 So, thank you. And thanks to the organizer. We're pleased to be here. So, the theorem that I will mention in the second part of the talk is a theorem with Bob Gerard regarding the leapfrogging phenomenon for the Gross-Bitaevsky equation. I've been working with Philippe Gravesger and Eveline Mio trying to extend this analysis or to adapt this analysis to the Euler equation and we have failed so far. But I will start with some basic things on the Euler equation and try to give the intuition. So first what leapfrogging is, what are the difficulties that are still there and then I will show the result for Gross-Bitaevsky and explain the proof and you will see also where the gaps are for the Euler equation and maybe that will give some ideas to some of you or some of you would have some ideas to make some progress. So, there is a pointer. So, I will start with the 2D Euler equation. So, the 2D Euler equation which I wrote immediately in a vorticity formulation. So, omega is a function here and say on R2 and V is a velocity field of the fluid which is recovered from the vorticity through the Biosavaro, so the gradient perp of the green function of the Laplacian and as you know this equation when in R2 is a transport equation. So, the initial vorticity field is just transported by the flow which is given by the velocity field. So, you think of particles, you evolve them with the velocity field and then you just transport the vorticity by this by the solution of this flow. This is known to have an infinite number. First because it is a transport equation, all functions of the vorticity are conserved in particular the total mass but there are other interesting conserved quantities in particular the center of mass which I have risqué here the angular momentum which is sometimes written in terms of the center of mass center that the center of mass already and the energy. There are many much more conserved quantities and one of the consequences of these conserved quantities among the many consequences of these conserved quantities is the fact that when you have a positive vorticity at initial time then it remains somehow concentrated for all times. So, if it has compact support say at times equals zero then most of its mass will stay in a in a bounded region for all time because that depends only on conserved quantities and very easy way to see it but it of course it requires that omega that the vorticity is a constant sign otherwise you see that it doesn't work so I split the mass as a part which is in a ball around the center of mass of omega and the the complement now the the part which is on the complement you just since it is x is a distance that at least are from the center then you can minimize one by this and and and there you see the the one of the conserved quantities appear the angular momentum and so therefore you have this lower bound on the mass of vorticity contained in the balls of radius r centered at this center of mass which does not move for the for which is a conserved quantity for the for the Euler equation in terms of the in terms of the total mass so if capital r is sufficiently big you can make that as as small as you wish so it is a weak form of concentration of the vorticity but of course this only works for this will have a an implication for later this only works if omega is a sign because otherwise you cannot do that okay and there are solutions of course which are known not to stay compact in in space you well known solutions are if you take a patch with a plus and a patch with a minus those two will behave like a traveling pair and so they will not stay localized say here close to zero in space but you could say well but they are still localized one close to each other but you could cook up a different thing like that and you would have two pairs that that escape one from each other so you take different signs and so there there is no way that that it would be localized then so so this is something which is for which the sign of the vorticity is is is crucial and now the kind of solution I will look at is precisely such kinds of initial data where I will put a scale epsilon and so I will consider patch which are this just to to put a parameter in the problem I could also use the distance between patch and make it big but it is somehow easier this way so they will behave with a diameter of size epsilon and so that they have positive mass the omega will be like epsilon minus two to the minus two into the uh in those patch so if you do that of course they and you rewrite this estimate for one patch then you you have that uh up to a factor epsilon square all the mass is still contained in a ball of radius of radius one but this of course only works if you take one patch because uh because the smallness the smallness of l that the smallness of l of course x square is small only if omega is concentrated close to a point but if you take two of them then if x square is small here it's not gonna be small there and so this this very simple heuristic does not work if you take if you take multiple patches okay so you cannot just use the invariant in that case to to tell that and the goal would be to say that such structure remain in time possibly with the motion that you will later study but but it's not sufficient to use the invariance to do this but what mark euro and pulverine t did uh in in the 80s is to say uh well um the what we are going to say so the the bio Savard law is a is a linear law that that that transports the omega to the v and since the Euler equation is a transport equation if initially your uh vorticity can be divided in two say i have one patch and a second patch i transport each of these patch then i have a way to split the vorticity at later time so i and this is just because it is a transport equation and so the bio Savard law is singular only at the origin okay and so when when this patch is going to move by the Euler equation i i can consider the the speed the speed which is generated by himself by bio Savard and look at it as the singular object but then there is also the speed which is uh which is implied by the other patch and since initially they are