A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description
Caglioti, E.; Lions, Pierre-Louis; Marchioro, C.; Pulvirenti, M. (1992), A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Communications in Mathematical Physics, 43, 3, p. 501-525. http://dx.doi.org/10.1007/BF02099262
TypeArticle accepté pour publication ou publié
Journal nameCommunications in Mathematical Physics
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Abstract (EN)We consider the canonical Gibbs measure associated to aN-vortex system in a bounded domain Λ, at inverse temperature β˜ and prove that, in the limitN→∞, β˜ /N→β, αN→1, where β∈(−8π, + ∞) (here α denotes the vorticity intensity of each vortex), the one particle distribution function ϱN = ϱ N x,x∈Λ converges to a superposition of solutions ϱ α of the following Mean Field Equation: ⎧⎩⎨⎪⎪⎪⎪⎪⎪ϱβ(x)=e−βψ∫Λe−βψ;−Δψ=ϱβinΛψ|∂Λ=0. Moreover, we study the variational principles associated to Eq. (A.1) and prove thai, when β→−8π+, either ϱβ → δ x 0 (weakly in the sense of measures) wherex 0 denotes and equilibrium point of a single point vortex in Λ, or ϱβ converges to a smooth solution of (A.1) for β=−8π. Examples of both possibilities are given, although we are not able to solve the alternative for a given Λ. Finally, we discuss a possible connection of the present analysis with the 2-D turbulence.
Subjects / Keywordstwo-dimensional Euler equations
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On the uniqueness of the solution of the two-dimensional Navier–Stokes equation with a Dirac mass as initial vorticity Gallagher, Isabelle; Gallay, Thierry; Lions, Pierre-Louis (2005) Article accepté pour publication ou publié