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Collision of almost parallel vortex filaments
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Title  Collision of almost parallel vortex filaments 
Title of Series  Trimestre Ondes Non linéaires  June Conference 
Part Number  15 
Number of Parts  23 
Author 
Banica, Valeria

Contributors 
Faou, Erwan

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/20828 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2016 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  We investigate the occurrence of collisions in the evolution of vortex filaments through a system introduced by Klein, Majda and Damodaran and by Zakharov. We first establish rigorously the existence of a pair of almost parallel vortex filaments, with opposite circulation, colliding at some point in finite time. The collision mechanism is based on the one of the selfsimilar solutions of the model, described in our previous work. We also extend this construction to the case of an arbitrary number of filaments, with polygonal symmetry, that are perturbations of a configuration of parallel vortex filaments forming a polygon, with or without its center, rotating with constant angular velocity. 
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00:02
around a it and in the world and in the end it was clear to me that the cell 1st I would like to send the organizers that his very nice syndication and have for organizing the statements of the minister said today I talk about the joint Warwick away there is 1 of Newell and that concerns us about experiments and the more particularly by the vortex Yemen's unity case not collide so what what is somewhat experiment it is in a truly where Sweeney fully aware there were city is concentrating on longer the curvy nursery and the costs of coalitions and there connection and hopeful vortex vehemently some single that happened so interwebbed afraid so it's important phenomenon and lots of course it is very difficult to capture this kind of of Singh's mathematically however it is expected expected that 1 situation which is rather simple to understand and use in the case of 2 almost filed what experiments with opposite the stipulations as it was suggested by prose work in the 70's so it is the case meaning to this situation that you can't see it actually in the sky if the weather is good so you see that for that to work from you cannot traces of the planar whites tracing the sky so these are bought experiment which will put it circulations that propagate and eventually as times goal but they finally they finally reconnect and the weather and start and started doing the vortex thing and others stocks was the call commentary on the I got the country's aircraft contractually actually coasted where appears in the American Journal of Food on Astronautica it's which has courses since agenda very hard to talk to my question however this this feature you can see from the sky and during your life that if still could you notice it at 1 point and you think about stock well OK so now all you want to say this is what happened in real life and you want to transcribe it mathematically and the courses through there it's true that the free mankind's equation this is something very difficult because the situation is very singular however are those that my knowledge 1 of more than that and with this kind of and modernization so far 1st the you should take a more text with OK let me call and so we are in that because the like this station OK and chic some higher level so we are in a confused look at case and then he should call the coordinates in displaying its J and White J but end then then few complex if I the planes you know which should just say you are looking what's what's happening on concluded last fight .period Jr 4 of once 104 jails relation government Jr there is a system that is supposed to monitor this evolution in Kuwait which is due to a client might and I'm about and actually hate was built prior to the Cardinals precisely in that case where any quit school and you have so you have to work apparel for vortex determined would opposite situation so it's precisely just like you had before the situation and before so that crowd who was interested in the equation for the antibiotic which is called the antiparallel peril for vortex Yemen's integer also this small of course climb and on and on real debate generalize to several more textured demands and deriving along here where the small cell in the city of few words about this modern so it is thank you for this special communicate it is the shredding of system so why running a system this can be aristocrat explained by the following 2 facts you have 1 billion are shredding their parties supposed to model their interaction the humans with said actually the national linger at the but is
