Collision of almost parallel vortex filaments
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Banica, Valeria
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 self-similar 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.
 So first I would like to thank the organizers for their very nice invitation and for organizing also all this semester. So today I will talk about a joint work with Erwan Fawou and Avilinho that concerns vortex filaments and more particularly parallel vortex filaments in case they are colliding. So what is a vortex filament? It is a fluid where the vorticity is concentrated along a curve in R3 and of course collisions and the reconnection of vortex filament is something that happens in turbulent fluids so it's an important phenomena but of course it is very difficult to capture this kind of things mathematically. However it is expected that one situation which is rather simple to understand is the case of two almost parallel vortex filaments with opposite circulations as it was suggested by Crowe's work in the 70s. So it is the case, I mean it is the situation that you can see actually in the sky if the weather is good. So you see the two trace of the plane white trace in the sky so these are vortex filaments with opposite circulations that propagate and eventually as times go by they finally reconnect and well then start doing vortex ring and other things. In English they are called con trails, condensation trails. Ah, con trails, aircraft con trails, yeah. Actually Crowe's paper appears in American Journal of Astronautics and Aerodynamics or something like this which of course is a journal very hard to obtain for a mathematician. However this picture you can see it in the sky and during your life now that you have seen this talk you'll notice it at one point and you'll think about this talk. Well, okay so now you want, this is what happened in real life, now you want to transcribe it mathematically and of course through the fluid mechanics equation this is something very difficult because the situation is very singular. However there is at my knowledge one model that do this kind of modelization. So first you should take a vortex filament, okay, let me call, so we are in R3, let me call Z the axis like this, okay, rotation, okay and fix some height, a level, so you are in R2 here, okay and then if you call the coordinates in this plane XJ and YJ then if you complexify the plane here which you just say you are looking what's happening for XJ plus IYJ for one filament J of circulation gamma J, there is a system that is supposed to model this evolution in fluids which is due to Klein, Maida and Damodaran and actually it was due prior to Zakharov precisely in the case where N equals 2 and you have, so you have 2 a pair of vortex filament with opposite circulation, so it's precisely the slide you had before, the situation you had before. So Zakharov was interested in collision for the anti-parallel, what is called the anti-parallel pair of vortex filament and he derived all of these models. Of course Klein, Maida and Damodaran really wrote a paper, generalized to several vortex filament and derived in a more clear way this model. So let me say a few words about this model, so it is lucky for us, the dispersion community, it is a Schrodinger system, so why a Schrodinger system? This can be realistically explained by the following two facts, you have the linear Schrodinger part is supposed to model the interaction of the filament with itself. Actually the linear Schrodinger part is actually a very rough approximation of the binormal flow or vortex filament equation which is a model for the evolution of one single filament and it's a model which is more complicated actually is in link with the cubic NLS. So in here one takes just the linear part, actually if you plug this ansatz epsilon, small spot perturbation into the binormal flow you will obtain as a leading turner in epsilon you will obtain the linear Schrodinger equation. So this is for the interaction of one filament, the J's filament with itself and then you have an extra term which is supposed to model the interaction, the influence of other filaments to the J's filament and actually this kind of interaction is precisely the one that appears in the case of 2D fluids when you have point vertices, so with the reduction to 2D and the, well of course one thing that one could criticize for this model is that roughly in some sense the influence on the J's filament from the other filaments is supposed to come only at fixed highs through the point vortex system. In some sense it's kind of you don't see the interaction from the other heights from the other filaments but I mean this is the model and this is the only one we have. So of course if you freeze the Z variable you get as a particular solution of this problem fragments that are located precisely in places in R2 evolving through the vortex point system. So these are particular solutions and we are studying perturbations, small perturbations near parallel at infinity of this particular solution of the system. Okay so of course I criticize quite a lot more than maybe I should have done on the model but as a counterpart except the fact that well this is the only model we have it has a very nice mathematical structure which probably means something physically. Okay so it is a Hamiltonian and it has several conserved quantities. In blue I put what corresponds to the point vortex system in 2D as conservation laws and I put it in here because of course you have to renormalize things because your system of Schrodinger equation is not with the vanishing condition at infinity. You are near some constant depending on time. Okay so before going further with filaments I will say a few words about what is known for the point vortex system because we shall work around straight parallel light located in the point vortex system solution. So the first thing is that if circulations are all positive, gamma j are all positive then the solution exists globally in time. In here in this context global existence means we want to avoid collision. Collision means that at some time well two point vortices just bump one into each other and at the level of the filament collisions will be there is a