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0:05
Tadmor from Universal Midland. He will talk about the existence and large time behavior in hydrodynamics warming. So first of all, thank you to the Professor, Ha. Thank you for the invitation. It's really my pleasure and welcome all hello from Maryland. Then wherever you are in the world. Here is the morning in Maryland.
0:32
OK, I see all the admissions and let me start. I will talk about the large time behavior and the existence questions in system of PDE is that arise from the models of swimming. So this is the second part, the hydrodynamics warming I will describe. I mean there's a lot of work that has been done in the last 10 years, which is pretty exciting and the. Most here can can is sharing this this excitement I will focus on. String of recent results which were being published these days,
1:16
so I would like to. To excuse myself for not mentioning references to every piece of work that was done in the last 10 or 20 years in this topic. So without further ado, let me start and this is a good point. Just second this. After a few things here.
1:39
OK, so let me start with the notion of alignment. So that's a good starting point with the work of cooker and smell from 2007. It's by now a classical model where we have capital NI believe that you use. You do see my my mouse, their capital N agents and they are identified by their positions X and by the velocity V and the key element is an alignment dynamic switches. Outlined here, where the change in the velocity of Agent I is balanced by a weighted average of the difference
2:16
between the velocity of each agent and the velocity of the neighbors which are indexed here by J. The dynamics takes place in a the ambient spaces Omega, which in order to avoid boundary conditions, I will take it to be either the whole space or just the Taurus. And this system of NOD is this is the starting point for discussion. Here are the the weights fire J which will be explored in a second. And the description here is a description of alignment.
2:51
Alignment is a statement where there are pairwise interaction outlined here, which Steeler tours in averaging heading. That that's that's basically the statement. And this is a key ingredient in self organization in many models. For social interactions, either in ecology. When you think about flocking of birds or swimming, or bacteria, or insects, cells in human interaction, the alignment of pedestrian. All the way to emergence of corn in opinion dynamics.
3:21
The imagine of consensus and sensor based networks swarming of mobile robots to macromolecules and there are many more examples. So a key feature in all these models is this alignment, which is written here, which I will try to explore during the lecture. Now in this in this dynamics everything is dictated by the communication kernel file.
3:48
Just one second I have to hide something. Now it works. Everything is dictated by the communication kernel, fi and the fire J. Here are the communication kernel between the position of the agent I and the other project. The other agent position J and I will normalize this kernel so that the integral will be one.
4:12
OK, so this is the general setup. We have capital and agents in little N dimension. And they are driven by this communication kernel. Little Phi, which depends on the position of myself. Agent I and my neighbors, Agent J we normalize it. Tao is some positive parameter which dictates the amplitude of this alignment process and the rules of engagement depends really on this choice of this kernel fight. This is not dictated from above and there are many origin for this kernel.
4:52
And the choice of the kernel is context dependent. Let me mention few examples. The phenomenological approach we just take here is this is in this example, which goes back to cocoa smell. It's something which decays algebraically with the distance between the position X and position X prime. So this is an algebraic decay which makes sense that the further we are from another agent, then the decay of the influence of the communication. There is some sort of decay which is described here.
5:25
Another source is empirical. For example, it's learned from the data and there are many works in this direction.
5:33
And I gave her only one reference that the Group of Maglione where they learn from the data. What is the shape of this kernel communication kernel 5 and then a third source for the kernel file, which is my favorite is driven from higher principle. So here is a model for anticipation which was originally introduced by Van Bergen and his coauthors, and the idea is that we see now the dynamics driven by the potential. But the reaction is for this potential. The reaction is given in terms of the.
6:11
In terms of the anticipated position of the neighbors. There are many things which are run on my screen, so I have to delete him constantly. Sorry. So the idea here is that the dynamics is driven by the anticipated position rather than by the position itself. Then dissipated position is the current position plus a small parameter taking into account the local velocity. What it means really is that mentality to 0. This is the usual N body problem when Tau is positive.
6:41
If we expand the right inside for small cow then the first time gives us. The term which is expanded here where you prime is has to do with the derivative of this potential, which is assumed to be radial. As you can see, and then the second term which is of interest gives us exactly the term of alignment that we had before. But now we have matrices here, so this and this. This term represents anticipation which leads to alignment. And these matrices have to do with some average hashian
7:13
of the original potential and first term is the usual repulsion plus attraction, so it's repulsive when you prime is positive, attractive when you attractive when you prime is positive and repulsive and you prime is negative. And here we have in some sense the new aspect which is the alignment and we had before in the matrix form. If we take the scalar case then we get the. So this is in some sense the derivation of the cucos male model from a higher principle.
