Concentrated vorticity in the Gross-Pitaevskii equations
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Jerrard, Robert
We will present recent results describing the behavior of certain solutions of the Gross-Pitaevskii equations, in a particular scaling regime, for initial data with concentrated vorticity.
 Let me begin by thanking the organizers for the invitation to spend some time here and to speak with you. It's been a great pleasure to be here. So let me first write down the equations I'll be studying. And so there are two equations I'm interested in. The first I'll call the gross spitievsky. It has a parameter epsilon and it looks like this. So the scaling is, here's the epsilon. I'll also consider the elliptic counterpart. So I'll call it Ginsburg-Landau with the epsilon. This is a bit of a misnomer. But OK. And then the setting for all this will be, say, in three dimensions in a cylindrical domain. So omega cross 0L, where little omega is a bounded open subset of R2, say, simply connected. In fact, we can take it to be a ball if we like for most of the talk. And notation I'll use rather consistently is I'll write points in omega in the form x and z where x is in little omega and z is the vertical variable. OK. So these are my equations. For concreteness, let's say I'll consider, for example, so this is the first one. So this is the more interesting one. I'll consider for here, say, Neumann boundary conditions on the lateral boundary and periodic in the z direction. And here I would want probably some Dirichlet data, for example, somewhere. OK. And so let me start by recalling some relevant quantities. So these would include, for example, the mass or rather density. There's a momentum, which I'll write in this way. So I'll consistently abuse notation in this fashion. So I'll write a dot product here between two complex numbers. I need to get the real dot product so I can identify the misvectors between those vectors in R2. So there's a real momentum density, the free energy. And so this is the quantity whose integral gives the conserved Hamiltonian for the gross Pity-Az equation. And let me write this as, I'll write little j to denote the momentum. I'll call this E sub epsilon of u. And then finally I'll define the vorticity, which I'll write as capital J of u. I can't call it omega because I called the domain omega. Capital J of u is 1 half of the curl of the momentum density. And it's a short calculation shows that this is in fact equal to, again, if I identify u as an R2-evaluated map with components u1 and u2, then this will just be the, in three dimensions, this is the cross part of, say, the real part, the grade and the real part and the grade and the imaginary part. And so what this tells us, incidentally, is that, of course, this grade and u1 is orthogonal to level sets of the real part. This is orthogonal to the level sets of the imaginary part. And so the vorticity is orthogonal to both of these normals, and hence it's parallel to the level sets of the complex value function. And so if you like, traditionally one defines a vortex filament as an integral curve of the vorticity vector field. Here because of this geometric property I've just described, such integral curves will just be level sets of the complex value function, where things are not regenerated and so on. And so there are a couple of conservation laws, a number of conservation laws that I'll recall. So for example, conservation of mass has the form. OK, so this is the continuity equation. The time of this is the divergence of that. There's a conservation of momentum. And I believe there's a 2 here. And then, OK, so I form the 3 by 3 matrix whose ij entry is the real inner product of the i-th derivative and the j-th derivative. I take the row-wise divergence. And then there's also a gradient term, which I won't care about much. The reason I won't care about it is I'm about to take the curl of this thing. And so to understand how the vorticity evolves, I just take the curl of this equation and I'll find that d dt, the vorticity, is, well, this disappears. And I have curl of divergence of grad u tensor, grad u. OK, so there's a nice evolution equation for the vorticity. And finally, the energy is also conserved formally. And it'll be conserved with the boundary conditions I have here. And so, for example, this tells me that the total energy is constant. When I study the gross pedigree equation, and actually also when I study the elliptic equation, I'll be interested in situations where the energy is a border log epsilon. This is natural for reasons we'll see in a second. And so if the energy is a border log epsilon, then in particular this u squared minus 1 is quite small in l2. And so it's convenient to think of this in this way. U squared minus 1 over epsilon is, I guess, logarithmic in l2. And so this tells me that the divergence of j is small in the regime I'm interested in. It's a border epsilon log epsilon, maybe. It's also true, in fact. We'll see that in this situation, in this regime where the energy is a border log epsilon, that the vorticity has a huge amount of structure. And there are actually two ways of thinking of this. One is that when I study the evolution equation, I'll put this structure in my initial data. But more generally, it's true that merely under the assumption of small energy, the vorticity has a huge amount of structure. And so in particular, it'll be concentrated along one dimensional curves in 3D. And away from those one dimensional curves, it'll be close to zero. And so then this tells us what the divergence of the current is small. The curl of the current, the vorticity, again, has structure. It's basically zero in a lot of places. It lives on curves. And so we have a very good understanding of both the divergence and the curl of the current density. Or we will when I tell you more about what this is doing. OK. So then let me describe a special solution of both of these equations. That's simplest to say what one refers to as a vortex solution. And so a special solution will be one who depends only on the horizontal x variable. So this is dependent of z and of t. And so for every integer d, let's write u upper d depending only on x. Depending on the parameter x long, we'll have the form of a modulus writing in polar coordinates. So d i d theta and then a modulus who depends on d and scales like r of reps long. And so if you make this ansatz, then you get an ode, let's say f d of zero is zero f d increases to one as the radial parameter goes from infinity. And so if you make this ansatz, you get an ode for f d. You can easily solve it. You can find this in a number of ways. Let's then remark some properties. And so the energy density of this, so this away from the origin, this looks like it's homogeneous of degree away from the origin. It's basically this right. So on a scale of order epsilon around the original, the modulus will be less than one away from the origin is basically e to the id theta. And so the energy of that will be like, say, a cut off on a ball of radius epsilon times d squared over, so the, if you just compute the energy of this, you get d squared over r squared. I think there's a factor of two. And so for example, the energy, if I integrate this over a ball of radius epsilon, over a ball of radius r, excuse me, over a macroscopic ball, I'll get something, I guess, d squared times log r over epsilon plus big O of one. Okay, this is the reason, this is, if you like, the basic reason I'll be interested in this logarithmic energy scale. This is the scale on which one sees these. So one thing, so this is being a basic vortex. And this logarithmic scale is the one on which vortices appear. Any questions? No? Excuse me, there's no z dependence on your special solutions? Here, no. That's right. And so this is really a 2D solution. It's really 2D. Just one. Yeah. And so, and so you can think of this being a cylinder, translation in the z direction and in the t direction. Another attribute of these solutions is, if I look at the vorticity, well, this is really going to be, I guess, pi times d. And then what's left will be an approximate identity. It'll have the form. And so this is really an approximate identity. Okay. It is a smooth function who will be, okay. So this is determined by the exact form of this f. And maybe it'll depend on d. But this is a non-negative function with integral 1. And we scaled in this fashion. Okay. And so we can see here that for this special solution, the vorticity is basically a smeared out delta function around the origin with multiple d pi times the energy d. And the energy scales like d squared. And what follows will always be interesting. d equals 1. And we can see very naively from this that if I call this a vortex, then a