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Desingularization of vortices for the incompressible Euler equation on a sphere
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Abstract
In this paper, we construct a family of global solutions to the incompressible Euler equation on a standard 2-sphere. These solutions are odd-symmetric with respect to the equatorial plane and rotate with a constant angular speed around the polar axis. More importantly, these solutions ``converges" to a pair of point vortices with equal strength and opposite signs. The construction is achieved by maximizing the energy-impulse functional relative to a family of suitable rearrangement classes and analyzing the asymptotic behavior of the maximizers. Based on their variational characterization, we also prove the stability of these rotating solutions with respect to odd-symmetric perturbations.
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In this paper, we study the stability of two-dimensional steady Euler flows with sharply concentrated vorticity in a bounded domain. These flows are obtained as maximizers of the kinetic energy on some isovortical surface, under the constraint that the vorticity is compactly supported in a finite number of disjoint regions of small diameter. We prove the nonlinear stability of these flows when the vorticity is sufficiently concentrated in one small region, or in two small regions with opposite signs. The proof is achieved by showing that these flows constitute a compact and isolated set of local maximizers of the kinetic energy on the isovortical surface. The separation property of the stream function plays a crucial role in validating isolatedness.
This article is devoted to stationary solutions of Euler’s equation on a rotating sphere, and to their relevance to the dynamics of stratospheric flows in the atmosphere of the outer planets of our solar system and in polar regions of the Earth. For the Euler equation, under appropriate conditions, rigidity results are established, ensuring that the solutions are either zonal or rotated zonal solutions. A natural analogue of Arnold’s stability criterion is proved. In both cases, the lowest mode Rossby–Haurwitz stationary solutions (more precisely, those whose stream functions belong to the sum of the first two eigenspaces of the Laplace-Beltrami operator) appear as limiting cases. We study the stability properties of these critical stationary solutions. Results on the local and global bifurcation of non-zonal stationary solutions from classical Rossby–Haurwitz waves are also obtained. Finally, we show that stationary solutions of the Euler equation on a rotating sphere are building blocks for travelling-wave solutions of the 3D system that describes the leading order dynamics of stratospheric planetary flows, capturing the characteristic decrease of density and increase of temperature with height in this region of the atmosphere.
We study the evolution of solutions to the two-dimensional Euler equations whose vorticity is sharply concentrated in the Wasserstein sense around a finite number of points. Under the assumption that the vorticity is merely L p integrable for some p > 2 , we show that the evolving vortex regions remain concentrated around points, and these points are close to solutions to the Helmholtz–Kirchhoff point vortex system.
This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.
We investigate the dynamical behaviours of the n-vortex problem with circulation vector Γ on a Riemann sphere S2, equipped with an arbitrary metric g. By mixing perspectives from Riemannian geometry and symplectic geometry, we prove that for any given Γ, the Hamiltonian is a Morse function for C2 generic g. If some constraints are put on Γ, then for such g the n-vortex problem possesses finitely many fixed points and infinitely many periodic orbits. Moreover, we exclude the existence of perverse symmetric orbits.
In this paper we consider the incompressible Euler equation in a simply-connected bounded planar domain. We study the confinement of the vorticity around a stationary point vortex. We show that the power law confinement around the center of the unit disk obtained in [2] remains true in the case of a stationary point vortex in a simply-connected bounded domain. The domain and the stationary point vortex must satisfy a condition expressed in terms of the conformal mapping from the domain to the unit disk. Explicit examples are discussed at the end.
We study the time evolution of an incompressible Euler fluid with planar symmetry when the vorticity is initially concentrated in small disks. We discuss about how long this concentration persists, showing that in some cases this happens for quite long times. Moreover, we analyze a toy model that shows a similar behavior and gives some hints on the original problem.
In this article, we prove nonlinear orbital stability for steadily
translating vortex pairs, a family of nonlinear waves that are exact solutions
of the incompressible, two-dimensional Euler equations. We use an adaptation of
Kelvin's variational principle, maximizing kinetic energy penalised by a
multiple of momentum among mirror-symmetric isovortical rearrangements. This
formulation has the advantage that the functional to be maximized and the
constraint set are both invariant under the flow of the time-dependent Euler
equations, and this observation is used strongly in the analysis. Previous work
on existence yields a wide class of examples to which our result applies.
We study the existence of stationary classical solutions of the
incompressible Euler equation in the plane that approximate singular
stationnary solutions of this equation. The construction is performed by
studying the asymptotics of equation -\eps^2 \Delta
u^\eps=(u^\eps-q-\frac{\kappa}{2\pi} \log \frac{1}{\eps})_+^p with Dirichlet
boundary conditions and q a given function. We also study the
desingularization of pairs of vortices by minimal energy nodal solutions and
the desingularization of rotating vortices.
In this note, we give a general criterion for steady vortex flows in a planar bounded domain. More specifically, we show that if the stream function satisfies “locally” a semilinear elliptic equation with monotone or Lipschitz nonlinearity, then the flow must be steady.
We study 2D Euler equations on a rotating surface, subject to the effect of
the Coriolis force, with an emphasis on surfaces of revolution. We bring in
conservation laws that yield long time estimates on solutions to the Euler
equation, and examine ways in which the solutions behave like zonal fields,
building on work of B.~Cheng and A.~Mahalov, examining how such 2D Euler
equations can account for the observed band structure of rapidly rotating
planets. Specific results include both an analysis of time averages of
solutions and a study of stability of stationary zonal fields. The latter study
includes both analytical and numerical work.