at a finite distance say o of one distance then as long as they remain like that i can treat this other term as a as a smooth or as a leapshitz term okay so what they did was just now even if you have multiple patch just consider one of those patch it will it will end and move it with a flow where there is a velocity which is given by the bio Savard law of itself and then the bio Savard law implied by the other patches you just see it as a forcing term as a leapshitz forcing term and it is leapshitz as long as the two patches do not come close one to the other you reformulate this is an abstract as an abstract flow and then you compute the the time derivative of the invariance or what was invariance for the pure Euler equation and you just get that the time and the time derivative of the center of mass now is just moved by the by the force f and and the important thing is the the angular momentum here satisfies this equation and if you notice that this quantity is zero just because this does not depend on x so if you integrate this on x by definition of pt you get zero forget about this put it out of the out of the integral so this is zero now you compare this with this and you make the difference you have f at x and f at the center of mass so since f is leapshitz the distance you can bound it by x minus pt and then you get the x minus pt at the square times omega which is which is LT so the the angular momentum under such kind of flows is a good candidate for for a grand valley inequality so if it was epsilon square initially it remains it remains of size of order epsilon square for times of of order one but of course there was an a priori assumption is the fact that there is no interaction between the two and and and so so you need kind to to bootstrap and to be able to say that if they are initially at finite distance then at least for an order one time that will remain the case and and this is a difficult that is a something which is not easy to prove and it is actually the difficult part in all the work of of marquero and pulverentine in rough terms this is exactly what they proved so if you start with patches that time that have size epsilon whatever their their their form their precise shape they do not need to be disks and at at at a distance which is order one then for times of order one they still remain separated and they are no longer of size epsilon after a positive time they cannot prove that by the way this is kind of an open problem called the the filamentation problem if you start with something of radius one how far can the the vorticity go and and one you once you want they are able to prove that the the vortex patch remain at order one distance one from the other then deriving their motion is just looking at the at the at the PDE and and and exchanging the omega by a delta and the fact that they are concentrated just say that you approximate this is just a consistency if you want and so once you know that the vort the patches are concentrated you derive the motion low the the so-called point vortex systems for the for the for the patches but so this is the difficult part of the proof and and this is the content say this is the consequence okay this is what i noted there and this system is an Hamiltonian which is which i wrote there the Kirchhoff Hamiltonian here and actually of course it has to do with the energy of the the Euler energy of such configurations and which i wrote which i wrote there and the reason why i wrote it here was that the the Hamiltonian of the point vortex systems here only depends on the relative positions of the patch one with each other so x i is the the center of mass of the patch i and x j is the center of mass of the patch j and they are small so so at the limit you consider them as only as points and the the energy as a main term if x i x j as distance one this is the term which is order one as a main term which is logarithmic diverging in epsilon but which does not depend on the position of the points okay and so that's the main that's the the main ingredient in the in the fact that you can use this this analogy here somehow you can treat the different patches separately okay that can be reflected in a way in the fact that the main term of the energy is somehow independent of the position of the point that would be something will be different in in the in the axis symmetric 3d in in Gospitaevsky so this is something i wanted to so this is just the the integral of one over x cut off at at epsilon oh that one has been thoroughly studied depending on the on the science well if the if the science of the lambda i lambda j are all the same then then you don't have collision problem it's no if you have different signs then you know you have those solutions that that go live and so this is a problem that they've been the long time we have this flow is not necessarily right this is finite time so yeah there is a double scale there that should be okay yes one remark also is that this this theorem by by marco and polverenti works for for arbitrary arbitrary shape of patches in particular it does not use this at all the some optimizing property of the patch okay it could be an ellipse it could be well an arbitrary patch and so somehow it's not if you want to to make a parallel with say multi-solid turns and so on it has nothing to do with that in the sense that it does not use is it does not use is the fact that for example one would be an optimum of the energy at a constant momentum and so on it just uses the fact that if they are sufficiently far apart then they do not interact it's just a long-range non-interaction and what could say yes what can we say more if the patches somehow have some solitonic behavior in particular if there are some optimum of some conserved quantities and there are some constraints and so those objects are are known and the the optimum are known so for example if you if you consider the circular patch then its ability has been has been studied and in particular one and polverenti have proved that if you consider a patch which is initially sufficiently close to the circular patch in l1 then it's it says close in n1 for all time