time and allocation such that when your two filaments get one into another like this. Okay so I was saying that when circulation is all the same the point vortices system evolves but you never have collision for a simple fact that on the first line if by any chance you have collision meaning xj getting too narrow to an xk then your logon will blow up and then in order to have conservation you have to have another pair at least of j of k such that the difference goes to infinity and then you have a contradiction from this part which says that all differences must be bounded uniformly in time. So when you have a circulation of same sign you have global existence and you don't have collision for the vortex point system. Well in the particular case n equals 2a you actually have that the distance is preserved between two vortices so the point translate in the case of opposite circulation and rotate otherwise and the case of the aircraft at the beginning is the case where the two point vortices circulate translate with uniform velocity depending on the initial distance and we're looking for a collision of perturbation of straight filament located in that pair of vortices. Okay things even at the 2d level start being more and more complicated of course when you increase the number of elements of points so already when n equals 3 of course if you take a circulation that are not the same there is a simple example of 3 points well chosen such that the whole picture will eventually collide at all the 3 will collide at the same point the triangle will turn and shrink at the same time. And the last two things are some special configurations special solution of the point vortex system that I will use in my talk so it is known that an equilibria for the point vortex system is the one where you place you take the vertices of n-poligon regular polygon for instance hexagon and you put the same circulation and then it is known that this configuration if you let it evolve to the vortex point system it will just rotate with some angular velocity which is determined by the circulation and the radius of the hexagon. And also it is known that you can allow also to take the center to put a vortex point in the center of the configuration with other circulation than the one in here and the angular velocity is computed again in terms of these. So these are the results for the point vortex systems and now maybe half of my talk will be about what is known about the Klein-Mind the modern Zakharov system for vortex filaments that we saw on the second slide and I will insist on what was done before for two reasons. First in some sense to convince you that it is a nice system because precisely we will see that it has many classes of particular solutions and also because there are not so many results so we can list them I hope all of them if I am not skipping some. The reason there are so few results is two-fold. One from the fluid dynamics community of course if you are trying to work directly in the fluid equation the situation is too singular. We are working on this model which can be criticized a lot of course but in any case the situation from fluids is too singular and on the other hand why this hasn't been studied too much for instance by the dispersion community because eventually you end up with a Schrodinger system. It is that I mean at least from my point of view of course many people didn't know about the model but also because it is not just a Schrodinger usual system it is a Schrodinger system with non-vanishing condition at infinity which means more Gross-Petitivsky equation and if you follow what is happening already in the Gross-Petitivsky equation community it is concerning really evolutionary time solution not just a stationary solution and started just around the years of 2000 I guess. So I think that are some explanation so let me point out what are the results we have. So first the results obtained by people that derived this model when deriving this model they conjecture that precisely one have to look for collision in the case of two anti-parallel vortex filaments so it was a conjecture and also they predicted that in case of same circulations one should have global existence on a collision and only numerical computation were given to to to comport these these conjectures. Okay another result due to Lyon's and Maida is concerning the statistical theory for this this configuration of several clusters of of vortex filaments but of course in their case they they took in account a very weak formulation of the equation which doesn't allow you to look precisely for collisions and then König-Ponsevega looked at the system and they consider small perturbation in at the H1 level they prove on one hand the local existence in for any initial configuration Xj and any circulation and they manage to extend the local existence of course basing themselves on an energy argument so the energy they use is the one here which is the Hamiltonian plus actually this part in the case where the distance between any two point vertices is the same this part is precisely the separation momentum that I show initially divided by this guy or equivalently this guy in this setting and then once you have this energy conserved you you manage to to obtain a global existence results of course this this condition is very restrictive because if you have points in the plane that are constrained to have same distance from one to each other then of course you can have only two points any two points or the vertices of an acrylatorial triangle okay also there there are work that have been done and are currently done for nearly parallel vortex filaments in the 3d Ginsberg-Landau equation and I think Roger I will maybe try to ask more about this tomorrow and then I had several results with Evelyn Mio that I will explain in the next slides and the last but not the least there is a result recent of Garcia-Spacia Craig and Young which treated the time and space solution periodics close to the correlating pair of filaments so same circulation at opposite okay so these are the known results and now I said that I will recall what we have done with Evelyn so first since you have in mind that you are in a Gross-Petyevsky setting you you try to solve to see everything