7:45
In this case it's a model for anticipation.
7:51
So I should say here that what we have is essentially a model which includes three rules of engagement, alignment, repulsion, and attraction. In this talk I will focus mainly on alignment, but I should mention that everything began with alignment, repulsion and attraction with the work of Craig Reynolds, really a pioneering work in 1987. Where he wanted to make a realistic description for flocking of what he called boys and the best way to describe it will be simply by a movie.
8:28
So what you see here are the boys of Craig Reynolds, driven by alignment, repulsion and attraction, which is a pretty realistic dynamics to something which looks like flocking. And if we wait a little bit and we see that we turn off the alignment even though we leave the attraction repulsion, then you see that the behavior is fundamentally different.
8:57
And now the alignment is turned off and you can see that you get a completely different behavior. So clearly the alignment is a. Key component in cell phone can self organize dynamics and this is the main feature that I would like to focus in this work. So as you can see here, now they're wondering around alignment is turned on, and then there is some order in some self organization is coming back into the model. But I should mention that alignment is not the only feature.
9:31
The work of Reynolds, what is called the Trees on protocol alignment. Attraction and repulsion, so this is that reason, protocol and. Today there is still a mathematical theory, in particular to the local aspect of this tree zone modeling. OK, So what are the questions that I'm interested or the questions that are being asked and still not clear? Well one question that are very much interested in the last few years is in the hookers male model. There is 1 / N YN.
10:08
No one counting anything, so N is really is not plugged into the system and is nothing but a scaling factor. And over the years there are various models where there are different factors instead of the end being uniform for the whole crowd. The end now depends on the agent. I will not. I will not explore that. In fact there is not just a question of scaling. There's a question of multi scaling, which is very fundamental which leads to multi flocking.
10:38
That's that's the topic of a separate lecture. If we have time then I will briefly mention something about this and so I want to share with you my concern about the weakness of this scaling 1 / N in some sense it just for convenience. Another question is the different classes of interaction for interaction kernel since they are not dictated from above. The question is what is the fact that we should use so in the literature there are various classes.
11:07
Let me mention few. The 1st is a metric Colonel where the five depends on the distance between the position X&X Prime X being typical agent and X prime is a typical neighbor. For example, in the original Cookus may model I mentioned already there was this kind of. Algebraic decay. With some parameter gamma here and what is important in the cucos male model are two things. The behavior actually the the important thing is the behavior is our tends to Infinity.
11:39
This is in sharp contrast to a singular kernel lycoris potential. Here we're now we're interested in the behavior for agent which are very close are is very close to one or R tends to Infinity depending on the power gamma. So these are two examples for metric kernels, but this does not include everything. Here's another class of topological kernels now Fi there, namely the rule of engagement between agent at position X&X prime depends on the number of agents between some enclosing
12:13
region between X&X prime. The statement here states that the interaction depends on the crowdedness, which is very reasonable in many situations where the propagation of information is not just in a. Vacuum, but it really depends how crowded the region. How many agents are between X&X prime in some every sense. So these are topological kernels. And then another class of kernels are cut of kernels which were introduced into another classical model option, not talk about the vicsik model,
12:48
which is a characteristic function. Let's say over ball. And the important thing here is that we are completely uninterested. What happens in our when art is much larger than one. We are just interested at moderate art. So these are local or cut of kernels. So here are different classes of kernels and there are many more. I will not cover, but I would like to raise the following fundamental question. Here is the key questions. How different classes of kernels affect the large time behavior?
13:17
That's the question I'm interested in. To address this question. Let me the question of large time behavior. Let me go back to the original Coca smell. So these are the velocities. This is the intensity of the alignment fire. J is the rule of engagement is a kernel which here I take. Actually it could be any symmetric kernel and we associated with dynamic. This dynamics are weighted graph. The graph consists of the verticies where the agents are and the edges. Whenever these two agents talk,
13:56
namely whenever Fire J is greater than 0, so we have here a graph and we would like to look now at the energy fluctuation for this dynamics and the energy fluctuation is defined here. It's the sum of all the VI minus VJ squared. This is the L2 energy fluctuation and the key statement is here. In the statement says that the rate of change of the energy fluctuation equals to the quantity here, which is called enstrophy. And we discussed this, and this quantity is upper bounded by a scalar factor
14:33
multiplying the the same energy fluctuation that we began with. So we can close this inequality by a global approach. So we just have to understand this equality won an inequality 2. Equality One has to do with the fact that the term fire JVJ minus VI is is antisymmetric. So the assumption is that the Fire JS symmetric they need not be even function of the distance any symmetric interaction. Will do because if the fire just symmetric the Vijay minus via antisymmetric.
