The N-Vortex Problem

Newton, Paul K.

doi:10.1007/978-1-4684-9290-3

Cited by 455 publications

(275 citation statements)

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“…We will elaborate on the connections between our equations and particularly the LandauLifshitz equation in the next subsection. A convenient reference for the LIE equation can be found in Newton [21]. The equations of motion for the elastica exhibit a rich variety of solutions.…”

Section: Related Geometrical and Physical Problemsmentioning

confidence: 99%

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On the Cauchy Problem for a Dynamical Euler's Elastica

Burchard

Thomas

2003

Communications in Partial Differential Equations

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The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order elliptic equation for its tension. A Hasimoto transformation is used to rewrite the equations in convenient coordinates for the space and time derivatives of the tangent vector. A feature of this elastica is that it exhibits time-dependent monodromy. A frame frame parallel-transported along the elastica is in general only quasi-periodic, resulting in time-dependent boundary conditions for the coordinates. This complication is addressed by a gauge transformation, after which a contraction mapping argument can be applied. Local existence and uniqueness of elastica solutions are established for initial data in suitable Sobolev spaces.

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Section: Related Geometrical and Physical Problemsmentioning

confidence: 99%

“…By definition, κ is nonnegative, and θ is real-valued. The curvature can be expressed in the various frames as 21) and the torsion as…”

Section: The Standard Normal Framementioning

confidence: 99%

On the Cauchy Problem for a Dynamical Euler's Elastica

Burchard

Thomas

2003

Communications in Partial Differential Equations

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“…Basically, a gas of point vortices constitutes a Hamiltonian system described by the Kirchhoff-Hamilton equations of motion where the coordinates (x, y) of the point vortices are canonically conjugate [13,14]. A statistical mechanics of this system has been developed by several authors [15,16,17,18,19,20,21,22,23] following the pioneering work of Onsager [24].…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional Brownian vortices

Chavanis

2008

Physica A: Statistical Mechanics and its Applications

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We introduce a stochastic model of two-dimensional Brownian vortices associated with the canonical ensemble. The point vortices evolve through their usual mutual advection but they experience in addition a random velocity and a systematic drift generated by the system as a whole. The statistical equilibrium state of this stochastic model is the Gibbs canonical distribution. We consider a single species system and a system made of two types of vortices with positive and negative circulations. At positive temperatures, like-sign vortices repel each other ("plasma" case) and at negative temperatures, like-sign vortices attract each other ("gravity" case). We derive the stochastic equation satisfied by the exact vorticity field and the Fokker-Planck equation satisfied by the N -body distribution function. We present the BBGKY-like hierarchy of equations satisfied by the reduced distribution functions and close the hierarchy by considering an expansion of the solutions in powers of 1/N , where N is the number of vortices, in a proper thermodynamic limit. For spatially inhomogeneous systems, we derive the kinetic equations satisfied by the smooth vorticity field in a mean field approximation valid for N → +∞. For spatially homogeneous systems, we study the two-body correlation function, in a Debye-Hückel approximation valid at the order O(1/N ). The results of this paper can also apply to other systems of random walkers with long-range interactions such as self-gravitating Brownian particles and bacterial populations experiencing chemotaxis. Furthermore, for positive temperatures, our study provides a kinetic derivation, from microscopic stochastic processes, of the Debye-Hückel model of electrolytes.

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“…The equations of motion for a system of point vortices on a sphere have been known for over a century; many references can be found in Newton [1] and Meleshko et al [2]. The derivations of the following paragraphs are presented for the convenience of the reader.…”

Section: Stationary Configurations Of Point Vortices and Vortex Sheetmentioning

confidence: 99%

“…Stationary configurations of vortices are interesting as critical points of the interaction energy, and have been the object of much investigation; many references are found in Newton [1] and Meleshko et al [2].…”

Section: Introductionmentioning

confidence: 99%

Stationary states of identical point vortices and vortex foam on the sphere

O’Neil

2013

Proc. R. Soc. A.

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Stationary configurations of identical point vortices on the sphere are investigated using a simple numerical scheme. Configurations in which the vortices are arrayed along curves on the sphere are exhibited, which approximate equilibrium configurations of vortex sheets on the sphere. Other configurations (found after starting from random initial conditions) exhibit net-like distributions of vorticity, dividing the sphere into many cells that contain no vorticity or diffuse vorticity and forming a stationary 'vortex foam' on the sphere. They may be viewed as intermediate-energy elements in the set of all identical point vortex equilibria on the sphere. In the continuum limit, these foam states may correspond to stationary states of multiple intersecting vortex sheets. Stationary configurations of point vortices are not found to have this character when vortices of opposite circulations are included.

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The N-Vortex Problem

Cited by 455 publications

References 0 publications

On the Cauchy Problem for a Dynamical Euler's Elastica

On the Cauchy Problem for a Dynamical Euler's Elastica

Two-dimensional Brownian vortices

Stationary states of identical point vortices and vortex foam on the sphere

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