Concentrated vorticity in the GrossPitaevskii equations
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Formal Metadata
Title 
Concentrated vorticity in the GrossPitaevskii equations

Title of Series  
Part Number 
22

Number of Parts 
23

Author 

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  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2016

Language 
English

Content Metadata
Subject Area  
Abstract 
We will present recent results describing the behavior of certain solutions of the GrossPitaevskii equations, in a particular scaling regime, for initial data with concentrated vorticity.

00:00
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08:35
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17:11
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24:10
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38:13
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44:31
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00:02
as and if you have to do so let me begin by thanking the organizers for the invitation I spent some time here and is viewed with great clarity OK so let me on 1st leg equations or be studying and so on the 2 equations and adjusting official Khalid respectively as primary on and inputs like this so the scaling here's the a month will also consider the Unitech counterpart smallcollege converted land and with that comes a misnomer location in setting for all this will be a say in 3 dimensions In a cylindrical domain several major crossing the government also were little ago is abounded opens substantive to say simply connected but in fact it was reported involvement for most stocks and and notation OU's rather consistently is all right .period donated in the form of texts can proceed with the next season and the lawmaker and the vertical delivery OK so these migrations of for a country necessary all consider for example so this is the more regional observers here say and Normand obligations on the lower boundary and periodic Monday direction and here I would want always during resemble somewhere Britain and so let me start by recalling some the relevant quantities so these are included for example the masses whatever density there is a momentum which already honestly so all of all this abuse tensions versions all right and adopt right here between 2 cumbersome resented the real upward identified inspectors between his and unsolicited men's density the free energy and so this is the the quantity Houston relatives and conserve held Tony and for the ghost is the equations and and let me write this sense of alright ability to deliver the momentum call this so that's kind of you and then finally all defined the vortices which operate under cabbages viewer can call because according to the woman covered debut is on 1 hand at the close the momentum since the end of it's a sure calculation shows that this is in fact equal to identify identify you as an order to evaluate now with components you want you to that this was a speedy in 3 instances across parts of central part of the greater the apartment measured OK and Saudis tells us that the Museum of course this great new ones orthogonal to level says of the real hard this is rough on levels of Nevada and so the vortices are affordable to both of these models enhances parallel to losses because of and circulated here traditionally 1 defines a war Hicks phone and as the interval curve of the importance vector field I'm here because of this genetic problems just described such report just the level sets Of the college life functions OK .period where we Worthington Onegin and some his other the undercover conservation laws and the Constitution was blown call and so for example on conservation of mass and the form and came to this is the kind of equations that time this is the worst of that and there is a accomplished of momentum and use it to hear then OK so from the 3 . 3 metric Sousa Vijay entry is the the real in a part of the active measures to undertake the role diversions and there's also a agreement terms which I won't care about much of there is no cameras on them but civic role this thing is about answer to understand how the water table the crawls equation and I'll find it you time keep autistic is well this disappears and I have "quotation mark of divergence of credit due to the credit of cases most of the agreed with autistic and finally the energy also conserve formally and they will be concerned with the With the boundary conditions ahead of you and so on and affordable this sort of little interest costs but all the when I study the gross polluters equation actually also isolated equation all the interested in uh in situations where the energies were Lord of slums natural resources and the images were a lot of fun in the middle of this you squared minus 1 is is quite small and also to end and and so it's continue to think of that In this way the U.S. women's 1 on is isn't is his McConnell's and so the that the diversions of Jerry is small In the regime and interest in this for a long it's also true that was seated in this in in this situation it this region where the energy is of order on hopes that the water still has a huge amount structure and I was in a very direct situation means was that when I studied the evolution criminal put destruction but more general it's true that under Angela merely under the assumption of smaller machine the institutional structure and so there is that it'll be concert along 1 of recruits in 3 D and a wave of 100 refers to 0 and so on this tells us what the versions of the current is it is it is small Of the coral the current work history again has structured 0 along faces a slew of hunkering down and so we we have evidence showing what the diversions in the growth of the current really well when I want to tell you more about what this is doing OK so then and let me describe in a special session of these equations simplest saying what 1 forces of aboard solution so special solution will be 1 of depends only on the horizontal exrebels as the end of the and silver alienated dealers right you offered depending only on Texas In the final will have the form of a tumultuous writing public on its illegality and models who depends on and skills like or at home OK so he made his hands outstanding at ODU if they do not deserve 0 indeed increases to 1 this has the the real progress 1 OK and so if you make it sound as already forever even solid union funds and in a number of ways let's been remark sound problems and so the