15:09
And then if we take this equation and we multiply by VI and saw it, then by the antisymmetry we get the term that we have here and the team here takes into account the fluctuations but only those fluctuations between VI and VJ that talk to each other, namely when I&J belongs to the having a an edge. If there is no communication, then there is no decrease on these fluctuations and the question now is how do we go from this quantity, which is the enstrophy to this inequality?
15:46
Or put it differently, how we can we can tell the fire J outside? And paying the price of a scalar multiple scalar multiplier. To do this, we have to define the graph Laplacian. A graph Laplacian is nothing but the original adjacency matrix 5, which is given here. Where we place in the diagonal the sum of each row. So we put minus the term for each by J and then we put the some of the terms in the diagonal. We get here a symmetric matrix which is positive definite,
16:20
not semidefinite, and in fact this dynamics on the talk is nothing but heat equation with this graph Laplacian and the minus here is because this graph Laplacian is positive and like the usual Laplacian in mathematics which are negative. And then if we look at this dynamics of the graph Laplacian, it follows that this equality, this this term on the right hand side, is bounded from above by minus Tau and then here they dis quantity in red ETA is precisely the second
16:55
eigenvalue of the graph Laplacian. As we know in usual case from the usual classics. So I will not elaborate here because we will have to do it. In great detail later on. So we get a sharp estimate in terms of the second eigenvalue of this matrix, which is the graph Laplacian, and then once we plug in the graph of class and we integrate these ground with inequality and we conclude that the energy fluctuations are bounded from above by this
17:28
scalar multiplier, multiplying the initial energy fluctuations. So now the question is whether or not this integrand tends to Infinity if it tends to Infinity. Actuation, we tend to 0 if the fluctuation tends to 0, then the large time behavior will be that all the velocities would like to approach a common limit. So. The question is now what happens with this second eigenvalue and the second eigenvalue here is called the figure number and what it reflects is the the connectivity of
18:04
the graph, the connectivity of the graph in the usual sense that every agent or every node is connected to every node through a path where the information propagates. So if the graph is connected in fact slightly more than connected, if the graph is integrally connected. Then we get unconditional flocking in the sense that all the velocities will approach a limiting behavior. So this is the last time behavior of this cucos male model and the only if here is the behavior
18:39
of the second eigenvalue of the graph Laplacian. In the second value of the graph Laplacian still leaves us with the open question about the connectivity of the graph. Now before I'm leaving the discrete case, I should say that I stop here becausw this question. This is similar in PZ. This question is basically open. Is basically open. In order to move from something is opened with something. I can say more. I will move now from the discrete case to the continuum and this is done by
19:19
taking the large crowd dynamics. There are many works that were done in this direction. I just mentioned the few. The important thing at the most recent one are by Natalie Natalie Knee and Poland right? Koi and the idea is the passage from the agent based description to the hydrodynamic description. So let me do it. In some not precise way, the idea is that we define the empirical distribution which is given here and we end up with a kinetic formulation.
19:52
And the kinetic formulation tells us the transport of this empirical distribution, so it's being transported. And it's also subject to this by Agent Interaction, which is defined by this quadratic term and this quadratic term takes into account all the binary interactions, the alignment, possibly repulsion, noise, and everything else. In the case of alignment, the idea is that the agent that look at position X with velocity V interacts with the agents.
20:23
This is a probability measure with agents with position X prime with velocity V prime. We take into account all the other agents and the important thing is that they are weighted by this communication kernel.
20:37
Now this large crowd dynamics is captured by two moments. The first 2 moments, so I will assume that the integral overall velocities of the zeros in the first moment exists without further. Description or words. I will just assume that this is true under certain assumptions and then we end up with that dynamic description in terms of the density roll and the momentum roll U, and this is the beginning of the PD talk when we end up with a mass equation,
21:09
the usual mass equation and the usual momentum equation. So this is the momentum. This is the pressure tensor and the only change from the usual hydrodynamic description is of course the alignment term. And they are given as follows. The pressure is is a symmetric positive definite matrix given by the assumed to exist limit of the second moments of F. This is a rank one matrix. And the alignment term is a factor 10 which is given here, which takes into account the velocity of an agent at
21:47
position X. It interacts with all the other agents at position X prime. It takes into account the density at position X prime we take into we integrate all the other neighbors with this communication, wait and here's the alignment. So there are two terms in this header. Dynamic description, the pressure and the alignment. The alignment is really what carries the information about the agent based description, the alignment model and the pressure of course requires the closure.