In this paper, we consider the planar vortex patch problem in an incompressible steady flow in a bounded domain Ω of . Let k be a positive integer and let be a positive constant, . For any given non-degenerate critical point of the Kirchhoff–Routh function defined on corresponding to , we prove the existence of a planar flow, such that the vorticity ω of this flow equals a large given positive constant λ in each small neighborhood of , , and elsewhere. Moreover, as , the vorticity set shrinks to , and the local vorticity strength near each approaches , .
The theory of the vortex motion of two-dimensional incompressible inviscid flow on a sphere is presented. Vorticity and stream function, which are related by the Laplace-Beltrami operator, are initially outlined. Green's function of the equation is obtained in which the stream function is expressed as integral form. The equations of motion for two vortex models on a sphere are derived. In particular, the equation for vortex patches with constant vorticity is given in terms of the contour integral appropriate for the contour dynamics.
It is shown that a steady two-dimensional flow, in which a finite vortex is in equilibrium with the irrotational flow past an obstacle, can be obtained as the solution of a variational problem in the class of rearrangements of a fixed function in Lp. The main step is to establish a bound on the support of the vorticity. The advantageof this approach, as in the recent works of Burton, Benjamin, and Auchmuty, is that the profile function of the vorticity is determined by the rearrangement class in which solutions are sought.
We are concerned here with steady, inviscid flows in two dimensions which have concentrated regions of vorticity. In particular, we study such flows which {open_quotes}desingularize{close_quotes} a configuration of point vortices in stable equilibrium with an irrotational flow. The vorticity functions in these flows are obtained as solutions of a variational problem for energy set in classes of rearrangements of fixed functions. It is inherent in the formulation of this problem that constraining sets which isolate different t regions of vorticity must be introduced. The existence of solutions of the variational problem follows from results of Burton, but in order for this to provide a flow with a continuous pressure, additional properties must be established. The simplest way to accomplish this is to show that the support of the vorticity is bounded away from the boundary of the constraining set. The major step in our analysis is a {open_quotes}local support lemma{close_quotes} which shows that this is possible under certain circumstances. This generalizes our earlier work for one vortex which in turn builds on results of Turkington. As we show below, it follows from the theory of Burton that for each vortex region Ω{sub i} there is a profile function Ï{sub i} such that the vorticity Ï{sub i} on that region satisfies Ï{sub i} = Ï, ο Ï, where Ï is the stream function for the flow. Turkington and Eydeland have given an iterative method for computing vortex flows with multiple vortex regions when the profile functions, rather than rearrangement classes, are prescribed in advance. 14 refs, 4 figs.
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
\left\{\begin{array}{l@{\quad}l} -\varepsilon^2 \Delta u = \sum\limits_{i=1}^m \chi_{\Omega_i^{+}} \left(u - q - \frac{\kappa_i^{+}}{2\pi} {\rm ln} \frac{1}{\varepsilon}\right)_+^p\\ \quad - \sum_{j=1}^n \chi_{\Omega_j^{-}} \left(q - \frac{\kappa_j^{-}}{2\pi} {\rm \ln} \frac{1}{\varepsilon} - u\right)_+^p , \quad \quad x \in \Omega,\\ u = 0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad x \in \partial \Omega,\end{array}\right.
Kelvin [13] studied steady planar flows of an ideal fluid, confined in a bounded region with solid boundaries. He supposed the vorticity to have a given value in a region of fixed area, outside which the flow was irrotational, and considered possible configurations of the vortex. In the case of a circular region, he observed that infinitely many steady configurations exist, and one can easily verify that any radially symmetric configuration is steady. Less obviously, he claimed that in a dumb-bell shaped region, infinitely many steady configurations can be obtained by dividing the vorticity between the ends of the dumb-bell in an arbitrary proportion, and seeking the configuration that maximises the kinetic energy subject to this restriction. In this paper we prove two results on the existence of infinitely many steady configurations for a vortex, based on Kelvin's principle of stationary kinetic energy. We admit flows in which the vorticity may be nonconstant in the vortex core (the region of non-zero vorticity), and seek steady flows in which the vorticity is a rearrangement of a given function; this more general formulation is based on a theory of 3-dimensional vortex rings proposed by Benjamin [1]; additionally, we prescribe the circulations of the velocity around the connected components of the boundary. We first show that in a bounded planar region of arbitrary shape, for a given non-negative function ~0, the kinetic energies of steady ideal fluid flows whose vorticities are
The article discusses the two-dimensional flow of an incompressible liquid between two infinitely close concentric spheres, due to an initial distribution of the vorticity differing from zero. The concept of point singularities (vortices, sources, and sinks) at a sphere is introduced. Equations of motion are obtained for point vortices, as well as invariants of the motion, known for the plane case [1]. The simplest case of the mutual motion of a pair of vortices is considered. Equations are obtained for the motion of point vortices at a rotating sphere. Integral invariants for the continuous distribution of the vorticity are obtained, having the dynamic sense of the total kinetic energy and the momentum of the liquid at the sphere. The effect of the topology of the sphere on the dynamics of the vorticity is noted, and a comparison is made with the plane case.
We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in L
p
for p>4/3. A related stability theorem using conservation of angular momentum in a circular domain is also proved.
We study the time evolution of a non-viscous incompressible two-dimensional fluid when the initial vorticity is concentrated inN small disjoint regions of diameter ε. We prove that the time evolved vorticity is also concentrated inN regions of diameterd, vanishing as ε→0. As a consequence we give a rigorous proof of the validity of the point vortex system. The same problem is examined in the context of the vortex-wave system.
We study the evolution of a two-dimensional, incompressible, ideal fluid in a case in which the vorticity is concentrated in small disjoint regions and we prove, globally in time, its connection with the vortex model.
We study the evolution of a two dimensional, incompressible, ideal fluid in a case in which the vorticity is concentrated in small, disjoint regions of the physical space. We prove, for short times, a connection between this evolution and the vortex model.
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