also because this just follows from inequalities on conserved on sorry on yes on this conserved quantity here just the the the energy and here also the remark that I would like to make having in mind parallel with interaction of soliton and so on this looks like a very good so omega not is the circular patch and omega is here is the is the the candidate patch this looks like a non-degeneracy condition the fact that there is a spectral gap it actually isn't and the reason is the lack of differentiability of the structure the manifold on which you look at the the manifold of vortex patches is not something which is a nice vector structure and perhaps the easiest way to see it is to consider the vortex patch so this is a quantity date now another quantity date would be there to dig a hole here and to put it here of course so you're in this you can do in one way you can do it with a plus but you cannot do it with a minus so if this is a permitted direction of variation the opposite of this variation is not permitted okay so somehow there is not a it's not a there there's no tangent space and and you cannot see this inequality as a non-degeneracy inequality of a second derivative of something okay so this is the main the main difference and one other way to to see that this would be observed to think in in this way is to replace the l1 norm by the l2 norm you would you would just get a square here and a power four here and then you would say oh this is not at all a non-degeneracy it is a degenerate it is a degeneracy okay so this is one of the reasons why Euler with respect to Gross-Betaevsky that I will present later is is badly behaved is the fact that the the co-adjoint orbit of a vortex patch so the set of of candidates for Euler flow starting for for a patch does not have at least for the l1 norm which is the one that has been studied does not have a good manifold structure so this was a stability theorem in l1 for initial vortex patch that were close to close to the circular vortex patch and Sideris and Vega in in a short and elegant paper proved that it is actually a global non-linear stability estimates but they have a proof which relies on on different conserved quantities than E so this was based on the energy and they used the conservation of momentum and angular momentum and again here I have I have put this factor two which somehow is the main one of the main trouble and also the l1 norm is is something you would like to avoid I will explain later so now I go to so leapfrogging is something about vortex strings so it's not in 2d it's in 3d but we'll look at the analysis in in 3d axis symmetric flows without swirl here so I wrote the the Euler equation in in 3d again you recover the velocity from the vorticity through the BOS avarlo and I will only consider vorticity that are independent of so I look in cylindrical coordinates so z r and theta and I will always use this so this is r z this will be called h so this is the half plane which you obtain in cylindrical coordinate and I forgot about the theta so if omega is is independent of theta and aligned with e theta in in that case the velocity is is a vector field in this in this half plane h and and the the Euler equation translates into an equation for omega theta which again is a transport equation but the transport equation not for omega theta but omega over r so if you divide I should have written it differently if you divide this equation by r you would have d dt of omega theta over r is is v gradient of this quantity and therefore this is this is transported the fact is that it is transported by a velocity field which is not divergence free okay divergence free if I look at it as a velocity field in h okay so so this makes things a little bit different but but still you have the fact that this quantity is is transported so if you go to higher values of r the omega is increasing okay now I wrote the now the question is assume I take a patch a little patch like this over t ct and what's going to happen to it so here was a patch so here it corresponds to a to a to an annulus okay if I draw a little patch and I wrote to this is just heuristic I wrote the the Euler equation in in weak form for here for an arbitrary omega which is axis symmetric and and you can play a little bit with the right hand side using the so this is the expression of the fact that the omega is the curl of v it means of this of this form as and this is the fact that the divergence of v as a three-dimensional vector is zero which translates in this way in the in the in the r space h and then you get an equation a weak formulation for the earlier equation where on the right hand side you only get the v okay you don't have the omega anymore and and so what can we guess about the this is the the way I prefer to understand the motion of a vortex strings this is the formulation I think it is the clear to understand because we know it is a transport for omega r so so this is a patch it's going to stay a patch this please locally concentrated and we want to to to guess its motion so to guess its motion it suffices to take a function phi which is an f a fine a fine function okay so if I take phi to be a fine then the here those two terms that have second derivatives of phi will not be present or and I cut I cut them off sufficiently far away where there is no where v has decreased already say and so I have to to understand these two forcing terms in order to guess what will be the motion of a of a small of a small vortex patch here so of a of a thin vortex strings and the fact is that the the the the biosavarlo in 3d looks locally like the biosavarlo in 2d well for for axis symmetric flows it looks like the biosavarlo in 2d and so if I have a small patch the the the velocity at least close to it will look like that like like 1 over r times times e theta well r no say 1 over rho where rho is the distance to the center of patch and not this r okay and so if you look at if you look at it you see that this would be like 1 over r square times a cosine square of theta and so this will integrate to something non-zero and it will integrate precisely to log of the of the cutoff