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 system
under 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">Γier hidden"> 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"> 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">HΩer hidden"> 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">Γier hidden"> 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">|Γi|er hidden"> 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">HΩer hidden">, if f&_user=111111111&_pii=S0022247X15007933&_rdoc=1&_issn=0022247X&md5=4de040f86afd2b92f2f0129374118e94">f" data-inlimgeid="1-s2.0-S0022247X15007933-si38.gif">er hidden"> 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">, 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"> and Ω is not simply connected.