in terms of the energy instead of just H1 and because of course small H1 implies small energy but not conversely and also I mean just the initial system is invariant under rotation and if you take a rotation the differences of course are just constant and are not in H1 and moreover also in the spirit of Gross-Petyevsky equation the energy space contains the gray soliton meaning traveling waves with the finite energy which are important solutions for the Gross-Petyevsky equation and also in here we'll see that we have something similar and the proof of course uses a fixed point argument that allowed the growth of the L2 so you don't stay small in H1 you allow the L2 norm to grow in time of course now if you want to pass from local to global existence you need the energy as before and actually you have seen it in the slides before it is conserved for the equilateral triangle and for two points by simply dividing some concepts quantity over the distance between the points but actually you can go further and compute precisely how the energy for instance if you take the square if you take a square or a hexagon here if you can express it in terms of the non-conserved quantities and to see what is left aside and see if by some symmetry imposed to the system you don't have conservation of the energy and actually you realize that if you want this you really have to impose very much symmetries either I mean not both on the initial configuration of the xj and also on the perturbation and actually this kind of symmetry is preserved if you just at each height you at each height let's say we are at height z you just do multiplication by a complex number so you delay it by the modulus of the complex number and you rotate everything by the argument of the complex number so your hexagon will still be a hexagon at each height z and this is something which is preserved by the system and the ansatz is the one there so you multiply all the stride filament by the same thing at each height and then actually you see that okay the result is that you have global existence you see that the equation the system all the system initial system reduce to this very nice equation omega is the angular velocity of your regular polygonal configuration of point vertices and of course it is a really a gross p-type equation you have a natural energy and actually realize that the energy of the system is n time this energy and then of course you extend to global existence so now that we have this very nice equation which is gross p-type like you can you can derive many classes of solutions and of course the first of them you can obtain as is done in the gross p-type setting a little bit more complicated but not too much really not too much you can obtain a traveling waves and you have actually you can transcribe I mean you don't even try scrub you just say it works I mean it's very simple to transcribe everything from gross p-type skip so then this means to view so okay let's say it is a square so this means that on each filament you have something that is moving and this is one single and also if you use the Galilean invariance of the system Schrodinger system initial Schrodinger system you realize that you can you can transform these things into four helices that I want to draw very nicely okay whatever no this is not good okay okay so on each helices helix helix you have something some some distortion that moves along it so why talking about helices because precisely Hopfinger and Bruand wrote a paper in nature when they showed that you can really it is in superfluids you can exhibit helices with with the distortion that that evolves like this for several vortex filament there are paper words to comfort the fact that hasimoto's work on one filament gives you the picture when you have only one helix with distortion moving on it but actually in the lab experiment you have several so this fits with what is seen at least in superfluids okay so then collision let's not forget that the exciting point is collision for vortex filaments well in this case of taking regular xj configuration and symmetric perturbations so ending up with that gross petyevsky equation when the angular velocity of your initial configuration is zero your gross petyevsky equation is just the linear Schrodinger equation and collision means that you want everybody to collapse meaning that you are looking to to a place time and place where v tz equals zero okay so then you can just just write by hand a very simple linear Schrodinger solution that cancels at time zero and space height zero which means that your all your vortices collide of course for having the angular velocity equals zero actually you have to put a point in the middle of your polygonal configuration so this means that you have at least three point vortices so you don't cover the case of pairs of filaments so now we are getting back to the problem of two filaments which is the aim of my talk okay so Zakharov suggested that solution should be searched under some that symmetry okay of course along the anti-parallel vortex filament and then the the problem reduces to this Schrodinger equation with a quite exotic term in here and what you are interested in are solutions for this Schrodinger equation that initial time don't vanish and that finite time vanish at some point so it is a quite non-non-usual study of the Schrodinger equation however I did a pleasure to quote a paper by Marla and Zag which is not so quotey in general but yeah yeah we are the first one so they had the same type of problem so Quensic for the vortex air connection in another situation which ended up with a with a problem of this type of finding finding some solution vanishing at the point in finite time for the nonlinear wave heat heat sorry heat equation and actually for this purpose we try to follow of course the thing the paper but there were some major issues so the idea in the paper is to replace your unknown function by one over your unknown function so you are looking for a solution that blow up at one point of course in the case of the heat equation there is another previous paper of Marla and Zag that gives you such a dispersive blow up wave a heat and so sorry heat solution and that they manage then to put it as an answer in their