distance here of the patch okay 1 over i square integrated whereas this one here is a 1 over r times cos theta 1 over r sine theta and that one integrates to 0 at main order okay so the the the main so this will give me the the time derivative of the position if omega is essentially a dirac and this is an affine function this is the main term it will go in the z direction with a speed which is proportional to 1 over the distance to the to the center so the radius of the annulus here and multiply by the logarithmic of the width of the vortex so this is if you take into account the the mass of the the mass of the patch which is at power 1 here and at power 2 here you get that the main order in the in the speed is this lambda over r naught so the intensity divided by the radius of the annulus multiplied by the log of the thickness of the patch which we assume to be epsilon at least at at initial time okay and this is something that elmol's already knew but he didn't do the computation with the equation in vorticity formulation he just used the he just computed the very precise asymptotics for the velocity v close to the tube and he argued that the main part this one over r had to cancel in some in some sense because it it only makes the the vortex rotate on itself but he already had precise computations and and even to the next order which was which were accurate actually the existence of exact vortex rings can be so exact in the sense that exact traveling wave solutions those solutions that do not deform in time except for the for the vertical translation can be constructed by maximizing the Euler energy which I wrote here in the vorticity formulation under constraint that the momentum is fixed sufficiently big if it's too small there there is nothing and the fact that the omega over r the transport of what it was initially this is also something which you have to impose otherwise the maximization problem becomes trivially infinity and okay so this was a long known it was an idea of Arnold to use such such optimizing constraint minimization to get exact traveling waves and there's a lot of work which in the end makes a very precise the fact that this problem is a solution provided p is larger than actually the the momentum of the ill vortex and and it has a shape of a vortex ring and so on on on different so this is for an exact traveling wave solution on a different direction benedeto caliote and mark euro proved that almost maximizers of this problem behave like a like a vortex ring in a in a weak sense in the sense that there there is a point close to where we where you expect and and a patch which has most of its mass there so therefore it is a weak weak form of the earlier equation and this form is going and this patch is going with the with the right speed asymptotically okay so that has been done for one vortex rings and the extension to the case of two or more vortex rings which would be the the exact equivalent of the point vortex system that mark euro and polarity derived has not been done for Euler and so this is still a and an open question so this is something which I wanted to show just a video of working extreme to show that this is an extreme this seems to be an extremely stable solution of Euler equation it was a Kelvin who proposed to build it just with a cardboard and and which I which I tried myself and as you can see it looks very stable it can go up to 10 or 20 meters in the lab without changing changing from there is nothing very precise in the disc inside I just I it was actually my daughters who built who built the hole with a pair of scissors and it goes really it goes really far without without changing shape and in the second half so this was the study of vorticity for 3d Euler flows was in this 1858 paper of elmoles in the second part he has a paragraph which is not preceded by computations it's it comes rather oddly but he forces what will be the behavior of the interaction between vortex rings and so he claims that if you take two of them so this one is going in this direction and that one will come close what's going to happen is that the exterior one is going to grow its radius the following one will shrink since it will shrink it will have a smaller radius and so it will be quicker because the speed is one over the radius and so we catch that one pass in front and then they will exchange their role that one will be behind and so they will play this leapfrogging one will pass inside the other actually I have a video that was made by Lim in Singapore it seems okay so this is a this is a lateral view of what's happening and you can see at least one what leapfrogging motion I thought this was a very very delicate things to reproduce in the lab but I have some students from Grande Colle close to Jussieu that came to me because they had to do a tape a and and we tried together to to reproduce this and to to measure some and so in the end they were able they were at SPCE where they were given many many facilities and you can see almost see one one of the leapfrogging motions with an apparatus which was built with the hand and with very few money it is less less beautiful than the previous one so so this is the phenomenon that we would like we would like to explain or to to derive rigorously and as I said we we have failed for the for the Euler equation and I will show you how it goes for the gross pitaya equation and try to to to to mention the link between the two so the gross pitaya equation is a is a non-linear defocusing Schrodinger equation it has a parameter epsilon which is fixed in the equation that's one of the main difference with Euler well in the previous one into the Euler you can fix it in the initial data and because it is transported it will it will remain inside but it is also a length scale epsilon so you will see that vortex tubes for gross pitaya ski will have somehow a fixed epsilon scale here whereas for Euler it was the difficult thing proving that concentrated vorticity remains concentrated somehow that will be the easy part in the gross pitaya ski equation but there there will be some parts that are more tricky to deal with too so u is with