vortex of degree 2 costs order 4 pi log 1 of epsilon energy. And so 2 vortices of degree 1 are much cheaper energetically than 1 vortices of degree 2 is worse. So vortices of degree plus minus 1 have sort of strong stability properties which are not possessed by other vortices. Okay. And so the ansets I'm interested in is the following. So I want to look for solutions of these two equations. What I would like is I want to, my ansets is, I want to look for u epsilon depending on who looks something like, well, a product of, I take the basic say, degree 1 vortex on a scale epsilon on each horizontal slice. And then I translate it by an amount depending on the vertical and time variables, f, i, of z and t. Okay, and so then if I look at a single product, this is just the basic vortex translated. And I multiply them together, I'll get a, and so the picture is this should look like, and sorry, I also want to put a, I want it to be a small translation, so it'll be a small scale h epsilon. Okay, and so the picture is. Sorry, all these, I've got a little smear, is there all these d equal 1? Yeah, these are all d equals 1, that's right. Okay, so these will all have the same sign in particular. One, yeah. So the picture is on the small scale h epsilon. You hear there are these, these fortices, and if I look at the certain height z, I will see these structures centered at points f1 of z up to f sub n of z scaled into the small tube. Okay. So this is the, this is what I am interested in, this is the situation I'd like to study. And let me say at the, so one might imagine writing a solution as, and also let me add if I have boundary conditions, I may want to multiply this by a global phase function in order to satisfy those boundary conditions. Okay, so one might imagine writing a solution of either of those equations as something of this form and then an error term and then doing a linear analysis. For whatever, for reasons, well it's just an empirical fact that this argument has not been carried out successfully for any of the problems, for essentially any of the problems I'm considering here, especially for any three dimensional question for any of these, for either of these problems or any, or any similar problem. And so somehow linearizing has not been successful for these questions. But this is the, this is the picture to have in mind. So this ansatz will not actually appear in any proofs. Okay. So in some way the story I want to describe begins with work of Dolpino, this was mentioned by Valerio Banica yesterday and Kowalczyk from 08, 2008, it was published. And so they show essentially, I'm going to state this a bit imprecisely, but they show that, well let's look at the following quantity, I'll define this G sub E of, so let this be the integral over the domain of the total energy density. And I want to rescale in a certain way, so I'll subtract off, in terms of the energy, and we'll divert on a logarithm of scale, pi n minus, and I guess I need l here as well, l being the length, being the vertical height of the domain, and pi l log epsilon minus n n minus 1 pi l log of the smaller scale h epsilon. So in fact they show that in some sense the correct scale to take is it will be h epsilon minus log epsilon to the minus 1 half, I'll tell you why this is an interesting scale in a moment. Then there's some constant that depends on n and the domain omega. Okay so essentially they define this and they said that if we consider G epsilon of the above ansatz, and so built into this rescaling is the fact that I'm considering something with n, n vortex lines, the parameter n is there. G epsilon of the ansatz is equal to I guess pi over 2 fi, okay something over i, and then logarithmic interaction terms, so some involving i and j different, I think there's a pi here also, log of fi minus fj of z, the whole thing in red with the z variable. And so their theorem was that this is true up to a little over 1 error terms, and they showed also that these little over 1 error terms in some way depend smoothly on f on the lines f here. And I'll call this G0, this is the, so okay and just one or two remarks about this, of course the choice I make for the phase here will affect the constant I get there, and part of the point of the paper of Dilpean and Kvalchik was to make choice of the phase which is, I guess they didn't prove upper bounds, this is only lower bounds, they're commuting the energy of a test function, but to make a choice which is presumably a good one in close to optimal, that makes this term as small as possible. And so in some way they identified the right constant here which is connected to the right phase there. Okay, and so based on this one can conjecture that, well they, I think they either conjectured or became close to conjecturing that there should be solutions of the gross Petyazdy equation, well I will say for the elliptic equation, one would expect one might hope at least to see solutions to the elliptic equation which have roughly this form for F, a map from 0L into N fold copy of R2 for F being a critical point of this energy much more, or for F a critical point here, and one might further hope to have solutions of the gross Petyazdy equation of this form where F solves some kind of shorting or equation associated with this Hamiltonian. Okay, so this is the, these, I think the first conjecture was explicit in the paper of Del Pino and collaborators and the second one may have been implicit. Sorry, I should say, Kovaljic and collaborators. So and let me remark also, there's earlier, I think they were in some way motivated by an earlier paper of Montero, Sternberg, and Zemur who showed that for suitable geometries, so they were considering the elliptic problem in certain 3D domains, the domain looked like this, they imagine a surface of revolution generated by this, by this figure, and so, it doesn't have to be a surface of revolution, or a 3D domain of revolution, but the point is that you want a geometry force, there's a local minimizer of the arc length functional, connecting two points on the boundary. And so what they showed is that there exist solutions of the Ginsburg-Lando equation for small epsilon, such that the energy density divided by the rescaling divided by the derogatory factor converges to n pi times how short one dimensional measure restricted to the segment, say L here. And similarly the vorticity converges to a weld vector 0, 0, 1 if you will forgive abduse limitation times the same thing. And so one understands this as saying that for these solutions one has n quanta of vorticity concentrating around this curve. And so I think part of the motivation of these authors was to ask, well, on smaller scales, what's the fine scale structure of these solutions? Okay, and let me say also a, I won't write this down, but the main tool in the Sternberg and in the Monterres-Sturmbergs-Zehmer theorem were earlier results of myself and Sonner and Alberti Bordeaux-Orlandi. Okay, so then let me state the main results of this talk, which are the following. So the first is joint work with Andres Contreras, which addresses this. So really let's say as a corollary it will address the elliptic problem. And let's say, so assume that I have a sequence u epsilon in h1 omega complex value functions with the following two properties. So the first will say that I want to see say n vortex filaments concentrating on this. So from now on h sub epsilon always denotes this log epsilon to the minus one half length scale. And so I'd like to see, let's write it this way. So at each, if I look at the, all right over here I guess J sub x of u is just the Jacobian with respect to the x components. And so this is the horizontal gradient of u1. If you like it's the determinant of the horizontal gradient of u, if you would say a real value as a larger value map. So I want to, I'd like this to look like a bunch of delta functions to leading order concentrated near the origin. So w minus one, one in omega, d z. Let me ask this to be, okay. And so this tells me that if I look at, so theoretically if I look at a given slice then on the average I'll see n vortices within this h epsilon scale of the vertical axis. We assume this, and I'll also assume that this g epsilon of u epsilon that the energy rescaled in the above fashion by subtracting off the correct diversion part that this is bounded by some constant c2. So I want these to hold uniformly in epsilon. So the conclusions are that after passing to a subsequence we have the, let me do the following. So I'd like to rescale in the horizontal directions. So let's let v epsilon of x and z be u epsilon of h epsilon x and z. And so v, and so v is, I've expanded the domain, I've moved the h epsilon scale out to order one. This satisfies that the horizontal