equation in our case well you replace psi 1 by 1 over f then you end up with the Gross-Pitevsky type again equation nonlinear equation with some nasty term derivative terms and everything well you will need some object around which you want to do your your ansatz such an object that you can find it now in the literature is a paper by Bono, Ponce, Sparber, So and Sparber so Gross-Pitevsky solution that displays a dispersive blow up at the point but of course the regularity is such that you don't manage in our case to write an ansatz in this equation transform with some derivative terms so it doesn't work so we get back to Zakharov work which said just this one can look for a self-seminar solution with that conditions at infinity so straight line and of course there is no proof in Zakharov's paper and in Maida and Bertotzi's book it's put as a major open problem to find such a similar solution and that's what we have done with Juan Faway and Evli Mio so we constructed first a self-seminar solution that will give you collisions of filament so it is something like this and of course as times go to zero they will end up colliding and that collision time it will be just across cross lines of course you are not in the case you want it initially because your your filaments are bent at infinity but then what we do is that we manage to unbend them because now we have the ansatz the ansatz will be precisely the the self-seminar solution and actually up posteriori this is a very natural thing you are looking to some singularity formation well you have lots of chances to find it by using some self-seminar feature so we managed to find a self-seminar solution for our model then we cut it and we completed to an equation of the of the initial system so eventually your solution is really like this so you have collision in finite time at one point so I will say in the five minutes I have left we say some words about the proofs so okay okay okay you can do the same thing so here it was for two vortex filaments actually you can take the symmetric configuration and then write a self-seminar solution for the that gross pitevsky equation that governs the the evolution and then you will end up by the same argument by constructing also a configuration that will collide at at the middle in finite time okay so some words about the proof so for the self-seminar profile you just look at the profile equation you you cut it in to I mean you rewrite the unknown function v you call it this and then you have some equation in v where you you can express it in terms of v so you end up by doing a fixed point which is not so complicated in a space that allows you to control the denominator so it's it's not so complicated so this is for for the self-seminar profile you start from infinity and you just construct it and now for the for the for the system for two nearly parallel filaments well you put these answers as we said then you write the equation for your reminder r you want that this little r won't ruin what's happened at the collision time meaning that remember that you want a collision meaning that you don't have collision before the collision the expected collision time and collision at the expected time and this is what you already have for the self-seminar solution now when you add that little r you don't want it to give you some cancellation before I mean yeah if you give I mean you you you don't you want to keep your self-seminar behavior leading with respect to to little r so for doing this you have to impose that little r will go to zero faster than the solutioner profile just this okay so you have an equation for little r with some some terms whatever and then you will do a fixed point as usual decay in in h1 with some rates of decay and moreover you will have to ensure yourself that locally at at zero at the place you want to see the collision you are decaying to zero less faster than the self-seminar profile at the constant the constant of of decay will be smaller okay and then of course this this kind of sinks one can see it also has been done for instance in the case of blow up at critical mass for cubic NLS on to the domains by butchers and that cough you take the the self-seminar solution blowing up on our to you cut it and then you you construct a little reminder are that goes to zero so it's it's the natural thing to do and then you for treating the equation for doing the fixed point you just get point-wise estimates near zero and that infinity and then also I'll to and h1 estimates for the reminder terms in the equation of are and then you just use trick arts estimates and some commutator with the localization for the and infinity estimates near zero and that's all okay thank you for attention so okay so when you look at it just one way you think I don't know there well actually if you look at what is the solution at time zero okay at time zero you'll have and in zero you'll have this guy is real this is zero okay and then actually you have several depending on how you've chosen your your cutoff yeah yeah yeah that's for sure but then the shape you have several I mean you yeah you'll have you'll have a corner but then you'll have some this part I mean yeah but so can you do that you know our self-summar solution are not unique and actually we can talk about it but there should be another one also okay so of course that's what we are why we are working on models I expect this will work but to to see I mean this model is supposed to be valid up to distance between the filaments is much larger than the core size of the tubes so of course game over at some point before this beautiful picture happens but but but but but my dad but what is book says that it is seen in numerical simulation in in fluid dynamics equation that when when the you you are at the balancing point where separation distance and core section are equivalent then it you will have a reconnection no I don't know I really don't know and you'll from free dynamics you just have numerics and well they should be interesting to see with this kind of of solution watch will give us through numics in free dynamics that that will take I mean I you have to have super computers of what the money is not okay thank you
