2022/03/25 by Mohameden Ould Ahmedou, Thomas Bartsch, Ahmedou, Mohameden +3
Mathematics · Physics and Astronomy · #35J15 #35J25 #35J60 #35R01 #37J12 #76B47 #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2203.13566
openalex publication_date 2022/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove the existence of critical points of vortex type Hamiltonians H(p1,…, pN) = ∑_i,j=1,i≠ jN ΓiΓjG(pi,pj)+ψ(p1,…,pN) on a closed Riemannian surface (Σ,g) which is not homeomorphic to the sphere or the projective plane. Here G denotes the Green function of the Laplace-Beltrami operator in Σ, ψ:ΣN→ℝ may be any function of class C1, and Γ1,…,ΓN∈ℝ∖\0\ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to ψ= -∑i=1N Γi2h(pi,pi) where h:Σ×Σ→ℝ is the regular part of the Laplace-Beltrami operator. We obtain critical points p=(p1,…,pN) for arbitrary N and vorticities (Γ1,…,ΓN) in ℝN∖ V where V is an explicitly given algebraic variety of codimension 1.