vorticity of v converges say in w minus one, one to a measure who is concentrated on the graph over the vertical axis of n h1 curves. And so this converges to I guess pi times the sum. Okay. And so what I mean here, I guess this is an expanding domain. I mean for every, if I fix a big ball then for epsilon small enough the domain, the rescaled domain will contain that ball. And so I mean, for example, on w minus one, of a big ball across zero L for every, for every, for every R. Okay. So after passing to a subsequence there exists a F such that this holds. And moreover, whenever this holds, then the, the limit for the energies g epsilon is greater than g zero of F. Okay. And so this is, so one can, one can view this as basically a lower bound to complement the upper bound of del p no and kv logic. This shows that the upper bound of del p no and kv logic is sharp. In other words, for any F I can construct a sequence u epsilon such that this holds in such a heavy quality here. And a few other conclusions which I won't write down at the moment. Okay. And so this is the first theorem as a corollary. Let's, let me state this not too precisely. So the corollary would be that in our, in our geometry that there exist solutions of the Ginsburg Landau epsilon equation. Okay. So here to be, to be precise, I'm considering dirichlet data on the top and bottom and Neumann data on the sides. And the point of the dirichlet data will be to force the presence of this, of n vortex lines near the origin. Okay. So there's just solutions of this equation such that if you like the, these, such that after rescaling in this fashion, the rescaled, the rescaled Jacobians converge to, and I mean, so I guess I mean, right. And so this, this measure has this form of a sum of delta functions on every z and then I integrate in z direction. Okay. And so this tells us for these solutions, the vorticity is indeed, sorry, and fi is a critical point of the limiting function and indeed a minimizer, subject to suitable boundary conditions. Okay. And so this says that indeed, while not giving the kind of sort of precise point wise description in terms of the ansatz that Kovalje and Delpino may have had, may have had in mind, this does say that for these equations, the vorticity is concentrating around, around curves who are, who minimize this limiting function. Okay. That's a theorem one. And the second theorem is a dynamic theorem. And so and this is joint work with D. Day Smith's, which, so these are different 16s that 16 means it was, it was submitted, I mean, it was, it was posted recently. And the 16 means it's quite underway at the moment. And so this is, we've, we've, and so I have to thank the organizers for this. I mean, so this, this stay in Paris has given us the opportunity to make progress on this problem. Okay. And so theorem two says, let me consider, oh, I guess I should make remarks about some remarks about this. And so, so, okay, without writing down names, let me just mention there's a huge amount of related work on the Ginsburg-Landau equation in 3D. And so in some, I guess I have to, so a lot of this follows from, say, seminal work of Bethel, Brezes, and Elan in the early 90s on the Ginsburg-Landau equation in 3D. People who can, and so the general picture in 3D is that one, is that in the level, one epsilon is small, energy and vorticity concentrate around, around lines. And more generally in, in, in dimensions where n greater than or equal to 3, energy and vorticity concentrate on co-dimension two minimal surfaces. This is the general picture. And so this has been studied, for example, a major conversion food in more or less chronological order, Bethwell in Riviera, Linn in Riviera, Bethwell-Brazise, Orlandie, Bethwell-Borghain, orlandie, Bethwell-Branie, Smets, et cetera. So these, these all made important contributions to this analysis. And so, of course, in 3D, a minimal surface is a line. And what this is doing is giving a description of the, of the fine scale structure in the case of higher degree concentration of vorticity around the, around the line. This is also related in a way to, so another parallel is with the scalar analog of the Ginzburg-Land equation, so the, the Allen-Con equation. And so there one sees, there's a phenomenon of what one calls interface, interface clustering. And so there are two interfaces are connected in some way to a minimal surface. And one can have multiple interfaces clustering around a single minimal surface. There's a lot of rather reasonable work on this by people including Del Pino-Covaltric way, well let's say Del Pino-Covaltric way, Packard, and, and basically various subsets of, in fact, subsets of those permutations. A few other, a few other authors have contributed also as well as a paper of Covaltric Liu and Packard, or a paper of Del Pino-Covaltric way and Yang. And so a lot of recent work on interface clustering, this is the, the parallel, say first parallel result in hard dimensions showing vortex clustering. Okay, so, theorem two then would be, and so henceforth I'm going to always state things in terms of rescaled variable v rather than u. And so let me write down what the equation, I'm going to start at the gross pity of the equation, let me write down what it looks like after it's rescaling. And so I'll have, say, minus, I guess I want, so the rescaling is anisotropic with respect to x and z. So, this will give rise to, giving rise to certain log, logarithmic factors in certain places. Okay, and so this will, for example, give rise to rescaled versions of these, of these conservation laws. And rescaled versions to conserve quantities. So, I'd like to assume this, I'd like to assume that, I'd like to assume that the vorticity initially concentrates, well it has these properties. I guess it's not written here. So, the, this converges, we can make a conversion, it's a precise essence, doesn't matter, but let's say w minus 1, 1, or this topology over here. This will converge to, okay, so, okay, so we have, something like this, and again I keep on forgetting this. Okay, so at every height I have n vortices and then I integrate it in the z direction. This is true, and also I want to then assume that I have the minimal energy possible given this conservation addition, and the minimal energy is given by this. So, g epsilon, after rescaling, when I rescale I get some logs in certain places, that this should converge to g naught of this, f0. And here f, f, f naught denotes a, is a smooth solution and, say, h4 is enough, continuous into h4. Of exactly the, exactly the Hamiltonian system associated to g naught, which is exactly the model that Valerian Banach has spoke about yesterday in his talk, and so this is, so f naught solves i, d, t, f naught minus the second, and there are components j. Okay, so I assume I have a smooth solution of this limiting system, and I assume that at time zero the energy and vorticity are concentrating around the curves who give the initial data for this solution. The conclusion is that then for t, here, again the vorticity converges to the vorticity associated to the solution, here, f0 at time t, tensor d z. This is true, and similarly the energy also converges the same thing, in fact. Okay, let's leave it at that. Okay, we have more conclusions which I've written down. And so what this says is the derivation of the equation studied by Valerian Banach yesterday. And so let's remember from Valerian's talk, this was first formally written down by guys in the fluids community by Klein, Maida, and Damodar in the mid-90s, following earlier work of Zakhrov in the late 80s. And basically the way these guys argue is by matched formal asymptotics. And so this is, as far as we know, the first, or I mean it's not even the first, because it's not finished yet, but we hope this will be the first rigorous derivation in any setting of this model for the dynamics of thin, nearly parallel vortex filaments, starting from an equation with fluid dynamical rodentite. I should have said the gross piti-fc equation in principle describes superfluids, and so quantum mechanical fluids. And so rather than looking at, let's say, classical ideal fluids as described by the older equations, we're considering their quantum counterparts, which in some ways the analysis is much easier in the quantum case in the gross piti-fc case. Okay, and so let me note here in this setting that all the vortices have the same sign, and so one would like to be able to do this for vortices of opposite sign. That's what one sees in these trailing vortices of airplanes, for example. I mean, vortices of the same sign actually appears to be more relevant in the quantum context, but who knows. And then, so concerning this equation, Valeria summarized yesterday a lot of history, and