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actually a very rough approximation all and there by normal flow of water vortex become integration which is a model for the evolution of 1 single Anderson model which is more complicated than actually singing Khaleda the CU and here we want takes just another part actually if you plug issues like this and that's exit on smaller sport perturbation by Norman for Europe Dana as a 1st step coming up as a leading Pepsi manual obtained National question OK so this is for the interaction of 1 Federman the JC government with itself off and then you have an extract which is supposed to model the interaction and the influence of other governments to the GAC government and they actually can so this kind of the interaction is precisely the 1 that appeared in the case soulful 24 awaits when you have .period tortoises with the election to to and then well of course once in that 1 would criticize for this morning is that the roughly we had some sense of the the influence on the Jays filament from the other few dominance is supposed to come only at high at Hicks highs through them .period what exceeds that in some sense it's the kind of fuel and oxygen correction from the other guys from the bad I think this is the motherland this is the only 1 we have cell of forests issue if you freeze up them and is inviolable and can you get as a practical matter solution of 4 this problem but In the Germans but I look at it that are located precisely precisely he even played since cannot told avoid being Trudeau what explains the case these are particularly solutions and we are studying perturbations small variation near Parliament the features of this particular the solution of this system OK so of course I criticize the appointment of modern media should have them in the morning and but says account apparently accepted the fact that when it is that you know that we have as all long as it has a very nice mathematical structure which probably means something politically OK so it is a it is a from dynamic and it has several conserved quantities as in I put what corresponds to the point of law Texas them into the Aceh conservation laws and they put it in here because of course you have to analyze things because your system of struggling Guerrero the equation is not with the vanishing conditions at infinity your nearest summer some cost settled before going further ahead with the filaments I resale of few words about what is known for that .period Bartek system before fault because we show work around strike fire and enlightened located in the .period Vortex Systems solution so the 1st thing is that the firm stipulations are all policy key Norman J. are all positive then the solution Giggs's globally dining here in this call in globally at global existence means we want to avoid collision collision means that at some time off work toward point of Waters's suggest that moment 1 preacher and at the level of the and collisions will be on duty climbed and allocation of such that consumption of 20 curative get this 1 OK so as I was saying that when circulation are on the same day .period Waters's system evolves about but you never have collisions for a simple fact that on the 1st line of EFA by any chance you have collision meaning the XJ getting too narrow to warrant K then your it will blow up and then in order to have conservation you have to have another apparently is over jails case such that the difference goes to infinity and then you have a of contradiction borrow from despite which says that all differences must be monitored uniformly so when you
11:36
have a circulation of same sign that you have a but existing don't have collision affordable vortex .period system while in the particular case Antigua's school where are you actually have adopted the stance preserved between 2 were marked his soul and and then the .period translated in the case of what was it circulation and rotate the otherwise and weaken the case so far .period crafted the beginning is the place where does that to work .period vortices circulate translate directly from velocity depending on the English ignition starts and will be looking for collision all that division of straight filament located in that area for .period 6 well casings a human at the 2011 was studying mom on complicated of course when you increase the number of mobile phone of elements home for the points so already when an was Street Of course you should take a situation that I'm not the same and there is a simple example I'm all for it it's 3 points was chosen such a death the whole picture will eventually qualify them at the whole industry when will collide at the that the the the triangle will turn and shrink the same on the day of the last 2 sinks are some specialized configuration special solution although on vortex that I will use you know the thing
13:26
like talker so it is known that the nicotine level the point where turned instead to the 1 where you place you take the reticence and bodyguard regular polygon finds the exception and the case and you put the same circulation and then it is known that this configuration if you let it evolve so that would explain system it would just rotate with some umbrella Venice city which is determined by that circulation and the direct use of Mexico and also it is known that you cannot allow also to take the the stand to put the vortex .period in the center of the configuration which other circulation than the 1 here and the anger over the city's completed began maintenance of this is flexible OK so I the surgery results for them .period Vortex Systems and no 1 maybe half a puff of my talk will be about to what is known about the kind of demand and supply of system for vortex students that on the 2nd slider and I will insist on what was done former for 2 reasons firstly in some sense to convince you that it is a nice system because precisely where you can only see that it has many classes of particular solutions and also because they are not so many results so we can use them on I hope all of them and that skipping some and the reason there are so few results is set to fall and 1 from them freedom dynamics community and of course if you're trying to ward directly in the leader equation that that this situation is to single we are working on this model which can be criticized a lot of course mechanic is the situation from which is tool to single