 So first I would like to thank the organizers for their very nice invitation and for organizing  also all this semester.  So today I will talk about a joint work with Erwan Fawou and Avilinho that concerns vortex  filaments and more particularly parallel vortex filaments in case they are colliding.  So what is a vortex filament?  It is a fluid where the vorticity is concentrated along a curve in R3 and of course collisions  and the reconnection of vortex filament is something that happens in turbulent fluids  so it's an important phenomena but of course it is very difficult to capture this kind  of things mathematically.  However it is expected that one situation which is rather simple to understand is the  case of two almost parallel vortex filaments with opposite circulations as it was suggested  by Crowe's work in the 70s.  So it is the case, I mean it is the situation that you can see actually in the sky if the  weather is good.  So you see the two trace of the plane white trace in the sky so these are vortex filaments  with opposite circulations that propagate and eventually as times go by they finally  reconnect and well then start doing vortex ring and other things.  In English they are called con trails, condensation trails.  Ah, con trails, aircraft con trails, yeah.  Actually Crowe's paper appears in American Journal of Astronautics and Aerodynamics or  something like this which of course is a journal very hard to obtain for a mathematician.  However this picture you can see it in the sky and during your life now that you have  seen this talk you'll notice it at one point and you'll think about this talk.  Well, okay so now you want, this is what happened in real life, now you want to transcribe it  mathematically and of course through the fluid mechanics equation this is something very  difficult because the situation is very singular.  However there is at my knowledge one model that do this kind of modelization.  So first you should take a vortex filament, okay, let me call, so we are in R3, let me  call Z the axis like this, okay, rotation, okay and fix some height, a level, so you  are in R2 here, okay and then if you call the coordinates in this plane XJ and YJ then  if you complexify the plane here which you just say you are looking what's happening  for XJ plus IYJ for one filament J of circulation gamma J, there is a system that is supposed  to model this evolution in fluids which is due to Klein, Maida and Damodaran and actually  it was due prior to Zakharov precisely in the case where N equals 2 and you have, so  you have 2 a pair of vortex filament with opposite circulation, so it's precisely the  slide you had before, the situation you had before.  So Zakharov was interested in collision for the anti-parallel, what is called the anti-parallel  pair of vortex filament and he derived all of these models.  Of course Klein, Maida and Damodaran really wrote a paper, generalized to several vortex  filament and derived in a more clear way this model.  So let me say a few words about this model, so it is lucky for us, the dispersion community,  it is a Schrodinger system, so why a Schrodinger system?  This can be realistically explained by the following two facts, you have the linear Schrodinger  part is supposed to model the interaction of the filament with itself.  Actually the linear Schrodinger part is actually a very rough approximation of the binormal  flow or vortex filament equation which is a model for the evolution of one single filament  and it's a model which is more complicated actually is in link with the cubic NLS.  So in here one takes just the linear part, actually if you plug this ansatz epsilon,  small spot perturbation into the binormal flow you will obtain as a leading turner in  epsilon you will obtain the linear Schrodinger equation.  So this is for the interaction of one filament, the J's filament with itself and then you  have an extra term which is supposed to model the interaction, the influence of other filaments  to the J's filament and actually this kind of interaction is precisely the one that appears  in the case of 2D fluids when you have point vertices, so with the reduction to 2D and  the, well of course one thing that one could criticize for this model is that roughly in  some sense the influence on the J's filament from the other filaments is supposed to come  only at fixed highs through the point vortex system.  In some sense it's kind of you don't see the interaction from the other heights from  the other filaments but I mean this is the model and this is the only one we have.  So of course if you freeze the Z variable you get as a particular solution of this problem  fragments that are located precisely in places in R2 evolving through the vortex point system.  So these are particular solutions and we are studying perturbations, small perturbations  near parallel at infinity of this particular solution of the system.  Okay so of course I criticize quite a lot more than maybe I should have done on the model  but as a counterpart except the fact that well this is the only model we have it has  a very nice mathematical structure which probably means something physically.  Okay so it is a Hamiltonian and it has several conserved quantities.  In blue I put what corresponds to the point vortex system in 2D as conservation laws and  I put it in here because of course you have to renormalize things because your system  of Schrodinger equation is not with the vanishing condition at infinity.  You are near some constant depending on time.  Okay so before going further with filaments I will say a few words about what is known  for the point vortex system because we shall work around straight parallel light located  in the point vortex system solution.  So the first thing is that if circulations are all positive, gamma j are all positive  then the solution exists globally in time.  In here in this context global existence means we want to avoid collision.  Collision means that at some time well two point vortices just bump one into each other  and at the level of the filament collisions will be there is a time and allocation such  that when your two filaments get one into another like this.  