so this includes work of what Klein might have done with Arne who derived it from Kenick Ponson Vega. Lots of work of Banek and Mio, also recent work of Walter Craig and Carlos Garcia as Piatia, and I think there's also work of Garcia Piatia and Isé, and so one can sort of mention all the, essentially everyone has worked on this in a sentence or two. Okay, and so in the remaining eight minutes, I'll say a bit about the proof. So here's how this goes. And so the overall point is that the proof relies incredibly strongly on, well, say on things in the spirit of theorem one, which is here, I think. So in theorem one, we assume only some kind of sort of crude knowledge about the vorticity. We say that we have n vortices concentrating on a certain scale with the origin about this line, and we assume that an energy balance, and then we deduce various conclusions. And so in the dynamical setting, we will choose initially such that these hypotheses hold, and if we can show these persist in time, then we have access to these conclusions, and further conclusions in the same spirit. And so a large part of this, there's a theorem three, so to speak, which is, again, we'll be in the paper of myself in DDA, which is sort of a refinement of these results under the same or similar hypotheses. We extract still further information about the behavior of the vorticity in these solutions. And so, but about the proof. So if we consider, let's call this the interacting vortex filmeth system. So here's the point. If, so suppose f and f naught solve this, then it's an easy computation to show that this is bounded by a constant times this. Okay, so what I want to do, I like to do the same thing if possible for f naught solving this, and f obtained from the gross PTIFG equation in the limit as f's long goes to zero. And so let me suppose that I've somehow arranged that at every time these conditions hold at every time in some interval. And so at every time I can pass limits along some subsequence and get some limiting f. That will depend on time. If I have some kind of aquacontinuity in time, I can do this. I can find a signal subsequence for taking conversions along a time interval. And so if we try to mimic this computation, we'll find that well for reasons that will become apparent in a moment. I get this, but then I get errors arising from the fact that this is not the right equation. I have to do some approximations. And these errors look like, and when I say f comes from this, I mean under my hypothesis, under my hypothesis of rather sharp energy bounds, what we're at data. Okay, there's some extra energy here. And so the idea is that if it's possible that, so I have a lower bound here, the limiting vortex elements can have at most this much energy. They may lose their energy. Energy may leak from the vortexes into the rest of the fluid. And that's kind of an enemy you have to be controlled. And one thing we can do however is we can replace this by the first variation by a linear term. Let's write it like this. And then I get quadratic terms which turn out to involve only the L2 norm, not the derivatives. And so I can absorb the quadratic terms into here. And I get a linear term. Okay, but then I'm still left controlling something who's linear in f by something who's quadratic. That doesn't look very good. However, it's also true if I go back to this system that for the linear term in question. One has this estimate. And again, this is a straightforward. And so the point is then, well, when I go over here and again do the same approximation arguments and so on, I'll find that for f coming from the gross pretty easy limit, this is bounded by the same expressions. And so at this point, I'm able to do a grung wall inequality and proceed. The, of course, all of work has been obtained in this assertion that I can do this. And so let me say a little bit about that in the remaining three minutes if I can. Okay, good. So then, so what I want to do then is to use the, is to use this equation. The equation tells me how the board is to move. Remember, the board is to tell me where the curves are. And so let me multiply this rescale, I write things in terms of v instead of u. I multiply the vorticity by test function phi. And after the rescaling, I'll find, well, there'll be a term phi sub t. There'll be a term epsilon. So here i and j are, well, it's sort of like this. I have the horizontal derivative of phi and then Hilbert-Schmidt inner product with something, the tensor product of v perp with v. And then I have dz, the horizontal derivative of phi and the dot product with, okay, the anisotropic rescaling gives rise to log ups on here. I'd like to pass the limits in this expression. And what I'll do is I'll take a, I'll take a phi who, for example, looks like 1 half x minus f not of z t squared. Okay. And so then I better sum over the components and multiply by cutoff functions. Okay. And so then formally, at least, well, actually, then it's really true that the phi times the Jacobian converges to as long as the limiting curves are supported in the region where these cutoff functions are 1. This will really converge to the L2 norm, the L2 norm squared. And similarly, this will converge to what we know what this is doing. This will give us some different pieces of that computation. And so we have to, in a bit, we have to know what this is doing. And so it's the fact that this is converging. So this is one of the things proved in the theorem 3. I didn't write down. This is converging to dz. So if I fix a time t, then this will converge to, so, it'll converge to measure supported along the curve, along the vortex curves involving the vertical derivative of the limiting curve. Okay. So these come from the limit. And whereas this involves, if you look at it, this involves the, this is just where the cutoff function is equal to 1. This is just equal to minus dz of f0 of the solution. Okay. And so here we have, so this will give rise to a term involving the orthogonal gradient of dz of f0 and dz of f, right, the thing I obtained from the limit. And this is the hardest term by far, and I don't have time to talk with that. So let me stop here. Thank you. Yes. So is there another geometric context of this? Maybe some domain which, not cylindric one. Maybe some other geometric, interesting geometric context. Well, all right. So I'd say for the, for Grosby Tkowski cylindrical or an unbounded cylinder or all of our three are the natural context. And so the, as I think Valerius said, this limiting system has been studied most on, on the real line. And so with that we'd need a domain which is unbounded in the z direction. That would be possible. For the elliptic equation, yeah, so there, there what's much more interesting is exactly the interplay between the geometry of the domain and, and the geometry in the limiting solutions. And so for example, one could presumably do the same thing, it would be harder. And so one could look for, if I have a three dimensional manifold with geodesic, a closed geodesic, one could look for solutions with, with, and filaments concerting on small scales around the geodesic. That's, that would be a bit harder. I actually have the same, same question, but just slightly different, so for my own purposes, you put diracley-Vonnician conditions at, at zero and L. Yeah. Can you put periodic? You can. I mean the, right, so we, we have a, I mean this, this, this corollary says really if we have a, if we have a local minimizer of the limiting functional, then we get local minimizing solutions of the epsilon functional. And so in, in the, if I just have, if I just have n curves, if they're not, if they're not linked in some way, then, then they'll always want to spread apart and lower their energy. So they're, they're presumably are, you know, let me mark also this is. So the diracley-Vonnician's make you fit, allow you to fix the curves at the, at the end. Yeah, that's right. In principle, they could also be fixed by, by nodding. If I, so the limiting equation in the elliptic case is just this without the time derivative. And if you look at it, this is, this is really a planar and body problem where Z plays the role of the time variable. And, and so people have constructed solutions. We, we didn't have to, however, to, to, the thing we didn't do was to identify nodded solutions who are local minimizers of some energy, which is what we need to do to, to relate them to these. But the time dynamics, I mean, the time dynamics allows you to give initial data and follow them for a period of time. And that wouldn't cost the, you wouldn't need a minimizer. No, that's right. Any other questions? Thank you.