08:37
energy identity this is so this summer away from the origin this looks like it's homogeneous and agree with your hands busy this response scale water epsilon working over a the models the less than 1 .period Norton is basically you said so the energy of that will be like on say a call from of all ovaries epsilon times D squared off over so and if you just completed areas discover our squared I think there's a factor of 2 1 and so forth double energy provided by investors over overboard but Warriors are excused from an interesting you can call against something that is D squared times what are we perhaps nonplussed 1 OK this is the reason for this is the life of the busy really interested in is Loren McGarry skills as the scale on which 1 sees this this is the 1 thing being a basic vortex and analyst Lawrence Hill is the 1 which forces appear in question no both he and some of his solutions here no attracted services related to decision really interested yeah and so on and so it can be misleading as long as the transition during direction and interior on another actually of these solutions is violated the water well this is really going to be this part times DUE and then what's that will be an approximate identity we'll have formed so this is really an appointment and the players who function who will be OK so this is determined by the exact form of this this Canadian and the but that is not a function within a 1 man we scaled such OK and so we can see that the vote for this study solution the vortices is basically a Maradona function around worsened with more was steep items in India In the end he scolded the square and waterfalls all readers intervals 1 we can see very naively from this that the I of support extended 42 degree to costs in order 4 Thai lauded warrants on the and so the answer to horses at 3 1 are much cheaper and injected them 143 to use words so horses reports mines 1 have have has strong properties which are not possessed by on the horses OK and so they on the incessant addresses the following summer look for solutions of these 2 equations delighted as I wanted my ancestors I want to look for you epsilon depending on who will look something like well a product of that is the basic said the degree 1 Bordick Summerskill of lying on each was almost lace and then I translated by not amount pending on the vertical and time various I want the see team OK and so then and developed a single Singaporeans it's just as the the busy Morrison translated and I'm I'm over here organizations of pictures this week should look like and that's also want to protect 1 small translations will be a small skillet just 1 the locations of the picture is for all those years only be it is royalty was 1 please answer these all of assigned 1 so that the pictures on a on a small scale Egypt's long period these these horses and violated certain hazy also I will see you the structure I'm senator points everyone yet to and the scale in the small too OK so this is the this is what I am interested in and this is the situation like to study and let me say as 1 might imagine I'm writing solution as and also the if I had done initials and they want more the vital phase function in order to satisfy this boundary conditions and so 1 might imagine writing in a solution of you those equations as something of this former and then there's Sherman and doing a that analysis and for a lot of room for a further reasons I was just as empirical fact that this argument has not been carried out successfully for many of the problems for forfeited for essentially involves understanding such a any threedimensional question for any of these for communities holds on rental and so somehow rising has not been successful for these questions but this is that this is the picture of mind so service and that will not actually hearing look and so on so in some way story I want to describe begins with work of don't you know this this is attention by by the end of this year and culture From 0 8 thousand published and and so they show essentially I'm willing to stake has been a bit imprecisely precisely but they show that people will to 1 finest said energy subsidy of so with the speedy and roll over the remainder of the total energy and intensity and I worries given a certain weight also dragged off and the terms of the energy was overdone Lawrence Taylor part of it minus and it 10 here as well as holding the lead that in the vertical and Maine and by a lot of minus and and 1 high and all Lord of the smallscale attacks on service vacation that is in some sense that the Cresskill it will be in each of slammed his lawyer that's come to mind 1 but I'll tell you why this additions Gilman and then there's some concern that depends on and in the domain of area OK so that they reminiscent of a defined as said that if around G strongly about concepts and built under this retailing is the fact that citizen within and for instance primary and is gentle Kansas is equal to guess Hi over to the case something times and then a lot of that interaction terms something and I enjoy different I think there's a pioneer also log of clever minds to receive all the countries which 0 the and so the theory was that
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this is true up to a total of 1 year terms and they showed also that these lower nearterm Missouri in some way the government smoothly on and
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on on the London 1 is