and on the other hand why this hasn't been studied too much for instance for or by the special communicate because eventually with the shredding of system it is that's coming at this from my point of view will of course many people didn't hold hope to women but also because it is not just the strutting their usual see is shortening the system would a non non vanishing condition at Infinity which means more onerous protects the equation and if you follow what is happening already in the gross protects the equation communicate the results concerning got really evolutionary find solution just pensioners solutions started to adjust the wrong way the years of 2 cups Inc yes so I think that's what are some explanation so let me point down to work .period um went under results we would have gone so far as to the results obtained by people that the Derenzis model wondering this morning they conjectured that precisely 1 have to look for a coalition in
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case of toward an undue prominence vortex filaments so it was a conjecture as the end of also on the predicted that in case of same circulation and so on should cover what existence college and dynamite competition was given told tool to call for today said this conjecture OK another result due to violence and my dad is concerning the status of the dairy farm In this and this configuration of several classes of vortex filaments but of course that in the case of the year they took into account the very weakened formulation of the equation which doesn't allow you to walk look precisely for for collisions and then
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I knew I couldn't possibly adopt look that the system and that they consider small perturbations in at the H 1 led by make fools on 1 hand the local exist financing for an initial configuration and Exjade and then stipulation and they managed to weak stand the local existence of course basing themselves on an energy argument so the energy the user is the 1 here which is dependent on young last actually the spot in the case where the distance between any point .period vortices is the same the spot is precisely the separation momentum and that I shall initially divided by these like Oracle when it is like this in this thing and then once you have this energy concern you will you manage tools to work then underwent existence results of course and this this condition is very restrictive because if you have instance in the playing that constraint will have worsened distance from want to each other then of course you can't have only 2 points in 2 points or 0 the itself like other OK all it's all there there are 1 that could have been done and Ira currently done a former nearly vortex filaments industrial begins the land of migration and ice in the region with maybe it is too small and this tomorrow and then I found that several results with everything you wanted by way of explaining the next lies In the last 1 abilities that there is a result of recent off against Astacio Craig and younger which treated to that diamonds very space solution periodic close toward the core operating itself since relations with OK so disorder known results and I said that they will uh GrecoRoman what we have them with Evelina so 1st since you have in mind that you are in address that's dissenting you'll hear tried to sort way to servicing in terms of the energy instead of just H and because of course monitoring enticements energy about another conversely and also coming just the initial system means environment on the rotation and the should take a rotation the differences of course is that just constant and that nothing nature and Marlboro also in the spirit of this approach is that if the equation the energy space contains and that and that gracefully only time meaning traveling waves with the feet energy which are important solutions for the most basic equations and also in here we see that we have some Sing Sing and the proof of course users of fixed point argument that allowed the gross of that is to say no you don't space morning each renewal and awarded them and demand to grow in times of course now if you want to pass from local development existence you need the Indian energy as before and actually you will have seen it in the side before it is considered foreign Atlanta turned and for 2 points by simply dividing some themselves wanted you aware the distance between the ports but actually you can go further and that computer precisely how the energy in for instance the fuel would take the square if you take a square hexagons here if you can express it in terms of the known conserved quantities and to see what is list but aside and Steve by some synergy imposed this system you don't have a consideration of the Energy and Natural You realize that if you on this you really have to impose very much symmetries either and a boasts on their on the initial configuration of the exchange and also on the perturbation and actually this kind of symmetry preserved if you just the at each fight you'll the child single and we have found that is the you just bought litigation by the complex numbers so you'll be later by the mothers of the complex number and you rotate canvassing by the argument of the complex numbers so your effects and analysts to be a flexible at each high conceit and this is something which is preserved provided by the system and the and no 1 there so you going to buy all those strikes endowment by the same scene at each half and then actually you will you see that I'm hoping that resulted that you have global existence you the equation the system on the system shows his reduced to this summer in nice equation all my guys that he's then angular velocity of your regular political analyst and configuration of .period vertices and of course it is really a grossly testified equation and you have a natural energy and actually realize that the energy all of this system he is entitled to use these energy and then of course you'll the extent of global existence so now that we have these remnants of Haitian which is the ski like
