Equilibria for the N-vortex-problem in a general bounded domain

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• 作者：Christian Kuhl ^{christian.kuhl@math.uni-giessen.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Kirchhoff&ndash ; Routh path function ; Vortex dynamics
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：433
• 期：2
• 页码：1531-1560
• 全文大小：711 K

文摘

This article is concerned with the study of existence of stationary solutions for the dynamics of N point vortices in an idealised fluid constrained to a bounded two-dimensional domain Ω, which is governed by a Hamiltonian systemunder selected conditions on the “vorticities” formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007933&_mathId=si2.gif&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=9b646096b4c23550f1e6edf4a4165729" title="Click to view the MathML source">Γ_ier hidden">erflow="scroll">Γi and various topological and geometrical assumptions on Ω. Here f&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=333b8bf2dfdcb37b827cec9a7de13be6">f" data-inlimgeid="1-s2.0-S0022247X15007933-si3.gif">er hidden">erflow="scroll">HΩfalse">(zfalse">):=∑j=1NΓj2hfalse">(zjfalse">)+∑i,j=1,i≠jNΓiΓjGfalse">(zi,zjfalse">) is the so-called Kirchhoff–Routh-path function. In particular, we will prove that formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007933&_mathId=si4.gif&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=8f6c8b3e2f8d0d08f442fd1d44c428f7" title="Click to view the MathML source">H_Ωer hidden">erflow="scroll">HΩ has a critical point, if it is possible to align the vortices along a line, such that the signs of the formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007933&_mathId=si2.gif&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=9b646096b4c23550f1e6edf4a4165729" title="Click to view the MathML source">Γ_ier hidden">erflow="scroll">Γi are alternating and formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007933&_mathId=si5.gif&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=4a4ef866d4e91b9b2fe618e763b2f194" title="Click to view the MathML source">|Γ_i|er hidden">erflow="scroll">false">|Γifalse">| is non-increasing. We are also able to derive a critical point of formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007933&_mathId=si4.gif&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=8f6c8b3e2f8d0d08f442fd1d44c428f7" title="Click to view the MathML source">H_Ωer hidden">erflow="scroll">HΩ, if f&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=4de040f86afd2b92f2f0129374118e94">f" data-inlimgeid="1-s2.0-S0022247X15007933-si38.gif">er hidden">erflow="scroll">∑j∈JΓj2>∑frac linethickness="0">i,j∈Ji≠jfrac>false">|ΓiΓjfalse">| for all formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007933&_mathId=si39.gif&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=e4304c7222f4612717bafbe26235b1a4" title="Click to view the MathML source">J⊂{1,…,N}er hidden">erflow="scroll">J⊂false">{1,…,Nfalse">}, formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15007933&_mathId=si40.gif&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=c0fa02fce39e8b614c6fc161e469f850" title="Click to view the MathML source">|J|≥2er hidden">erflow="scroll">false">|Jfalse">|≥2 and Ω is not simply connected.