 Let me begin by thanking the organizers for the invitation to spend some time here and  to speak with you.  It's been a great pleasure to be here.  So let me first write down the equations I'll be studying.  And so there are two equations I'm interested in.  The first I'll call the gross spitievsky.  It has a parameter epsilon and it looks like this.  So the scaling is, here's the epsilon.  I'll also consider the elliptic counterpart.  So I'll call it Ginsburg-Landau with the epsilon.  This is a bit of a misnomer.  But OK.  And then the setting for all this will be, say, in three dimensions in a cylindrical  domain.  So omega cross 0L, where little omega is a bounded open subset of R2, say, simply connected.  In fact, we can take it to be a ball if we like for most of the talk.  And notation I'll use rather consistently is I'll write points in omega in the form  x and z where x is in little omega and z is the vertical variable.  OK.  So these are my equations.  For concreteness, let's say I'll consider, for example, so this is the first one.  So this is the more interesting one.  I'll consider for here, say, Neumann boundary conditions on the lateral boundary and periodic  in the z direction.  And here I would want probably some Dirichlet data, for example, somewhere.  OK.  And so let me start by recalling some relevant quantities.  So these would include, for example, the mass or rather density.  There's a momentum, which I'll write in this way.  So I'll consistently abuse notation in this fashion.  So I'll write a dot product here between two complex numbers.  I need to get the real dot product so I can identify the misvectors between those vectors  in R2.  So there's a real momentum density, the free energy.  And so this is the quantity whose integral gives the conserved Hamiltonian for the gross  Pity-Az equation.  And let me write this as, I'll write little j to denote the momentum.  I'll call this E sub epsilon of u.  And then finally I'll define the vorticity, which I'll write as capital J of u.  I can't call it omega because I called the domain omega.  Capital J of u is 1 half of the curl of the momentum density.  And it's a short calculation shows that this is in fact equal to, again, if I identify  u as an R2-evaluated map with components u1 and u2, then this will just be the, in three  dimensions, this is the cross part of, say, the real part, the grade and the real part  and the grade and the imaginary part.  And so what this tells us, incidentally, is that, of course, this grade and u1 is orthogonal  to level sets of the real part.  This is orthogonal to the level sets of the imaginary part.  And so the vorticity is orthogonal to both of these normals, and hence it's parallel  to the level sets of the complex value function.  And so if you like, traditionally one defines a vortex filament as an integral curve of  the vorticity vector field.  Here because of this geometric property I've just described, such integral curves will  just be level sets of the complex value function, where things are not regenerated and so on.  And so there are a couple of conservation laws, a number of conservation laws that I'll  recall.  So for example, conservation of mass has the form.  OK, so this is the continuity equation.  The time of this is the divergence of that.  There's a conservation of momentum.  And I believe there's a 2 here.  And then, OK, so I form the 3 by 3 matrix whose ij entry is the real inner product of  the i-th derivative and the j-th derivative.  I take the row-wise divergence.  And then there's also a gradient term, which I won't care about much.  The reason I won't care about it is I'm about to take the curl of this thing.  And so to understand how the vorticity evolves, I just take the curl of this equation and  I'll find that d dt, the vorticity, is, well, this disappears.  And I have curl of divergence of grad u tensor, grad u.  OK, so there's a nice evolution equation for the vorticity.  And finally, the energy is also conserved formally.  And it'll be conserved with the boundary conditions I have here.  And so, for example, this tells me that the total energy is constant.  When I study the gross pedigree equation, and actually also when I study the elliptic  equation, I'll be interested in situations where the energy is a border log epsilon.  This is natural for reasons we'll see in a second.  And so if the energy is a border log epsilon, then in particular this u squared minus 1  is quite small in l2.  And so it's convenient to think of this in this way.  U squared minus 1 over epsilon is, I guess, logarithmic in l2.  And so this tells me that the divergence of j is small in the regime I'm interested in.  It's a border epsilon log epsilon, maybe.  It's also true, in fact.  We'll see that in this situation, in this regime where the energy is a border log epsilon,  that the vorticity has a huge amount of structure.  And there are actually two ways of thinking of this.  One is that when I study the evolution equation, I'll put this structure in my initial data.  But more generally, it's true that merely under the assumption of small energy, the vorticity  has a huge amount of structure.  And so in particular, it'll be concentrated along one dimensional curves in 3D.  And away from those one dimensional curves, it'll be close to zero.  And so then this tells us what the divergence of the current is small.  The curl of the current, the vorticity, again, has structure.  It's basically zero in a lot of places.  It lives on curves.  And so we have a very good understanding of both the divergence and the curl of the current  density.  Or we will when I tell you more about what this is doing.  OK.  So then let me describe a special solution of both of these equations.  That's simplest to say what one refers to as a vortex solution.  And so a special solution will be one who depends only on the horizontal x variable.  So this is dependent of z and of t.  And so for every integer d, let's write u upper d depending only on x.  Depending on the parameter x long, we'll have the form of a modulus writing in polar  coordinates.  So d i d theta and then a modulus who depends on d and scales like r of reps long.  And so if you make this ansatz, then you get an ode, let's say f d of zero is zero f d  increases to one as the radial parameter goes from infinity.  And so if you make this ansatz, you get an ode for f d.  You can easily solve it.  You can find this in a number of ways.  Let's then remark some properties.  And so the energy density of this, so this away from the origin, this looks like it's  homogeneous of degree away from the origin.  It's basically this right.  So on a scale of order epsilon around the original, the modulus will be less than one  away from the origin is basically e to the id theta.  And so the energy of that will be like, say, a cut off on a ball of radius epsilon times  d squared over, so the, if you just compute the energy of this, you get d squared over  r squared.  I think there's a factor of two.  And so for example, the energy, if I integrate this over a ball of radius epsilon, over a  ball of radius r, excuse me, over a macroscopic ball, I'll get something, I guess, d squared  times log r over epsilon plus big O of one.  Okay, this is the reason, this is, if you like, the basic reason I'll be interested  in this logarithmic energy scale.  This is the scale on which one sees these.  So one thing, so this is being a basic vortex.  And this logarithmic scale is the one on which vortices appear.  Any questions?  No?  Excuse me, there's no z dependence on your special solutions?  Here, no.  That's right.  And so this is really a 2D solution.  It's really 2D.  Just one.  Yeah.  And so, and so you can think of this being a cylinder, translation in the z direction  and in the t direction.  Another attribute of these solutions is, if I look at the vorticity, well, this is really  going to be, I guess, pi times d.  And then what's left will be an approximate identity.  It'll have the form.  And so this is really an approximate identity.  Okay.  It is a smooth function who will be, okay.  So this is determined by the exact form of this f.  And maybe it'll depend on d.  But this is a non-negative function with integral 1.  And we scaled in this fashion.  Okay.  And so we can see here that for this special solution, the vorticity is basically a smeared  out delta function around the origin with multiple d pi times the energy d.  And the energy scales like d squared.  And what follows will always be interesting.  d equals 1.  And we can see very naively from this that if I call this a vortex, then a vortex of  degree 2 costs order 4 pi log 1 of epsilon energy.  And so 2 vortices of degree 1 are much cheaper energetically than 1 vortices of degree 2 is  worse.  So vortices of degree plus minus 1 have sort of strong stability properties which are not  possessed by other vortices.  