 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 Eveline Mio 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 3D 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 and 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 Crow'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 they finally reconnect and well then start to start to doing a vortex ring and other things. Actually Crow's paper appears in American Journal of Astronautics and Dynamics 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 modellization so first if you 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 then if you complexify the plane here which you just say you are looking what what's happening for xj plus i yj 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 do prior to Zakharov precisely in the case where n equals 2 and you have so you have to 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 also this model 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 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 it's in link with the cubic NLS so in here we one takes just the linear part actually if you plug if you plug this ansatz epsilon smaller spot perturbation into the binormal flow you'll obtain as a first amina as a leading terminal in epsilon you'll obtain the linear Schrodinger equation okay 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 this kind of interaction is precisely the one that appear in the case of 2d fluids when you have point vertices so with the reduction to 2d and well of course one thing that one could criticize for this model is that roughly in some sense the the influence on the J's filament from the other filaments is supposed to come only at high 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 if you freeze the the Z variable okay you get as a particular solution of of this problem filaments that are located that are located precisely precisely in places in R2 evolving through the vortex point system okay so these are particular solutions and we are studying perturbations small perturbation 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 and 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 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 normalize things because your system of Schrodinger equation is not with the vanishing condition at infinity you are near some some constant depending on time okay so before going further with filaments I will say a few words about what is known for the for the point vortex system before because we shall work around stride parallel and 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 like globally 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 such that when your two filaments get okay so I was saying that when circulation are all the same the point vortices system evolves but 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 it 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 for from this part which says that all differences must be bounded uniformly in time so when you have a circulation of same sign the you have global existence don't have collision for the vortex point system well in the particular case n equals 2 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 we the case of the aircraft at the beginning is the case where the the two point vortices circulates translate with uniform velocity depending on the initial distance and we'll looking for a collision of perturbation of straight filament located in that pair of vortices okay since even at the 2d level start being more and more complicated of course when you increase the number of of of elements of points so already when n equals 3 of course if you take a circulation that are not the same and there is a simple example of three points well chosen such that the whole picture will eventually collide at all the three will will collide at the same point the the the triangle will turn and shrink at the same time and the the last two things are some special configurations special solution of the point vortex system that I will use in in my talk so it is known that an equilibria for the point vortex system is is the one where you place you take the vertices of n polygon regular polygon for instance hexagon okay 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 this this swaps okay so these are the results for the point vortex systems and now maybe half half of my talk will be about what is known about the Klein mind the modern Zakarov 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'll we'll 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'm not skipping some the reason they 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 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 too singular and on the other hand why this hasn't been studied too much for instance for 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 non vanishing condition at infinity which means more gross p type ski equation and if you follow what is happening already in the gross p type ski equation community results concerning really evolutionary in time solution not just a stationary solution and started just around the years of 2000 I guess okay so I think that's that are some explanation so let me point out what are the what are the results we we have so first the results obtained by people that derive 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 convert to these these conjectures okay another result due to Lyons 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 for collisions and then Koenig-Ponce Vega 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 vortices is the same this part is precisely the separation momentum that I show initially divided by this guy or equivalently this guy in this 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 Ginsburg-Landau equation and I think Roger I will maybe talk to us more about this tomorrow and then I had several results with Eveline 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 so now I said that I will recall what we have done with Eveline 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 more H1 implies most energy but not conversely and also I mean just the initial system is invariant on the rotation and if you take a rotation the difference is of course we are just constant and they're 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 conserved 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 a 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 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 delayed 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 by the system and the answer is the one there so you multiply all the straight filament by the same thing at each height and then actually you 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 is the the angular velocity of your regular polygonal configuration of point vertices and of course it is really a gross p-type equation you have a natural energy and actually realize that the energy of the system is and time this this energy and then of course you 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 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 transcribe 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 one thing 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 wanted very nicely okay whatever no this is not 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 their paper was to comfort the fact that has emotions work on one filament gives you the picture when you have only one helix with the 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 the regular xj configuration and symmetric perturbations so ending up with that gross pitaevsky equation when the angular velocity of your initial configuration is zero your gross pitaev 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 phi 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 z and space height zero which means that you are all your vertices collide of course for having the 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 vertices so you don't cover the the case of pairs of filaments so now we are getting back to the the problem of two filaments which is the aim of my talk okay so Zakharov suggested that solution should be search 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 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 it is a pleasure to quote a paper by Marla and Zag which is not so quoted in general but yeah yeah we are the first one so they had the same type of problem so quenching 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 non-linear wave hit hit sorry hit 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 managed then to put it as an ansatz in their their equation in our case well you you replace XI1 by 1 over F then you end up with the Gross-Pitevsky type again equation non-linear 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 Bona 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 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 Bertoltz's book it's put as a major open problem to find such a similar solution and that's what we have done with the one for when they've been me also we constructed a first a self-seminar solution that that 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 a cross cross lines of course you are not in the case you wanted initially because your your filaments are bent at infinity but then what we do is that we managed to unbend them because now we have the ansatz the ansatz will be precisely the the self similar 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 similar 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 similar 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 some words about the proof so for the self similar profile you just look at the profile equation you you cut it in to I mean you really 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 so complicated so this is for for the self similar 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 are you want that this little r won't ruin what's happening 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 is what you already have for the self similar solution now when you add that little r you don't want it to give you some cancellation before you don't you want to keep your self similar 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 self similar 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 similar 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 Bertrand Wettkopf you take the the self similar 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-wide estimates near zero and that infinity and then also I'll to and they try and 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 an infinity estimates near zero and that's all okay thank you for attention so okay so when you look at it just one way 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 yeah you'll have you'll have a corner but then you'll have some this part I mean yeah no our sub-summoner solution are not unique and actually we can talk about it but there should be another one would survive if you fatten the people a little bit if you think in which frame okay so of course that's what we are why we're working on models I expect this will work but to 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 but my time 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 quarter section are equivalent then it you will have a reconnection no no I don't know I really don't know and you will from free dynamics you just have numerics and well they else should be interesting to see with this kind of of solution which will give us through numerics in free that that that will take I mean I you have to have super computers so what I mean is not thank you