Okay so I was saying that when circulation is all the same the point vortices system  evolves but you never have collision for a simple fact that on the first line if by any  chance you have collision meaning xj getting too narrow to an xk then your logon will blow  up and then in order to have conservation you have to have another pair at least of  j of k such that the difference goes to infinity and then you have a contradiction from this  part which says that all differences must be bounded uniformly in time.  So when you have a circulation of same sign you have global existence and you don't have  collision for the vortex point system.  Well in the particular case n equals 2a you actually have that the distance is preserved  between two vortices so the point translate in the case of opposite circulation and rotate  otherwise and the case of the aircraft at the beginning is the case where the two point  vortices circulate translate with uniform velocity depending on the initial distance  and we're looking for a collision of perturbation of straight filament located in that pair  of vortices.  Okay things even at the 2d level start being more and more complicated of course when you  increase the number of elements of points so already when n equals 3 of course if you  take a circulation that are not the same there is a simple example of 3 points well chosen  such that the whole picture will eventually collide at all the 3 will collide at the same  point the triangle will turn and shrink at the same time.  And the last two things are some special configurations special solution of the point  vortex system that I will use in my talk so it is known that an equilibria for the point  vortex system is the one where you place you take the vertices of n-poligon regular  polygon for instance hexagon and you put the same circulation and then it is known that  this configuration if you let it evolve to the vortex point system it will just rotate  with some angular velocity which is determined by the circulation and the radius of the hexagon.  And also it is known that you can allow also to take the center to put a vortex point in  the center of the configuration with other circulation than the one in here and the angular  velocity is computed again in terms of these.  So these are the results for the point vortex systems and now maybe half of my talk will  be about what is known about the Klein-Mind the modern Zakharov system for vortex filaments  that we saw on the second slide and I will insist on what was done before for two reasons.  First in some sense to convince you that it is a nice system because precisely we will  see that it has many classes of particular solutions and also because there are not so  many results so we can list them I hope all of them if I am not skipping some.  The reason there are so few results is two-fold.  One from the fluid dynamics community of course if you are trying to work directly in the  fluid equation the situation is too singular.  We are working on this model which can be criticized a lot of course but in any case  the situation from fluids is too singular and on the other hand why this hasn't been  studied too much for instance by the dispersion community because eventually you end up with  a Schrodinger system.  It is that I mean at least from my point of view of course many people didn't know about  the model but also because it is not just a Schrodinger usual system it is a Schrodinger  system with non-vanishing condition at infinity which means more Gross-Petitivsky equation  and if you follow what is happening already in the Gross-Petitivsky equation community  it is concerning really evolutionary time solution not just a stationary solution and  started just around the years of 2000 I guess.  So I think that are some explanation so let me point out what are the results we have.  So first the results obtained by people that derived this model when deriving this model  they conjecture that precisely one have to look for collision in the case of two anti-parallel  vortex filaments so it was a conjecture and also they predicted that in case of same  circulations one should have global existence on a collision and only numerical computation  were given to to to comport these these conjectures.  Okay another result due to Lyon's and Maida is concerning the statistical theory for  this this configuration of several clusters of of vortex filaments but of course in their  case they they took in account a very weak formulation of the equation which doesn't  allow you to look precisely for collisions and then König-Ponsevega looked at the system  and they consider small perturbation in at the H1 level they prove on one hand the local  existence in for any initial configuration Xj and any circulation and they manage to  extend the local existence of course basing themselves on an energy argument so the energy  they use is the one here which is the Hamiltonian plus actually this part in the case where  the distance between any two point vertices is the same this part is precisely the separation  momentum that I show initially divided by this guy or equivalently this guy in this  setting and then once you have this energy conserved you you manage to to obtain a global  existence results of course this this condition is very restrictive because if you have points  in the plane that are constrained to have same distance from one to each other then of course  you can have only two points any two points or the vertices of an acrylatorial triangle  okay also there there are work that have been done and are currently done for nearly parallel  vortex filaments in the 3d Ginsberg-Landau equation and I think Roger I will maybe try  to ask more about this tomorrow and then I had several results with Evelyn Mio that I  will explain in the next slides and the last but not the least there is a result recent  of Garcia-Spacia Craig and Young which treated the time and space solution periodics close  to the correlating pair of filaments so same circulation at opposite okay so these are  the known results and now I said that I will recall what we have done with Evelyn so first  since you have in mind that you are in a Gross-Petyevsky setting you you try to solve to see everything  in terms of the energy instead of just H1 and because of course small H1 implies small  energy but not conversely and also I mean just the