24:42
you can now you characterize many classes of solutions and of course the 1st of them you can obtain analysis is that I mean that was potentially saving a bit more complicated but not too much in too much but you can obtain a traveling waves and you have factory transcriber menu and don't interest from you just say to watch very simple to transcribe arising from gross defective so and this mindset 2 you so don't think it's a it's a square so this means that on each filament you have something that is moving and the this is sworn 1 scene and also if you will he was the very landing variants of the system from their system missions within our system you realize that you can you can transform the into for his the only very nice OK and this is not OK and OK so on each is Hendricks and links you have something somewhere some distortion that moves along Canongate so talking about cases because precisely the halting and Bronte wrote the a paper in Nature where the shoulder that you can remains it is essential for influence Yukon exceed made palaces With and with the distortion that that involves like these have for several vortex filaments their paper were spoken for 2 the fact that the present autos work on 1 she laments gives you the picture when you have only 1 headaches we're done distortion moviegoers but actually in the lab experiment you caps ever so this fits with what is seen at least into OK so then coalition that's another forget about the exciting .period is a condition for wrote experiments while in the case of taking the regular XJ configurations and the symmetry :colon perturbations so and the GOP with that grossed that's the equation 1 younger velocity of your initial configuration is 0 euros divides the equations deliver natural equation and quality assurance it means that you want everybody to grownups meaning that you're looking toward cooler place time and the place where are the Tunisian team was 0 OK so then you can just the just the right to life and the various invite you naturally in the solution that cancers at times you once it is in the space of 5 0 0 which means that you are on your aboard this is what of course for having done that the angular velocity when Xerox and actually you have to put up for Inter in the middle of your body configuration so this means that you have at least 3 . Bautista so you don't cover .period that that case of pairs of students so now we are getting back to the problem of too few moments which is the name of my dog OK so it was that kind of some suggested that solution should be certain there's someone that human trait OK of course among and the Wood Dixieland and then there the program reduces tool this shredding the equation with quite exotic here and what you are interested in our solutions for the struggling question but that the initial time don't vanish and that finite time vanish at some point so it is a client's not unusual study of the shredding of equation however I present the Cold player by madam hands which is not so quickly to the Internet and but you know we are the 1st of all they had the same type of problem so when followed what exact relation in situation which ended Apple with them with the problem of despite calls for finding them finding some solution vanishing at the point and finite time the 1990 a walk and a new wave he said which and then actually for this purpose we tried to the full of course listings the paper and what about the day there were some of the major insurers so what that the idea that they praise to replace the your unknown function by 1 Hallberry on all financial and so we are looking for a solution that the blow up at 1 point that as of course in the case although he take wish there is another previous paper of amendments and that gives you such as disburses blow close wave will hit and sigh a shift solution and that they manage them to put a dozen hands at seeing that the equation in our case is where you'll you replace excited 1 by 1 all of our efforts there then you will end up with the gross Pitofsky type again equation in equation with some nasty turn derivative terms Neversink while a few women some objects around which you want to do your own urine and that's such an object that you can find it now or the literature is that they paraded by the reports aspired budget so inspired by sold most events social ocean that displays of his personal at the point but of course the regularity he such that you don't managing our case of tools to write and that in this equation transformative and the Swede doesn't work so we get