Okay.  And so the ansets I'm interested in is the following.  So I want to look for solutions of these two equations.  What I would like is I want to, my ansets is, I want to look for u epsilon depending on  who looks something like, well, a product of, I take the basic say, degree 1 vortex on  a scale epsilon on each horizontal slice.  And then I translate it by an amount depending on the vertical and time variables, f, i,  of z and t.  Okay, and so then if I look at a single product, this is just the basic vortex translated.  And I multiply them together, I'll get a, and so the picture is this should look like,  and sorry, I also want to put a, I want it to be a small translation, so it'll be a small  scale h epsilon.  Okay, and so the picture is.  Sorry, all these, I've got a little smear, is there all these d equal 1?  Yeah, these are all d equals 1, that's right.  Okay, so these will all have the same sign in particular.  One, yeah.  So the picture is on the small scale h epsilon.  You hear there are these, these fortices, and if I look at the certain height z, I will  see these structures centered at points f1 of z up to f sub n of z scaled into the small  tube.  Okay.  So this is the, this is what I am interested in, this is the situation I'd like to study.  And let me say at the, so one might imagine writing a solution as, and also let me add  if I have boundary conditions, I may want to multiply this by a global phase function  in order to satisfy those boundary conditions.  Okay, so one might imagine writing a solution of either of those equations as something of  this form and then an error term and then doing a linear analysis.  For whatever, for reasons, well it's just an empirical fact that this argument has not  been carried out successfully for any of the problems, for essentially any of the problems  I'm considering here, especially for any three dimensional question for any of these, for  either of these problems or any, or any similar problem.  And so somehow linearizing has not been successful for these questions.  But this is the, this is the picture to have in mind.  So this ansatz will not actually appear in any proofs.  Okay.  So in some way the story I want to describe begins with work of Dolpino, this was mentioned  by Valerio Banica yesterday and Kowalczyk from 08, 2008, it was published.  And so they show essentially, I'm going to state this a bit imprecisely, but they show  that, well let's look at the following quantity, I'll define this G sub E of, so let this be  the integral over the domain of the total energy density.  And I want to rescale in a certain way, so I'll subtract off, in terms of the energy,  and we'll divert on a logarithm of scale, pi n minus, and I guess I need l here as well,  l being the length, being the vertical height of the domain, and pi l log epsilon minus n  n minus 1 pi l log of the smaller scale h epsilon.  So in fact they show that in some sense the correct scale to take is it will be h epsilon  minus log epsilon to the minus 1 half, I'll tell you why this is an interesting scale  in a moment.  Then there's some constant that depends on n and the domain omega.  Okay so essentially they define this and they said that if we consider G epsilon of  the above ansatz, and so built into this rescaling is the fact that I'm considering something  with n, n vortex lines, the parameter n is there.  G epsilon of the ansatz is equal to I guess pi over 2 fi, okay something over i, and then  logarithmic interaction terms, so some involving i and j different, I think there's a pi here  also, log of fi minus fj of z, the whole thing in red with the z variable.  And so their theorem was that this is true up to a little over 1 error terms, and they  showed also that these little over 1 error terms in some way depend smoothly on f on the  lines f here.  And I'll call this G0, this is the, so okay and just one or two remarks about this, of  course the choice I make for the phase here will affect the constant I get there, and  part of the point of the paper of Dilpean and Kvalchik was to make choice of the phase  which is, I guess they didn't prove upper bounds, this is only lower bounds, they're  commuting the energy of a test function, but to make a choice which is presumably a good  one in close to optimal, that makes this term as small as possible.  And so in some way they identified the right constant here which is connected to the right  phase there.  Okay, and so based on this one can conjecture that, well they, I think they either conjectured  or became close to conjecturing that there should be solutions of the gross Petyazdy  equation, well I will say for the elliptic equation, one would expect one might hope  at least to see solutions to the elliptic equation which have roughly this form for  F, a map from 0L into N fold copy of R2 for F being a critical point of this energy much  more, or for F a critical point here, and one might further hope to have solutions of  the gross Petyazdy equation of this form where F solves some kind of shorting or equation  associated with this Hamiltonian. Okay, so this is the, these, I think the first conjecture  was explicit in the paper of Del Pino and collaborators and the second one may have been  implicit. Sorry, I should say, Kovaljic and collaborators.  So and let me remark also, there's earlier, I think they were in some way motivated by  an earlier paper of Montero, Sternberg, and Zemur who showed that for suitable geometries,  so they were considering the elliptic problem in certain 3D domains, the domain looked like  this, they imagine a surface of revolution generated by this, by this figure, and so,  it doesn't have to be a surface of revolution, or a 3D domain of revolution, but the point  is that you want a geometry force, there's a local minimizer of the arc length functional,  connecting two points on the boundary. And so what they showed is that there exist solutions  of the Ginsburg-Lando equation for small epsilon, such that the energy density divided by the  rescaling divided by the derogatory factor converges to n pi times how short one dimensional  measure restricted to the segment, say L here. And similarly the vorticity converges  to a weld vector 0, 0, 1 if you will forgive abduse limitation times the same thing. And  so one understands this as saying that for these solutions one has n quanta of vorticity  concentrating around this curve. And so I think part of the motivation of these authors  was to ask, well, on smaller scales, what's the fine scale structure of these solutions?  Okay, and let me say also a, I won't write this down, but the main tool in the Sternberg  and in the Monterres-Sturmbergs-Zehmer theorem were earlier results of myself and Sonner  and Alberti Bordeaux-Orlandi. Okay, so then let me state the main results of this talk,  which are the following. So the first is joint work with Andres Contreras, which addresses  this. So really let's say as a corollary it will address the elliptic problem. And let's  say, so assume that I have a sequence u epsilon in h1 omega complex value functions with the  following two properties. So the first will say that I want to see say n vortex filaments  concentrating on this. So from now on h sub epsilon always denotes this log epsilon to  the minus one half length scale. And so I'd like to see, let's write it this way. So at  each, if I look at the, all right over here I guess J sub x of u is just the Jacobian  with respect to the x components. And so this is the horizontal gradient of u1. If you like  it's the determinant of the horizontal gradient of u, if you would say a real value as a  larger value map. So I want to, I'd like this to look like a bunch of delta functions to  leading order concentrated near the origin. So w minus one, one in omega, d z. Let me  ask this to be, okay. And so this tells me that if I look at, so theoretically if I look  at a given slice then on the average I'll see n vortices within this h epsilon scale  of the vertical axis. We assume this, and I'll also assume that this g epsilon of u epsilon  that the energy rescaled in the above fashion by subtracting off the correct diversion part  that this is bounded by some constant c2. So I want these to hold uniformly in epsilon.  So the conclusions are that after passing to a subsequence  we have the, let me do the following. So I'd like to rescale in the horizontal directions.  