initial system is invariant under  rotation and if you take a rotation the differences of course are just constant and are not in  H1 and moreover also in the spirit of Gross-Petyevsky equation the energy space contains the  gray soliton meaning traveling waves with the finite energy which are important solutions  for the Gross-Petyevsky equation and also in here we'll see that we have something similar  and the proof of course uses a fixed point argument that allowed the growth of the L2  so you don't stay small in H1 you allow the L2 norm to grow in time of course now if you  want to pass from local to global existence you need the energy as before and actually  you have seen it in the slides before it is conserved for the equilateral triangle and  for two points by simply dividing some concepts quantity over the distance between the points  but actually you can go further and compute precisely how the energy for instance if you  take the square if you take a square or a hexagon here if you can express it in terms  of the non-conserved quantities and to see what is left aside and see if by some symmetry  imposed to the system you don't have conservation of the energy and actually you realize that  if you want this you really have to impose very much symmetries either I mean not both  on the initial configuration of the xj and also on the perturbation and actually this  kind of symmetry is preserved if you just at each height you at each height let's say  we are at height z you just do multiplication by a complex number so you delay it by the  modulus of the complex number and you rotate everything by the argument of the complex  number so your hexagon will still be a hexagon at each height z and this is something which  is preserved by the system and the ansatz is the one there so you multiply all the  stride filament by the same thing at each height and then actually you see that okay  the result is that you have global existence you see that the equation the system all the  system initial system reduce to this very nice equation omega is the angular velocity  of your regular polygonal configuration of point vertices and of course it is a really  a gross p-type equation you have a natural energy and actually realize that the energy  of the system is n time this energy and then of course you extend to global existence so  now that we have this very nice equation which is gross p-type like you can you can derive  many classes of solutions and of course the first of them you can obtain as is done in  the gross p-type setting a little bit more complicated but not too much really not too  much you can obtain a traveling waves and you have actually you can transcribe I mean  you don't even try scrub you just say it works I mean it's very simple to transcribe everything  from gross p-type skip so then this means to view so okay let's say it is a square so  this means that on each filament you have something that is moving and this is one single  and also if you use the Galilean invariance of the system Schrodinger system initial Schrodinger  system you realize that you can you can transform these things into four helices that I want  to draw very nicely okay whatever no this is not good okay okay so on each helices helix  helix you have something some some distortion that moves along it so why talking about helices  because precisely Hopfinger and Bruand wrote a paper in nature when they showed that you  can really it is in superfluids you can exhibit helices with with the distortion that that  evolves like this for several vortex filament there are paper words to comfort the fact  that hasimoto's work on one filament gives you the picture when you have only one helix  with distortion moving on it but actually in the lab experiment you have several so  this fits with what is seen at least in superfluids okay so then collision let's not forget that  the exciting point is collision for vortex filaments well in this case of taking regular  xj configuration and symmetric perturbations so ending up with that gross petyevsky equation  when the angular velocity of your initial configuration is zero your gross petyevsky  equation is just the linear Schrodinger equation and collision means that you want everybody  to collapse meaning that you are looking to to a place time and place where v tz equals  zero okay so then you can just just write by hand a very simple linear Schrodinger solution  that cancels at time zero and space height zero which means that your all your vortices  collide of course for having the angular velocity equals zero actually you have to put a point  in the middle of your polygonal configuration so this means that you have at least three  point vortices so you don't cover the case of pairs of filaments so now we are getting  back to the problem of two filaments which is the aim of my talk okay so Zakharov suggested  that solution should be searched under some that symmetry okay of course along the anti-parallel  vortex filament and then the the problem reduces to this Schrodinger equation with a quite  exotic term in here and what you are interested in are solutions for this Schrodinger equation  that initial time don't vanish and that finite time vanish at some point so it is a quite  non-non-usual study of the Schrodinger equation however I did a pleasure to quote a paper  by Marla and Zag which is not so quotey in general but yeah yeah we are the first one  so they had the same type of problem so Quensic for the vortex air connection in another situation  which ended up with a with a problem of this type of finding finding some solution vanishing  at the point in finite time for the nonlinear wave heat heat sorry heat equation and actually  for this purpose we try to follow of course the thing the paper but there were some major  issues so the idea in the paper is to replace your unknown function by one over your unknown  function so you are looking for a solution that blow up at one point of course in the  case of the heat equation there is another previous paper of Marla and Zag that gives  you such a dispersive blow up wave a heat and so sorry heat solution and that they manage  then to put it as an answer in their equation in our case well you replace psi 1 by 1 over  f then you end