values in the complex and and so it is the Hamiltonian flow for this Gainsbourg-Lando energy and the analogy which I propose is the following when you have a u here you compute its Jacobian in the following way so maybe the easiest way to understand this formula is to think of u as being a so since it's complex value think it as a modulus exponential of a phase then from u you compute we call it's current j of u which is u cross gradient u is just the gradient basically the gradient of the phase so rho square the gradient of the phase there is a you see here in the gross pitaya ski equation energy there is a penalization when epsilon is small this is a penalization of the fact that the modulus of u is not close to one so you should think that for for functions that have fields that have a bounded energy or controlled energy rho should be close to one in most of the domain so the current is essentially the gradient of the phase and and the the vorticity defined for the gross pitaya ski is the curl of this of this current and so you see in particular that if if you have a pure phase that is if rho is exactly equal to zero then the jacobian is the curl of the gradient is it is identically equal to zero so the main difference here with respect to Euler is that in Euler when you have the vorticity you recover the velocity the velocity is is completely defined by the vorticity through bio-savar this is not the case in gross pitaya ski so in gross pitaya ski you see if you are a pure phase the vorticity is identically zero but the but the the velocity can be an arbitrary gradient of a phase so there is some additional degree of your freedom you can think of it also i will mention that later on the fact that somehow gross pitaya ski as a as a non compressible component whereas Euler was incompressible as a compressible component whereas Euler was was incompressible this is something i mentioned that so i have written here the weak formulation of the of the gross pitaya not the weak formulation because it does not characterize this but i have written a weak form a weak identity for the evolution of the jacobian j of u when u is an is an axisymmetric solution of the gross pitaya ski equation this is the equation where this is the capital F the exact form does not have such interest here but the the point is that i have written it here in a form that resembles very much what i wrote before for the Euler equation and actually so much that i can write these two equations here this is for gross pitaya ski and this is for Euler with the exact same capital F just that it acts on different objects for the gross pitaya ski equation it acts on the full gradient of u which contains both the what i call the little j of u the current but also the non the non compressible part whereas in Euler equation it just acts on on v and so in in a sense to to close the equation for gross pitaya ski you need also an equation for the comp for the compressible part and this is the one i wrote here which is because j of u is not divergence is not divergence free for rj of u in in cylindrical coordinates but it is almost divergence free which you can see here it's i have put the epsilon on purpose this quantity is part of the energy well if you take its its square this is the potential part of the energy so this equation tells you that on average in time if you average in time the divergence of j is is epsilon so this is the link with the incompressible Euler but you need this additional equation this is the loss whereas the gain was the fact that the length scale was fixed also another analogy which you can see at the level of the of the existence of vortex rings here i just copied what i had for the Euler equation you can get exact vortex rings for Euler by maximizing the energy under momentum constraint for the gross pitaya ski you can get exact traveling vortex rings by minimizing the energy under the exact equivalent constraint so so this is an analogy but also a very big difference because one is a maximization problem and the other one is a minimization problem so so the analogy tells you somehow that there is a saddle for for Euler you are a maximizer in the in the set of fields that are incompressible and if you allow compressibility then you are a minimizer it's not it's an analogy but there is a somehow a sharp difference here one is the man is a maximization and the other one is a minimization okay so okay now how will will will we study interaction of vortex ring and and it will in which regime so as i said marquerel pulveranti did the analysis for arbitrary arbitrary vortex patch we will use the energy and so we will use greatly the fact that some objects are optimizers of energy under fixed momentum we'll use that in a crucial way so we cannot work with this very broad setting and so i have to explain precisely what our initial data which will look like and their their energy so to do that i consider an arbitrary curve uh oriented curve uh in in a tree which i see as a think of it as a as a line of current capital j and i consider the vector of the distribution which has a two pi this is something also which is important for gross pitaevsky one of the main difference with oiler there is this quantization for gross pitaevsky which you do not do not need for oiler so i consider two pi times the integration along the curve x of so this is a vector distribution this is just circulation of along the curve c now when you have a line of current i think i think in terms of electromagnetic you have an induction capital b which satisfies these divergence b equals to zero and the curl is is exactly capital j which you recover by biosavar and when you have the induction you can go to the vector potential which satisfies these sets of equation and then you get the the vector Laplace equation of the vector potential is equal to j now now if you are in cylindrical coordinates i will take just one point here in the in the half plane a little a and and i will tell you what is a a dirac the equivalent of a dirac mass for oiler is a dirac along along the circle and and so the capital a is just a solution of of a vector Laplace equation