22:22
So few words that I would like to mention here. This is not another dynamic model because I didn't tell you how I close this piece. Still depends on the second moment of the kinetic formulation and we have to do something from the outside in order to close the system. This is the point of departure in my talk from a lot of literature. Most of the literature closed this model either by mono kinetic closure. We just assume that P vanish is or some simple closure which is different than Maximilian.
23:01
In order to proceed. I will not do that. All my talk will be relevant to general system even though I will not close the pressure here. So that's the point number one, and I would like to bring here a quotation from from a very I think important paper by Victor can zephrys. And they mention of course this pressure has to do with the internal energy, which I will not elaborate right now. They said that in dynamics of flocking and swarming, the source of energy making the motion possible,
23:40
these are not relevant and therefore in other words they say we completely ignore this pressure term and there is no need even to conserve the momentum. It is true, however, my point of view is that the lack of thermal equilibrium and all these things of course are related to the fact that how do we close the pressure? The lack of thermal equilibrium, even though we let the qualities I would like to substitute this equalities by it is inequalities.
24:11
And in particular, I would like to close this inequality by decay of energy fluctuations. So even though for the people who are not familiar with this topic, I would like to say that this is a critical point. How do we close this system and my statement here? I will try to pursue as much as possible the analysis without closing the system at all, but due to the fact of the existence of alignment and we will see that alignment is a pretty strong forcing. So let's talk about energy fluctuations.
24:45
So the energy fluctuations I continue. Now we take the second moment and this of course is the usual total energy, which can be decomposed into kinetic and internal energies. According to this simple decomposition. So 1/2 the square is 1/2. You squared 1/2 -- 2 squared and then this inner product. Now when we integrate defeat this term drops out and the total energy here drops into two components. The 1/2 you squared. Newton this is the kinetic energy and then we have the internal energy.
25:19
So we have kinetic and internal energy. And now the energy fluctuations. I would like to consider now the energy fluctuation which correspond to the discrete energy fluctuation that we had before and here at the energy fluctuations there are there microscopic fluctuations in the macroscopic velocity that the difference squares between U&U prime and then I take a symmetric sum of the kinetic, the internal function energy fluctuations observed that this little E
25:53
is really the internal fluctuations around the average velocity U. So this will be the energy fluctuation. And just as we had in the discrete case we have the following statement, the rate of change of the energy fluctuation defined here is equals to the same energy fluctuated fluctuations rated by 5. This is a pretty strong statement which goes back actually to our original paper with her my original paper with her. Even though I have to admit we have to correct, there's some typos there.
26:32
So this is a pretty strong statement, and the important of this statement that this applies in dependently of the closure. In fact, if I equal to 0, is simply the conservation of the energy. This is just the conservation of energy and then in general in the case of a general interaction we have the decay of the fluctuation dictated by this quantity. Of course the question is how negative this quantity is, namely how negative dystrophies, namely, we are now looking for the.
27:02
Second question, we are looking for an inequality of the type that this is a scalar multiplier. We're looking for this scalar multiplier, multiplying the energy fluctuations. So the question is, is there such a scalar multiplier and that's what I would like to discuss next? And this is done by spectral analysis and much of what I'm discussing today is actually in in an upcoming article that will be on mathematics of swarming that will be appearing on the notices of the Ms.
27:37
So the first thing to notice is that we have internal fluctuations. Let me denote by 5 convolution Rd this interval, even if I is not metric, then it's easy to see simply because of the additive nature of the internal. The internal energy that it appears in the fluctuations that this integral rated with five is bounded from below, and that's what we're looking for. We are now looking for a lower bound for this sense trophy. Is bounded from below by the double integral of the
28:09
sum of the internal energies. So we have now this one scalar multiplier which has to do with the minifier convolution withdrawal. Of course more complicated term will be what do we do with this microscopic fluctuations? So we have to look at that are dynamic fluctuations given here this this is. This is means simply roar. Prime DXD explained to this and I introduce a weighted Laplacian operator. The weighted laplacian. This is a Hilbert Schmidt operator and
28:43
I restrict my attention now to the Taurus and the weighted Laplacian. It's weighted by the density is defined as follows, so this is a precise waiting as it should. It just fits into this theory and observe that this operator is symmetric. It's clearly symmetric and it's no negative Laplacian. The reason that it's not negative is when you act when it acts on a weighted. Vector functions, so W is a vector function weighted by the square root of the density. You get precisely a positive term.
