler flows whose vorticity is continuous and
supported in disjoint regions of small diameter, and each of them is
contained in a small neighborhood of , and in each of these small
regions the stream function satisfies a semilinear elliptic equation with a
given profile function.Comment: 21 page
Steady vortex patches near a nontrivial irrotational flow
In this paper, we study the vortex patch problem in an ideal fluid in a
planar bounded domain. By solving a certain minimization problem and studying
the limiting behavior of the minimizer, we prove that for any harmonic function
corresponding to a nontrivial irrotational flow, there exists a family of
steady vortex patches approaching the set of extremum points of on the
boundary of the domain. Furthermore, we show that each finite collection of
strict extreme points of corresponds to a family of steady multiple vortex
patches approaching it.Comment: 21 page
Desingularization of vortex rings in 3 dimensional Euler flows: with swirl
We study desingularization of steady vortex rings in three-dimensional
axisymmetric incompressible Euler fluids with swirl. Using the variational
method, we construct a two-parameter family of steady vortex rings, which
constitute a desingularization of the classical circular vortex filament, in
several kinds of domains. The precise localization of the asymptotic singular
vortex filament is shown to depend on the circulation and the velocity at far
fields of the vortex ring and the geometry of the domains. We also discuss
other qualitative and asymptotic properties of these vortices.Comment: 33 pages. arXiv admin note: text overlap with arXiv:1905.1034
Global solutions for the generalized SQG equation and rearrangements
In this paper, we study the existence of rotating and traveling-wave
solutions for the generalized surface quasi-geostrophic (gSQG) equation. The
solutions are obtained by maximization of the energy over the set of
rearrangements of a fixed function. The rotating solutions take the form of
co-rotating vortices with -fold symmetry. The traveling-wave solutions take
the form of translating vortex pairs. Moreover, these solutions constitute the
desingularization of co-rotating point vortices and counter-rotating pairs.
Some other quantitative properties are also established.Comment: 30 page
Existence and regularity of co-rotating and travelling global solutions for the generalized SQG equation
By studying the linearization of contour dynamics equation and using implicit
function theorem, we prove the existence of co-rotating and travelling global
solutions for the gSQG equation, which extends the result of Hmidi and Mateu
\cite{HM} to . Moreover, we prove the regularity of
vortices boundary, and show the convexity of each vortices component.Comment: 33 page
Rotating vortex patches for the planar Euler equations in a disk
We construct a family of rotating vortex patches with fixed angular velocity
for the two-dimensional Euler equations in a disk. As the vorticity strength
goes to infinity, the limit of these rotating vortex patches is a rotating
point vortex whose motion is described by the Kirchhoff-Routh equation. The
construction is performed by solving a variational problem for the vorticity
which is based on an adaption of Arnold's variational principle. We also prove
nonlinear orbital stability of the set of maximizers in the variational problem
under perturbation when .Comment: 21 page
Existence of co-rotating and travelling vortex patches with doubly connected components for active scalar equations
By applying implicit function theorem on contour dynamics, we prove the
existence of co-rotating and travelling patch solutions for both Euler and the
generalized surface quasi-geostrophic equation. The solutions obtained
constitute a desingularization of points vortices when the size of patch
support vanishes. In particular, solutions constructed in this paper consist of
doubly connected components, which is essentially different from all known
results
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