with a with a dirac on the right hand side which you can write this way you can you can tell what the capital a is even with a complete elliptic integral this is this is something that that is that can be done but that is not really the most important part the thing is that since there is a two pi here we can define the gross beta eski vector field u that will be related to capital a and in the sense that that its vorticity will be precisely this singular circle okay and what you do basically is you use this formula forgot about the little r this is something just because we are in cylindrical coordinates but what you say is that the gradient of the phase of u is just the gradient perp of the of the of the potential okay so this would behave like like a log close to the singularity and so the phase will look like a like a typical dirac vortex in 2d you get this equation this sets of equation for you so the curl is 2 pi delta and the divergence of j is 0 to this change of of matrix so this is the equivalent of vortex points now I have to say what is the equivalent of circular patches circular patches where the the optimizers for energy the optimizers for energy in the ginsburg-landau framework are known and they are called vortices as well so they have a they have this particular form they behave like exponential i theta so this is in 2d and they and they have a profile which is like this so you will get a profile this is the modulus of u will look like that on the singularity it will not be x over modulus of x but you regularize it by something which has a width of order epsilon and this is energetically optimal in a sense it is the best way to regularize a singular vortex rings in the context of the gross p type k equation and then you define your field just just in this way so this would be the equivalent of of disk of a vortex disk in the 2d Euler now if you want to take many of them since you are essentially unimodular maps in c you just multiply them so this would be the singular one you multiply the the complex fields and and the desingular ones you you come you you also multiply them so now you have defined this way a field which has a number a1 a2 which has a number of vortex rings at different points and you can compute the expansion for its energy i have written it here and also check that its vorticity is close to the sum of Dirac masses and here is the difference with respect to Euler equal in Euler the l1 norm was a natural candidate well the used candidate to measure the the spreading of vorticity in gross p type k it turned out that the the dual of Lipschitz function is is something which is well behaved and and which which also has some advantages because it tracks more the fact that small parts want to go to infinity for the w minus 1 1 norm if a small part goes to infinity you see the distance at which it goes whereas for the l1 norm you just see that it's not there but it does not make the l1 norm does not make the difference between being a distance 1 and being a distance 1000 from there so in some sense this is something this is kind of estimates that would be nice to have in Euler 2 rather than l1 distances and the second important thing here is that the main term in the in the energy now depends of course on the position of the points and not only this is the equivalent of the interaction in the log xi minus xj which we add into d now there is a main term here which depends on the on the position of the of the vortex of the vortex rings okay so since gross p type k is the Hamiltonian flow for the for the gross p type k equation at the limit one would expect that the position of the vortex rings obey the Hamiltonian flow for the limit energy which we computed which is h epsilon which I I just wrote here and this will actually be the the content of the theorem I just mentioned that this flow here which is which I call the leapfrogging ODE as a in addition to them to the Hamiltonian as a conserved quantity which is the momentum which is just the sum of the square of the radii of the of the vortex rings and actually not for the ODE but for the full gross p type k we have seen that there was the conservation of this which was the equivalent of the of the angular momentum for Euler and for patches you have the the same the same asymptotic so this is wait no no no no don't pay attention to 39 this sorry okay so now the statement of the theorem I claim that the right the right scaling to see the interaction between vortices is this one so when vortex cores are at the distance of order one over square root of log so I wrote them as a sent a main point a not and and the distance here of order one over square root of log somehow if they are closer then the effect of of being 3d will not be will not be seen if they are too close they would behave like the 2d like straight vortex filaments and one would recover the the equivalent of marquero and pulveranti result and if they are too far apart then they will just behave like vortex rings with with too few interactions between them and there will not be an interesting limit od and the theorem is differing so so this is an assumption on the initial data and I assume that I have a bound I let those points evolve by the limit what what I think to be the limit od and I in the statement of the theorem I I assume that the bi epsilon of s remains bounded okay this is something you can always have for sometimes and and we will see later even for times of order one maybe a little bit bigger in some regime I assume that initially my vorticity is concentrated close to those points this is what I call the concentration scale and I assume that the energy of my initial data is not too much bigger than that what I expect to be the optimal energy for such configuration of points I do not claim that this is the this is the optimal one it will be at little or or the little or of one this will be a consequence of the theorem but I just expect I assume that I have not too much energy and this is what I call the excess and the theorem says that if both the concentration scale and the excess are not too big initially then then they remain not too big