Negative temperature bounds for 2D vorticity compounds

TY - JOUR

T1 - Negative temperature bounds for 2D vorticity compounds

AU - Kiessling, Michael K.H.

PY - 1995/5

Y1 - 1995/5

N2 - If λ ≥ 0 and Λ ⊂ ℝ2 is a connected Lipschitz domain, with |Λ| < ∞, it is a matter of standard convex functional analysis to show that the nonlinear eigenvalue problem of Poisson-Boltzmann type {Mathematical expression} with 0-Dirichlet boundary data for ψ, has a unique solution ψ0 Ξ 0. Here, we prove the stronger result that ψ =ψ0 Ξ 0 is the unique solution also for λ ∈ (λ*, 0), where λ* < 0 is some critical value which depends only on Λ, but in any event with λ* < -8 π/5. This result settles a conjecture about negative temperatures of vorticity compounds in 2D turbulence which goes back to 1949 work of Onsager.

AB - If λ ≥ 0 and Λ ⊂ ℝ2 is a connected Lipschitz domain, with |Λ| < ∞, it is a matter of standard convex functional analysis to show that the nonlinear eigenvalue problem of Poisson-Boltzmann type {Mathematical expression} with 0-Dirichlet boundary data for ψ, has a unique solution ψ0 Ξ 0. Here, we prove the stronger result that ψ =ψ0 Ξ 0 is the unique solution also for λ ∈ (λ*, 0), where λ* < 0 is some critical value which depends only on Λ, but in any event with λ* < -8 π/5. This result settles a conjecture about negative temperatures of vorticity compounds in 2D turbulence which goes back to 1949 work of Onsager.

KW - Mathematical Subject Classifications (1991): 35Q53, 53A10, 76C05, 82D50

UR - http://www.scopus.com/inward/record.url?scp=0342696510&partnerID=8YFLogxK

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U2 - https://doi.org/10.1007/BF00739374

DO - https://doi.org/10.1007/BF00739374

M3 - Article

SN - 0377-9017

VL - 34

SP - 49

EP - 57

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

IS - 1

ER -