So let's let v epsilon of x and z be u epsilon of h epsilon x and z. And so v, and so v is,  I've expanded the domain, I've moved the h epsilon scale out to order one. This satisfies  that the horizontal vorticity of v converges say in w minus one, one to a measure who is  concentrated on the graph over the vertical axis of n h1 curves. And so this converges  to I guess pi times the sum. Okay. And so what I mean here, I guess this is an expanding  domain. I mean for every, if I fix a big ball then for epsilon small enough the domain,  the rescaled domain will contain that ball. And so I mean, for example, on w minus one,  of a big ball across zero L for every, for every, for every R. Okay. So after passing  to a subsequence there exists a F such that this holds. And moreover, whenever this holds,  then the, the limit for the energies g epsilon is greater than g zero of F. Okay. And so  this is, so one can, one can view this as basically a lower bound to complement the  upper bound of del p no and kv logic. This shows that the upper bound of del p no and  kv logic is sharp. In other words, for any F I can construct a sequence u epsilon such  that this holds in such a heavy quality here. And a few other conclusions which I won't  write down at the moment. Okay. And so this is the first theorem as a corollary. Let's,  let me state this not too precisely. So the corollary would be that in our, in our geometry  that there exist solutions of the Ginsburg Landau epsilon equation. Okay. So here to  be, to be precise, I'm considering dirichlet data on the top and bottom and Neumann data  on the sides. And the point of the dirichlet data will be to force the presence of this,  of n vortex lines near the origin. Okay. So there's just solutions of this equation  such that if you like the, these, such that after rescaling in this fashion, the rescaled,  the rescaled Jacobians converge to, and I mean, so I guess I mean, right. And so this,  this measure has this form of a sum of delta functions on every z and then I integrate in  z direction. Okay. And so this tells us for these solutions, the vorticity is indeed,  sorry, and fi is a critical point of the limiting function and indeed a minimizer, subject to  suitable boundary conditions. Okay. And so this says that indeed, while not giving the  kind of sort of precise point wise description in terms of the ansatz that Kovalje and Delpino  may have had, may have had in mind, this does say that for these equations, the vorticity  is concentrating around, around curves who are, who minimize this limiting function.  Okay. That's a theorem one. And the second theorem is a dynamic theorem. And so  and this is joint work with D. Day Smith's, which, so these are different 16s that 16  means it was, it was submitted, I mean, it was, it was posted recently. And the 16 means  it's quite underway at the moment. And so this is, we've, we've, and so I have to thank  the organizers for this. I mean, so this, this stay in Paris has given us the opportunity  to make progress on this problem. Okay. And so theorem two says, let me consider, oh,  I guess I should make remarks about some remarks about this. And so, so, okay, without writing  down names, let me just mention there's a huge amount of related work on the Ginsburg-Landau  equation in 3D. And so in some, I guess I have to, so a lot of this follows from, say,  seminal work of Bethel, Brezes, and Elan in the early 90s on the Ginsburg-Landau equation  in 3D. People who can, and so the general picture in 3D is that one, is that in the  level, one epsilon is small, energy and vorticity concentrate around, around lines. And more  generally in, in, in dimensions where n greater than or equal to 3, energy and vorticity concentrate  on co-dimension two minimal surfaces. This is the general picture. And so this has been  studied, for example, a major conversion food in more or less chronological order, Bethwell  in Riviera, Linn in Riviera, Bethwell-Brazise, Orlandie, Bethwell-Borghain, orlandie, Bethwell-Branie,  Smets, et cetera. So these, these all made important contributions to this analysis.  And so, of course, in 3D, a minimal surface is a line. And what this is doing is giving  a description of the, of the fine scale structure in the case of higher degree concentration  of vorticity around the, around the line. This is also related in a way to, so another  parallel is with the scalar analog of the Ginzburg-Land equation, so the, the Allen-Con  equation. And so there one sees, there's a phenomenon of what one calls interface, interface  clustering. And so there are two interfaces are connected in some way to a minimal surface.  And one can have multiple interfaces clustering around a single minimal surface. There's a  lot of rather reasonable work on this by people including Del Pino-Covaltric way, well let's  say Del Pino-Covaltric way, Packard, and, and basically various subsets of, in fact,  subsets of those permutations. A few other, a few other authors have contributed also  as well as a paper of Covaltric Liu and Packard, or a paper of Del Pino-Covaltric way and Yang.  And so a lot of recent work on interface clustering, this is the, the parallel, say first parallel  result in hard dimensions showing vortex clustering. Okay, so, theorem two then would be, and so  henceforth I'm going to always state things in terms of rescaled variable v rather than  u. And so let me write down what the equation, I'm going to start at the gross pity of the  equation, let me write down what it looks like after it's rescaling. And so I'll have,  say, minus, I guess I want, so the rescaling is anisotropic with respect to x and z.  So, this will give rise to, giving rise to certain log, logarithmic factors in certain  places. Okay, and so this will, for example, give rise to rescaled versions of these, of  these conservation laws. And rescaled versions to conserve quantities. So, I'd like to assume  this, I'd like to assume that, I'd like to assume that the vorticity initially concentrates,  well it has these properties. I guess it's not written here. So, the, this converges,  we can make a conversion, it's a precise essence, doesn't matter, but let's say w minus  1, 1, or this topology over here. This will converge to, okay, so, okay, so we have,  something like this, and again I keep on forgetting this. Okay, so at every height I have n vortices  and then I integrate it in the z direction. This is true, and also I want to then assume  that I have the minimal energy possible given this conservation addition, and the minimal  energy is given by this. So, g epsilon, after rescaling, when I rescale I get some logs  in certain places, that this should converge to g naught of this, f0. And here f, f, f naught  denotes a, is a smooth solution and, say, h4 is enough, continuous into h4. Of exactly  the, exactly the Hamiltonian system associated to g naught, which is exactly the model that  Valerian Banach has spoke about yesterday in his talk, and so this is, so f naught solves  i, d, t, f naught minus the second, and there are components j. Okay, so I assume I have  a smooth solution of this limiting system, and I assume that at time zero the energy  and vorticity are concentrating around the curves who give the initial data for this  solution. The conclusion is that then for t, here, again the vorticity converges to  the vorticity associated to the solution, here, f0 at time t, tensor d z. This is true,  and similarly the energy also converges the same thing, in fact. Okay, let's leave it  at that. Okay, we have more conclusions which I've written down. And so what this says is  the derivation of the equation studied by Valerian Banach yesterday. And so let's remember  from Valerian's talk, this was first formally written down by guys in the fluids community  by Klein, Maida, and Damodar in the mid-90s, following earlier work of Zakhrov in the  late 80s. And basically the way these guys argue is by matched formal asymptotics. And  so this is, as far as we know, the first, or I mean it's not even the first, because  it's not finished yet, but we hope this will be the first rigorous derivation in any setting  of this model for the dynamics of thin, nearly parallel vortex filaments, starting from an  equation with fluid dynamical rodentite. I should have said the gross piti-fc equation  in principle describes superfluids, and so quantum mechanical fluids. And so rather than  looking at, let's say, classical ideal fluids as described by the older equations, we're  considering their quantum counterparts, which in some ways the analysis is much easier in  the quantum case in the gross piti-fc case. Okay, and so let me note here in this setting  that all the vortices have the same sign, and so one would like to be able to do this  for vortices of opposite sign. That's what one sees in these trailing vortices of airplanes,  for example. I mean, vortices of the same sign actually appears to be more relevant  in the quantum context, but who knows. And then, so concerning this equation, Valeria  summarized yesterday a lot of history, and so this includes work of what Klein might  have