for for a time of order one and this this I can make as small as I wish by my initial data and we have this this loss which we cannot which comes from technicalities in the proof there there has to be something here and and the point is that we have an exponential here but with no singularity in epsilon there is no epsilon one over epsilon or something like that here whereas there was a one over epsilon square in the equation so for order one we can say that the behavior of the four times of order one we can say that the behavior of the Gospitowski equation is essentially the same as the behavior of the leapfrogging ODE now the leapfrogging ODE I I have depicted here it's its phase diagram when there are only two vortex strings in that case so there are four free parameters because each each A1 and A2 both have two degrees of freedom but because of conservations first you can well you have the momentum that fixes one leaf and there is also a translation invariance if I have those two points and I move them by a fixed a fixed vector in the z direction the the motion will be equivalent with this this this transformation so I can just look at the difference between the altitudes of the two annuli and use it as a as a function to describe the Hamiltonian and the other one as I said was the conservation of the momentum p so the and p was the sum of the the square radius of the two so I can define this new variable eta by these two quantities the sum is always p and so eta if eta is equal to one or minus one means that one of the two vortex strings is much bigger than the other one and when eta is zero it just means that they have the same the same radii and so if you plot the Hamiltonian h epsilon in terms of eta and z then you get this phase diagram and so you see that here in the center you have a region where solutions are periodic which means that so eta goes from positive to negative and z positive negative which is exactly what you you observed in the video in the sense that sometimes one annuli is the smaller the under and the other one is bigger and and they they they catch each other but there are different regions here so this would correspond to the place where the vortex strings have too much different radii initially and so they will interact you see the one is going closer to the other one but the it has the the difference is too big and it's not catched by the other one and and it will flow inside for example you take a very small one and a very big one in front it will pass through but it is not sufficient it is not sufficiently close to interact truly with it in a long time it will escape but there are also some very nice regions here where one will come closer that one will accelerate it will not pass inside but they will finish like like this so it come close and then that one goes in in this direction you can also at least numerically look at the at this that the leapfrogging systems for true or more vortex rings there are some very nice examples like like for the point vortex like for the point vortex system okay so i do not have time i had some slides on the proof but i think my time is over so i just finished to to say that one would like to do the same for Euler the difficulty is the concentration and so the main question i think the one good way to try to to tackle it is replace the l1 norm in the in the vega sidere so one pulverenti estimates by a norm which is a more adapted to catch the part of the vorticity that would that would want to go to infinity is it possible to do it with dual of lip sheets we've tried that and this is actually wrong but one may think of other way to measure the the expansion of vorticity in this filamentation problem in in Euler but this is a difficult question for on which we have failed if you see on the face so they are periodic on the side yes and so they they are which order because it means that so you can see many circles i mean in your yes so that you mean that this this require a time of order one so if if this is of size k times square root of log epsilon this will be essentially exponential of k so these are actually very flat things so the motion is is actually it's it's periodic in time but it does not look like something that does this but it models when k is big it's just something like that you don't see anything for a long time and then it exchanges you wait for a long time and then it extends again when you when you get closer to the to the etero clinic here and this is and it is you see many certain many periods in this in this theorem where was that in this theorem we can actually go to times that are log log of epsilon and and log log of this one it really depends on the constant which you have in front if you really want to have figures but but this requires a time of order one in this in this scaling so in principle we can if we take epsilon very very big very very small sorry we can see many many leapfrogings many of those yeah we we can follow it's not just one part of the branch no we can see many of it yes what happens if for you your unimodular map at each circle you change the degree say you take opposite degree for yes then then yes then i have a i have a video for that so so now that the two the two so in the leapfrogings they are both going in the same direction and interact now if you take different signs in the vorticity they will collide well or depending on where they are but and this is this is also a video by lim and the life seems to be much more complicated in that case yeah mathematically you would say there is something which you don't like because you you start in a in a cylindrically symmetric situation and you see this the solution is no longer cylindrical is symmetric but life is never similar cylindrical is symmetric so this is this video is very popular in the fluid mechanics community because you see you recover some vortex strings which have each of them as half of of a part which initially was and also this is a real experiment this is not a simulation you see another vortex strings which have these calving waves so this is probably much more difficult when you take opposite opposite vorticity