up with the Gross-Pitevsky type again equation nonlinear equation with  some nasty term derivative terms and everything well you will need some object around which  you want to do your your ansatz such an object that you can find it now in the literature  is a paper by Bono, Ponce, Sparber, So and Sparber so Gross-Pitevsky solution that displays  a dispersive blow up at the point but of course the regularity is such that you don't manage  in our case to write an ansatz in this equation transform with some derivative terms so it  doesn't work so we get back to Zakharov work which said just this one can look for a self-seminar  solution with that conditions at infinity so straight line and of course there is no  proof in Zakharov's paper and in Maida and Bertotzi's book it's put as a major open  problem to find such a similar solution and that's what we have done with Juan Faway and  Evli Mio so we constructed first a self-seminar solution that will give you collisions of  filament so it is something like this and of course as times go to zero they will end  up colliding and that collision time it will be just across cross lines of course you are  not in the case you want it initially because your your filaments are bent at infinity but  then what we do is that we manage to unbend them because now we have the ansatz the ansatz  will be precisely the the self-seminar solution and actually up posteriori this is a very  natural thing you are looking to some singularity formation well you have lots of chances to  find it by using some self-seminar feature so we managed to find a self-seminar solution  for our model then we cut it and we completed to an equation of the of the initial system  so eventually your solution is really like this so you have collision in finite time  at one point so I will say in the five minutes I have left we say some words about the proofs  so okay okay okay you can do the same thing so here it was for two vortex filaments actually  you can take the symmetric configuration and then write a self-seminar solution for the  that gross pitevsky equation that governs the the evolution and then you will end up  by the same argument by constructing also a configuration that will collide at at the  middle in finite time okay so some words about the proof so for the self-seminar profile  you just look at the profile equation you you cut it in to I mean you rewrite the unknown  function v you call it this and then you have some equation in v where you you can express  it in terms of v so you end up by doing a fixed point which is not so complicated in a space  that allows you to control the denominator so it's it's not so complicated so this is  for for the self-seminar profile you start from infinity and you just construct it and  now for the for the for the system for two nearly parallel filaments well you put these  answers as we said then you write the equation for your reminder r you want that this little  r won't ruin what's happened at the collision time meaning that remember that you want a  collision meaning that you don't have collision before the collision the expected collision  time and collision at the expected time and this is what you already have for the self-seminar  solution now when you add that little r you don't want it to give you some cancellation  before I mean yeah if you give I mean you you you don't you want to keep your self-seminar  behavior leading with respect to to little r so for doing this you have to impose that  little r will go to zero faster than the solutioner profile just this okay so you have an equation  for little r with some some terms whatever and then you will do a fixed point as usual  decay in in h1 with some rates of decay and moreover you will have to ensure yourself  that locally at at zero at the place you want to see the collision you are decaying to zero  less faster than the self-seminar profile at the constant the constant of of decay will  be smaller okay and then of course this this kind of sinks one can see it also has been  done for instance in the case of blow up at critical mass for cubic NLS on to the domains  by butchers and that cough you take the the self-seminar solution blowing up on our to  you cut it and then you you construct a little reminder are that goes to zero so it's it's  the natural thing to do and then you for treating the equation for doing the fixed point you  just get point-wise estimates near zero and that infinity and then also I'll to and h1  estimates for the reminder terms in the equation of are and then you just use trick arts estimates  and some commutator with the localization for the and infinity estimates near zero and  that's all okay thank you for attention so okay so when you look at it just one way you  think I don't know there well actually if you look at what is the solution at time zero  okay at time zero you'll have and in zero you'll have this guy is real this is zero  okay and then actually you have several depending on how you've chosen your your cutoff yeah  yeah yeah that's for sure but then the shape you have several I mean you yeah you'll have  you'll have a corner but then you'll have some this part I mean yeah but so can you do that  you know our self-summar solution are not unique and actually we can talk about it but  there should be another one also okay so  of course that's what we are why we are working on models I expect this will work but to to  see I mean this model is supposed to be valid up to distance between the filaments is much  larger than the core size of the tubes so of course game over at some point before this  beautiful picture happens but but but but but my dad but what is book says that it is  seen in numerical simulation in in fluid dynamics equation that when when the you you are at  the balancing point where separation distance and core section are equivalent then it you  will have a reconnection no I don't know I really don't know and you'll from free dynamics  you just have numerics and well they should be interesting to see with this kind of of  solution watch will give us through numics in free dynamics that that will take I mean  I you have to have super computers of what the money is not okay thank you
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