done with Arne who derived it from Kenick Ponson Vega. Lots of work of Banek and Mio,  also recent work of Walter Craig and Carlos Garcia as Piatia, and I think there's also  work of Garcia Piatia and Isé, and so one can sort of mention all the, essentially everyone  has worked on this in a sentence or two. Okay, and so in the remaining eight minutes, I'll  say a bit about the proof. So here's how this goes. And so the overall point is that  the proof relies incredibly strongly on, well, say on things in the spirit of theorem one,  which is here, I think. So in theorem one, we assume only some kind of sort of crude  knowledge about the vorticity. We say that we have n vortices concentrating on a certain  scale with the origin about this line, and we assume that an energy balance, and then  we deduce various conclusions. And so in the dynamical setting, we will choose initially  such that these hypotheses hold, and if we can show these persist in time, then we have  access to these conclusions, and further conclusions in the same spirit. And so a large part of  this, there's a theorem three, so to speak, which is, again, we'll be in the paper of  myself in DDA, which is sort of a refinement of these results under the same or similar  hypotheses. We extract still further information about the behavior of the vorticity in these  solutions. And so, but about the proof. So if we consider, let's call this the interacting  vortex filmeth system. So here's the point. If, so suppose f and f naught solve this,  then it's an easy computation to show that this is bounded by a constant times this.  Okay, so what I want to do, I like to do the same thing if possible for f naught solving  this, and f obtained from the gross PTIFG equation in the limit as f's long goes to zero.  And so let me suppose that I've somehow arranged that at every time these conditions hold at  every time in some interval. And so at every time I can pass limits along some subsequence  and get some limiting f. That will depend on time. If I have some kind of aquacontinuity  in time, I can do this. I can find a signal subsequence for taking conversions along  a time interval. And so if we try to mimic this computation, we'll find that well for  reasons that will become apparent in a moment. I get this, but then I get errors arising  from the fact that this is not the right equation. I have to do some approximations.  And these errors look like, and when I say f comes from this, I mean under my hypothesis,  under my hypothesis of rather sharp energy bounds, what we're at data. Okay, there's  some extra energy here. And so the idea is that if it's possible that, so I have a lower  bound here, the limiting vortex elements can have at most this much energy. They may lose  their energy. Energy may leak from the vortexes into the rest of the fluid. And that's kind  of an enemy you have to be controlled. And one thing we can do however is we can replace  this by the first variation by a linear term. Let's write it like this. And then I get quadratic  terms which turn out to involve only the L2 norm, not the derivatives. And so I can absorb  the quadratic terms into here. And I get a linear term. Okay, but then I'm still left  controlling something who's linear in f by something who's quadratic. That doesn't look  very good. However, it's also true if I go back to this system that for the linear term  in question. One has this estimate. And again, this is a straightforward. And so the point  is then, well, when I go over here and again do the same approximation arguments and so  on, I'll find that for f coming from the gross pretty easy limit, this is bounded by the  same expressions. And so at this point, I'm able to do a grung wall inequality and proceed.  The, of course, all of work has been obtained in this assertion that I can do this. And  so let me say a little bit about that in the remaining three minutes if I can. Okay, good.  So then, so what I want to do then is to use the, is to use this equation. The equation  tells me how the board is to move. Remember, the board is to tell me where the curves are.  And so let me multiply this rescale, I write things in terms of v instead of u. I multiply  the vorticity by test function phi. And after the rescaling, I'll find, well, there'll be  a term phi sub t. There'll be a term epsilon. So here i and j are, well, it's sort of like  this. I have the horizontal derivative of phi and then Hilbert-Schmidt inner product  with something, the tensor product of v perp with v. And then I have dz, the horizontal  derivative of phi and the dot product with, okay, the anisotropic rescaling gives rise  to log ups on here. I'd like to pass the limits in this expression. And what I'll do is I'll  take a, I'll take a phi who, for example, looks like 1 half x minus f not of z t squared.  Okay. And so then I better sum over the components and multiply by cutoff functions. Okay. And  so then formally, at least, well, actually, then it's really true that the phi times the  Jacobian converges to as long as the limiting curves are supported in the region where these  cutoff functions are 1. This will really converge to the L2 norm, the L2 norm squared. And similarly,  this will converge to what we know what this is doing. This will give us some different  pieces of that computation. And so we have to, in a bit, we have to know what this is  doing. And so it's the fact that this is converging. So this is one of the things proved in the  theorem 3. I didn't write down. This is converging to dz. So if I fix a time t, then this will  converge to, so, it'll converge to measure supported along the curve, along the vortex  curves involving the vertical derivative of the limiting curve. Okay. So these come from  the limit. And whereas this involves, if you look at it, this involves the, this is just  where the cutoff function is equal to 1. This is just equal to minus dz of f0 of the solution.  Okay. And so here we have, so this will give rise to a term involving the orthogonal gradient  of dz of f0 and dz of f, right, the thing I obtained from the limit. And this is the  hardest term by far, and I don't have time to talk with that. So let me stop here. Thank  you.  Yes. So is there another geometric context of this? Maybe some domain which, not cylindric  one. Maybe some other geometric, interesting geometric context.  Well, all right. So I'd say for the, for Grosby Tkowski cylindrical or an unbounded  cylinder or all of our three are the natural context. And so the, as I think Valerius said,  this limiting system has been studied most on, on the real line. And so with that we'd  need a domain which is unbounded in the z direction. That would be possible. For the  elliptic equation, yeah, so there, there what's much more interesting is exactly the interplay  between the geometry of the domain and, and the geometry in the limiting solutions. And  so for example, one could presumably do the same thing, it would be harder. And so one  could look for, if I have a three dimensional manifold with geodesic, a closed geodesic,  one could look for solutions with, with, and filaments concerting on small scales around  the geodesic. That's, that would be a bit harder.  I actually have the same, same question, but just slightly different, so for my own purposes,  you put diracley-Vonnician conditions at, at zero and L.  Yeah.  Can you put periodic?  You can. I mean the, right, so we, we have a, I mean this, this, this corollary says really  if we have a, if we have a local minimizer of the limiting functional, then we get local  minimizing solutions of the epsilon functional. And so in, in the, if I just have, if I just  have n curves, if they're not, if they're not linked in some way, then, then they'll  always want to spread apart and lower their energy. So they're, they're presumably are,  you know, let me mark also this is.  So the diracley-Vonnician's make you fit, allow you to fix the curves at the, at the  end.  Yeah, that's right. In principle, they could also be fixed by, by nodding. If I, so the  limiting equation in the elliptic case is just this without the time derivative. And  if you look at it, this is, this is really a planar and body problem where Z plays the  role of the time variable. And, and so people have constructed solutions. We, we didn't  have to, however, to, to, the thing we didn't do was to identify nodded solutions who are  local minimizers of some energy, which is what we need to do to, to relate them to these.  But the time dynamics, I mean, the time dynamics allows you to give initial data and follow  them for a period of time.  And that wouldn't cost the, you wouldn't need a minimizer.  No, that's right.  Any other questions?  Thank you.
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