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and I call the socalled G 0 this is the so OK and I just wanted to remarks this of course the UN the choice of make for the phase here will affect the there and and part on the paper document which it was to was to make a choice of it wishes no a stateowned did improve upper bounds is only about onethird of the community that evidence linking but to make a choice which is which is presumably a good 1 .period warning makes the students most of them so and so some with the right constant here which is connected to the right of the there OK so basin as 1 can conjecture that while they I think that you could write a written constitution that there should be solutions of viii Gross hideously equation there was a specific an equation 1 result 6 1 might 1 might hope at least see solutions to the growing which have Russia's Form 4 I'm not from 0 l into and copies were too being a accrued .period isn't much more aware it's for a credible here and what might or what might further hope To have solutions of the gross desecration this form where souls as some have shorter equation ready with OK so this is the and that is why I think the 1st conjecture was explicit in the paper of covering insignificant international culture controllers so I and let let Iraq also there's earlier but I think you're in someone would the paper of Montreal Sternberg the man who showed that the facility allergies so there are considering the litter problem inserted 3 new domains the main would like this said imagine made a service of revolution generated by this by figure and so it into the role and that it is not a service revolution review of 3 demand revolution but the point is that you want younger forces a local minimizes the Arctic functional Canadian DuPont and and so they showed is that there exists solutions some of the Ginsburg equation for small excellent and such that the energy density divided by the rescaling dividend divided during largely factored converges to and High Times house trough onedimensional measure restricting the segment on here and there summary of what is the the conversions to the world Vector 0 0 1 if you will forgive abuse tradition tendency and and so 1 understands as saying decisions 1 has a and quantum of water constant curve and so I think are the motivation of the these authors close to but was was said asked what 1 wants more skills so what's the fun the funds go structure of the solutions OK many Celso Ayala along with some of the main tool in the western bird and Vermonters to resume resume were earlier results of myself and Sonoran Alberto and Albertine Bordeaux on Monday on OK so OK then and let me state that mean results of his talks which are the following the 1st Israel could understand terrorists which addresses is so really I would say as a corollary will address the problem and let's say and so soon that's sequence of you that's 1 of the individual Vega ,comma started functions With the fall into look so forceful say that I want to see a stadium and experiments concerning on this tour from now on H sub epsilon always knows this a lot of slanted minus onehalf length scale and so that is the only answer this way so at each other if either of the all right over here it is adjacent X 1 you is just the surgical with parts and services the horizontal grid interviewed on this if you like the determinant the horse operated on you here is a real value as it has so I was like this look like want adult functions to leading order of concentrated near the origins of W . 1 1 in major deal is the linear Mississippi OK and so salty that if I look at it as a source security given insights from on the average all and vortices with Anastasia from skill of fertilizers and we assume this and also also assumed that this energy epsilon you have slandered the energy rescaling about fashion by subtracting off the correct version part that this is not a bison constancy to so I wanted to all the running of slump so
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conclusions are that the passengers subsequence we have let me let me just add to the fun so I would like to highlight to I'm rescaling on directions so let's let the slowness and the new slant of each as planned tax incidence of the millions of years have extended the main I've movie exceptional toward 1 but this it satisfies and horizontal forces steal the the converters sitting in and last 1 2 a measure who is concentrated on this is on the grass over the vertical axis of N H 1 curves he said some interesting worries to this time country look at and so I mean here and this is an expanding demanding for every hour if I fix a big ball then for UPS on small enough that the main musical doing what containable and so I mean for example on W. minus 1 1 of the big ball across 0 Alpha every year for every at the start of the presence of there exists a set of holes and that moreover whenever this holds then the villain in the energy as she gets 1 is greater than the OK and so this is as strong as you want this is basically a lower bound to complement the upper bound of Filipino and culture dishes of the about opium cultivation In other words ready African construct sequence you have plans such this discloses that quality and I I'm a few other conclusions which I won't right down at the moment OK and so this is the 1st as a corollary months when students not to precisely as of the counter quality in our in our geometry that there exists on the solution of the Ginsburg land out of long equation but here to be awarded to the prison considering and duration data along the top and bottom and mounted on the sides and and the .period defended to force the presence of this event vortex lines under the worst case of exists visions of this equation such that if you like the please some subsidies after rescaling in this fashion the skilled the recently convinced version 2 and so it is Nina right and this measure has this form of some adult classes honorees United reconsideration OK and so salsa for these solutions and the waters indeed in this area and Isaac 1 of the and function Indian analyzer the subject is to evaluations OK and so this is it indeed while not giving the kind of sir precise point was description intermediate and that's that the "quotation mark until the event may have had in mind and this doesn't 46 decision vortices answering around the around curves to borrowers who minimize this of this firm 1 and the 2nd caramel is identical and so on and this is 1 of India's Smith's as rich as it is a different 616 into as it was fascinated with aneurysm was posted recently and the 16 is a quite under way and services that we and so I have to thank you were introduced into this standards across the region make progress what is let me consider an old decision making remarks about some version of this and so on and so forth OK with running on names let me just mention that there's a huge operated work on begins Rolando equation in 3 D comments on some disasters it's so some of his falls from the answer some Oregon before disease millenarian ideas on the division to do the