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back to was are choral work which is said the just this 1 can look good for session our solution with that conditions had continued itself strike light and then of course there is no proof is a kind of State Bar and mine and that puts his bulk full as major Hoboken problem To find success in assertion and that's what we have done with them and and 1 for when the neon so we constructing a Forrester said Samina solution that said that will give you collisions of London was always to been something like this and of course Feinstein's bill 2 0 all day with a hand up colliding and that the coalition government would be just across the across lines of course if you are not in the case you want initially because your your filaments I meant it meeting but then what we do is that we managed to banned them because now we have the odds that the answers will be precisely that and that's assuming our solution and actually end up was the abuses of great natural singular looking so some singularity of formation where you have lots of chances to find itself by using some subsuming future so we managed to find the substantive solution for all modern then we cut and the wheel completed 2 of integration .period all 4 of the communications system so we eventually Europe solution is so you have collision in finite time at 1 point so I will sit the front means I have less mess as we say some words about proofs and so I don't know OK OK OK OK you can do the same scene so here it was foretold work experience actually you can't data that was geometric configurations and then write to us at seminars solutions for dead that gross the equation that governed the evolution and anyone and not go by leaving the same argument by constructing blues so the the configuration that we collided at the median finding of someone's about the true from cell of Florida sesame not profile of you just look at the office provided way Sharon you'll you will it 2 Armenian really really to the unknown function in the qualities days and then you have some equations and Ft where you know you can expressing terms all of these so you end up by doing the fix .period 1 which is not so complicated Fina status that allows you to control the denominator so it should be not so complicated so would this is for a Anthony Florida Institute of profiler you start from community and you just constructed and there are now 4 days for Florida for the system for it to work nearly punishing elements where you put to the side that says we said then you right that the equation for your reminder on you want to that this little are wont avoiding what's happening at the collision time meaning that remember that you want the cohesion meaning that you don't have collision before the creation expected collision time provision of expected back and this is what you already have for the deception in our solutions now when you add the detonator are you don't want it to give you some consolation before you you were you will don't you want to keep a the said similar the savior please leading with respect to 22 are sold for doing this you have to impose that little are will go to a 0 0 faster than the city OK said of immigration for little are with some some terms within and then
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can I you will do a fixed point as usual the chaos in mind H1 with summaries of decay and moral where you will have to to ourselves that locally at and 0 and the place you want to see him in the collision you will are indicating to Xerox less so fast and that the subsidy at the constant the constant of all became would be more of K and then of course this in this kind of seeing so no 1 can see also has been done for instance in the case of fraud a blowup at critical mass for QB Canada's to the domains by Bulger and that got you take down the Simpson a solution blowing up on tour you cut it and then you know you constructed at the reminder on that Lostwithiel is choosing to do
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and then you'll for treating the equation for entering the fixed fixedpoint you just get .period tried this teammates near 0 and that Infiniti and then so as to when they each won estimates Florida I reminded terms in the equation of art and then you just use the current system send them some commentators with the localization for and Infiniti estimates Nunez here that center for adventure if the end what question so that he would do you have to do police no 1 will unlike there there'll actually if you look at what is the solution of finds the wrong the OK at 10 0 you would have ended in 0 0 if you have this guy is really this is 0 OK and the that entry would have several depending on how you've chosen your yeah that was His demand and yet yeah yeah that's for sure but then the shape you have several if you have the you have acquired about the news absolutely it is this far In this case the not our system of social commentary and actually is and we can talk about it but there should be an the question it this was a move that is the view will the ones who don't support aid in which frame OK so of course that's what we have always wear a kimono say speaking this will work but that the 2 2 tours to see I'm in this model is supposed to be evaluated opt to work in distance between the few dominance is much larger than the cost size of the troops so of course of the game over at some point before beautiful picture have answered but but but but but and some of my then that purchase his books says that it is seen in no numerical simulation even being fully the dynamics equations that too when no I wonder you will you out the balancing point where separation distance and on a section of equivalent then happen connection to the host the but number this what was that all 4 no and I really don't know in the way the uniform through the suggest have numeric standard away dating analysis should be interesting to see with this kind of of solution watch will among from amazing prevent that that students would take I mean I have to have the confidence of 1 the if you look at them
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