A special class of stationary flows for two-dimensional euler equations: A statistical mechanics description. Part II

Caglioti, Emanuele; Lions, Pierre-Louis; Marchioro, Carlo; Pulvirenti, Mario

doi:10.1007/bf02099602

Cited by 252 publications

(189 citation statements)

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“…When this happens, we speak of ensemble inequivalence. At equilibrium, ensembles are equivalent for vortices in a disk (or other simple domains) when the Hamiltonian is postulated to be the only constraint [22,69]. However, they can be inequivalent in other kinds of domains [69] or also in a disk when the conservation of the angular momentum is taken into account [70,23].…”

Section: Resultsmentioning

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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Section: Resultsmentioning

confidence: 99%

Two-dimensional Brownian vortices

Chavanis

2008

Physica A: Statistical Mechanics and its Applications

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“…Related problems are studied by Carleson and Chang in [14]. Such equations on bounded domains of R 2 with Dirichlet boundary conditions play an important role in the context of statistical mechanics of point vortices in the mean field limit as discussed in Caglioti, Lions, Marchioro and Pulvirenti [12,13] and Kiessling [30]. In particular, it is proved in [35], when (M, g) is a flat torus with fundamental cell domain [− In view of our earlier work [31], we propose a different approach to study the existence of solutions of (E u ) λ .…”

Section: Introductionmentioning

confidence: 99%

Harnack Type Inequality: the Method of Moving Planes

Li¹

1999

Communications in Mathematical Physics

272

271

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Abstract:A Harnack type inequality is established for solutions to some semilinear elliptic equations in dimension two. The result is motivated by our approach to the study of some semilinear elliptic equations on compact Riemannian manifolds, which originated from some Chern-Simons Higgs model and have been studied recently by various authors. IntroductionLet (M, g) be a compact Riemann surface without boundary, V be a positive function on M , W be a function with M W dv g = 1. Throughout the paper dv g denotes the volume element of g, g denotes the Laplace Beltrami operator with respect to g. For λ ∈ R, we seek a solution ofClearly M W dv g = 1 is a necessary condition for (E u ) λ to have a solution. If we set ξ = u − log M V e u dv g for a solution of (E u ) λ , then ξ satisfies

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“…Equation (8) has been derived by Caglioti, Lions, Marchioro and Pulvirenti [7,8] from the mean eld limit of point vortices of the Euler ow, see also Chanillo Kiessling [9] and Kiessling [10]. Equation (8) occurs also in the study of multiple condensate solutions for the Chern Simons Higgs theory, see Tarantello [11,12].…”

Section: Logarithmic Trudinger Moser Inequalitiesmentioning

confidence: 97%