31:01
incidence of adoption treaty is that 1 is alone of 1 of Law's small energy and water still concentrate around and around the alliance and more generally in the end demands were angry over 3 energy forces Jakarta and coordination to middle surfaces and this is a general picture and so this is then studied for example a major interest in food and moralist ,comma more about well revered in Revere about 4 overseas Romandie the full board composition were 90 but no amendments etc. to these are all the important contributions to the office and so of course interregional services line and when what is doing is imminent as a description of the Of the fine skills structure in the case of a higher degree of concentration of water around around 1 but this also laid in wait to and this so another parallels with the scalar analog of they could rely on operations earlier the Hong Kong creation and so there season there's a phenomenon of Warhol's interface in this posturing and the 2 interfaces that are connected in some way to animals services and want to have multiple interfaces customers surface and this or rather recent work on this divide people including adoption of logic where all that a docking across equated the card come and visit the the various subsets of an invited a subset of those provisions that few other a few officer intervals work as well as a pillar of culture Liu and Packard or paper adoption of culture quake in Yunnan and so on malodorous working interest customers is the the pair apparel sales 1st peril result hard emissions showing the vortex clustering OK so their children it would be and so our henceforth I'm going to always say things in terms of resale variable the revenue and so let me right on with the creation of Minnesota grossly readily at its in Seoul have said when somebody so the isotropic with the Texans you're giving rise to a a certain logic largely vacuous concerned OK and others for example give rise to the versions of these these conservationists and recoveries because requirements so let's assume assumed that I'd like to assume that the vortices initially concentrates all has these problems this is not survey this conversion is where we can make a conversion for the visitors precise estimates of the policy of the mines 1 1 or or this topology over here and his conversion to said please we have 1 some news and I forgetting this OK so every had and more seasoning .period In is the stew and also I wanted assume that I have the minimal energy possible given this conservation additional money is given residents from the GOP slot History scaling and when I read latest and awards in certain places and that this should converge to GE not of this 0 and here have said that they had not denotes it was in attrition and but say they each forest enough continues in which for of exactly the exactly Hamilton system associated to gin on time which is exactly the model that the myriad anarchist local distribution services so that was sold to the National minus the 2nd and their components Jr OK so I assume I have a smooth of this system and the and I assume that that time 0 the energy and Bautista concentrating around the curves to give the initial data for this solution that inclusion is there then for 2 in here again the vortices conversion to and the vortices urgent decisions here 0 and 10 teams answer the as he and this is true similarly the Energy also conversion sending but it's but when he was listed in the beginning we have working and
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so on and so what this says it is that right now this is a summary of what is is this the derivation of the equations and by American history and so let's remember I'm from Blair's office 1st and formally written down by guys in the suits 20 by climate and Amadora in the midNineties the following are the workers are crossing the ladies and on and basically the way these guys are used by the matched form wasn't on and so this is hardly the 1st to remain it's not in the 1st that is not finished yet but 1 but we hope this will be the 1st reservation in any setting of of this small for the band to of the nearly Pirlo vortex filaments from starting from equation 4 with food and grog Federation said the gross .period equation principle describes superfluous and so will require mechanical flaws and so rather than looking at the classical ideal food destructive dispensed Strawberry origins words inadequate the quantum counterparts which in some ways the analysis is much here for inequalities in the Tigris visions came and so on let me know here in the setting of all the words have since signed and saunalike to words of of officers sign the vessel was seized in these trailing bosses of airplanes resentful of winning horses and actually appears to be more relevant in the quantum context for the protest and then sell concerning this equation and there is summarized yesterday a lot of history and so this includes work of booklined martyrdom the underwriters and composite Baker a lot of work of and also recent work of autocratic and Carlos Garcia as catcher and I think is also work of Viagra civilian and users the comments 1 considers the National Association of everyone understand in sensitive OK and so and so in the remaining it's also approved and the of goes and so that the overall point is that the the proof rising crime we strongly on and on well failed things in the spirit of fair 1 which is so here I think so In pheromone 1 we assume only some kind of so crude knowledge about the vortices we said that we have and continental scale with the origin of all about the line and we assume that an energy balance and then we reverse conclusions and so in the identical setting we will watches initiatives such as these these of officeholders and and if we can show these persistent time then we have access to these conclusions and I and up further conclusions in the instance of the and so large part of this is if there is a fair and 3 so speak which is again the board and revise of India which which is it's a refinement