2015/02/22 by Christian Kuhl, Kuhl, Christian
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS
paper · pdf · doi:10.48550/arxiv.1502.06225
35 pages, 3 figures
arxiv created 2015/02/22 · arxiv updated 2015/02/24
This article is concerned with the study of existence and properties of stationary solutions for the dynamics of N point vortices in an idealised fluid constrained to a bounded two--dimen\-sional domain Ω, which is governed by a Hamiltonian system \\beginaligned Γi(d xi)/(d t) =(∂ HΩ)/(∂ yi)(z1,…,zN)
Γi(d yi)/(d t) =-(∂ HΩ)/(∂ xi)(z1,…,zN) \endaligned \hspace2cmwhere zi=(xi,yi), i=1,…,N, . where HΩ(z):=∑j=1NΓj2h(zj)+∑i,j=1, i\not=jNΓiΓjG(zi,zj) is the so--called Kirchhoff--Routh--path function under various conditions on the "vorticities" Γi and various topological and geometrical assumptions on Ω. In particular, we will prove that (under an additional technical assumption) if it is possible to align the vortices along a line, such that the signs of the Γi are alternating and |Γi| is increasing, HΩ has a critical point. If Ω is not simply connected, we are able to derive a critical point of HΩ, if ∑j∈ JΓj2>∑_\substacki,j∈ J i\not=j|ΓiΓj| for all J⊂\1,…,N\, |J|≥ 2.