these results and under the same or similar policies with respect still further information about the behavior of the water skiing indeed the solutions and so on but about the proof so if we consider all its coldest the interacting vortex film systems so so here's the point if so and that not so that then is easy competition to show that the and this is bounded by constant what is what I wanted I like to do the same thing as possible for we have not solving the this endeavor obtained From the grossest yesterday equation in the limited steps long goes to 0 and so and so let me suppose that somehow the arrays that every time these things these conditions hold that in times in a row and so it every time I can I can pass them a slice subsequent and get some living after that will depend on time and if I have something like 100 time might induce had invited signals of humans fortunately versus a long time and so it is only 10 minutes hunters will find well for reasons of moments and this but then I get errors arising from the
43:55
fact that this is not the equation which approximations and fundraisers look like and then I say said Ofcom's mezzanine undermine policies in my bosses of rather Shaaban rebounds will OK there's extra energy here and so on and so it is aware that if I it's possible that in the years since I have a analog here the yield an inglorious funds had a most I at lost as much energy they may
44:33
admit they may lose their engineer leak from the bosses isn't the rest the fluid and and we have have the Nets have enviously controlled on 1 can do have we can replace despite the 1st by the near term maturity and then the quadratic terms Richards involves only the Altoona not that nothing rivers and seconds or the quality of the new year the little interest OK then I'm I'm still love me controlled zone within Europe and as far as the musical writers and readers however it is also true if I go back to the system that for the lantern questions 1 as this 1 is as president and Kansas assistant and so many points than what when I go where ended as a reflection of what assurances on all funding for the last coming from the ghost limited and this is done by the same expressions and so this 1 I'm able to do 0 Grunwald inequality the perceived the owner of course all work has been has been obtained in this assertion that I can do this until it's a little that in the remaining 3 minutes cab but it so then so I wanted and is to use the to use this equation Equation don't have water through the waters is somewhere the Kurds are insulin multiply this same resale things in terms of the interview and multiply the waters of a test match the and after the reselling of find probably turn conceptually cuddle determined epsilon so here I am Jr was for I have a horse owners of the In the horse metaphorically with something in the testified the publicity and then I have on DVD and CD In doctors with the giddiness about recent address on it and that it has limits in this expression and and well developed a chaotic if he who for example looks like 1 connects minus not key squared OK and so then and there are some of the the components and more public call functions OK and so then formerly listed whitey and is really shouldn't the fees and the Jacoby numbers to as long as as long as the learning curves for support region where these colorful here 1 will really can reveal to I'm the optimum squared and Samara this will converge to where we know what this is doing this will give us some different pieces of Over Of the computation and so we have to get in the retina where this is doing so it's fact this is conversions so this is 1 is proving that theorem 3 and write but this is converting to do see so I a fix a time a time then this will converge to so the 2 the measure support along a curve along with curves involving the vertical derivatives the learning curve His visit these these complement and nowhere is this involves if you look at it this involves viii and and this is just where the Court including 1 this signal to minus Deasy of if not on of the other solutions the OK and so here we have this is so this will give rise to germ involving of the words that were following rating of these easier modern DEC of reviving and influence and and this is the hardest hit by far and wide on a timetable for the subject many of the individual but the politics of this summer only in its most students and some of the people in the history of the Cold War for a source said I'd say for the for granted against the cylindrical or I'm not a cylinder or all of our 3 are the natural context and so the has Singletary said this living systems and studied most on on the realignment and that we need a dimension that it was actions that would be possible for them to let equation uh rule years ago there was like watching is very inefficient and the geometry Maine and then the geometry in both solutions and so forth I want producing thing I'll do be horizontal 4 and identify 3 threedimensional manifold with jitters because just 1 could look for solutions to live with and filaments ,comma constant small schools around the genetic but that's about and you have to say this "quotation mark Mary the European continent of Europe keep the period you can Indiana Wu rates so we know what we have In this discourse relief we have if we have a local minimize error of the inning functional and we get local minimizing solutions of the actual functional as in effect is that it hasn't and curves and if they're not going to be a final linked in some way then haven't always want to spread apart Lauder the and so there there presumably are you know you have a little let me recall so this is a list of your the love this versus the yen as rent increase would also be fixed by and by nodding if I was sort of an equation in the latter case justice without the the time or and if you look at this as this is really a plenary and body problem words he plays the role of the and I am so people have constructed answers we however I had attended 2 of the things we didn't do was too identify noted solutions to our local minima Jesus manager which is what we need to do to to related to this this time the time of year usually called for a period time and that was part of the In other questions you know few and