29:19
So that that's the key definition here. And then this operator and Hilbert Schmidt. It's a symmetric nonnegative. It has a discrete spectrum, Lambda once less equal than Lambda two, and so forth. And we know even what is the first eigenvalue. The first eigenvalue is 0. The reason is 0 is because we have an eigenfunction which is a constant function weighted by the square root row. If you plug here, the square root roll up to a constant vector function, you get 0. So the first eigenvalue is 0.
29:49
As it should with the Laplacian and therefore we come now to the second eigenvalue in the second eigenvalue by. Classic results is of course the infamous of overall vector functions which are orthogonal to the first eigenfunction, and you can see here this inner product already told you it gives us precisely what we get on the exstrophy. This is exactly the eltoro norm of W because the inner product vanish is better tonality, but this ratio is precisely what we're looking for in
30:25
order to bound the extra fee. So we have now a sharp lower bound. On the extra fee that this integral of the weighted fluctuation of the velocity is bounded by the 2nd eigenvalue, now it's not a discrete Laplacian. Now with this weighted at present that they just introduced. And so we have a bound on the energy fluctuations and our theorems therefore says that now the energy is decaying by a scalar multiplier multiplier, which has to do with the mean between the spectral.
30:59
This is the second eigenvalue of this operator and of course the minimal value of five convolution role you remember the internal energy. Now this statement applies to any mesoscopic pressure independent of the pressure, which is quite. I would say remarkable and it applies to arbitrary symmetric kernel. It may not be a metric. And of course, the question which is not trivial is now to find lower bounds for the second term, and for this second eigenvalue. And of course for the convolution.
31:33
Now let me start with the convolution for talking about long range interactions. The lower bound for the convolution. This bounded clearly take into account that we normalize far is bounded by the minimum of the. The the the Colonel. So I will take you in the middle of the kernel and the second eigenvalue of health also is bounded by the minimum value of this kernel. So this gives us a requirement of global communication. If the minimum of the kernel is bounded away from zero,
32:06
of course we will get exponential decay and hence we get the large time behavior for the models for the Euler alignment models that I mentioned before. So let me look now in the class of kernels with this perretta tail so. We will assume that this kernels decay. There need not depends on the distance, but they have a lower about which decays in this kind of operator tail behavior. So in this case we have to introduce the diameter of the crowd.
32:38
This is the diameter. Depending on the support of fraud and diameter of the support of Rowan, the immediate corollary for arbitrary pressure is if there is a strong solution driven by long range kernel language kernel means that this gamma is positive. Then we get this. Of course this kind of estimate cause this diameter is a lower bound for this kind of behavior. So now we have to see what happens with the diameter of the crowd. And we get a flocking,
33:10
which is a prototype of swarming behavior and the following statement is true if the diameter does not spread too much, say bye. Range of order Beta and his baited at times Gamma is less than one. Then the fluctuation decades to zero. These are the microscopic fluctuations and the intended fluctuations as well. So there's a if here the question is whether or not we can control the the Spurs dispersive effect of the crowd and to what range. And let me just mention 2 examples for long range
33:46
locking. For example, if the velocities are uniformly bounded, clearly the diameter of the flow can be not larger than T, and then we get fractional exponential decay. In fact, if we have a model kinetic closure, if we now assume that there is no pressure in this case, it is known there's a lot of literature about that, and the diameter remains bounded and there is exponential decay. So I think I mentioned few. I mean, I particularly should mention here the result of hiring you.
34:21
It goes back to. There are many results in this direction. I would like to mention another result in this direction. After example, let's take the matrix valued kernels of Coca smell. That's fundamentally different, because now we have couple system. In this case, if the matrix is bounded is decaying by desperate retail. So this averaging means this. This this definition it has this kind of decay then. It can be shown that the diameter does not grow faster than some power beta,
34:50
which has to do is 2 /, 2 minus gamma and therefore we get flocking if gamma is less than 2/3. So OK, so these are example of long range. And the flocking behavior that we get in this case are either traveling wave. This is a Canonical example or we get if we had a potential, we get our money consolator. And here is just another simulation just to show you that it's more interesting than just traveling wave in a straight path and what you see here is the concentration
35:23
of the group along harmonic oscillator. OK, since time is running short, let me move now. Too short range interactions and in the case of short range interaction we cannot assume that file is uniformly bounded from below. So now I'll restrict attention to metric kernel and I would like to try to estimate the enstrophy here along the lines of what I've done before. What will be a lower bound for the extra fee? So a lower bound for the internal energy is easy
35:54
because five convolution role is now bounded by the minimum of the density. What about the enstrophy? It turns out that in this case, the second eigenvalue of the operator introduced is bounded from below by a constant, which is specifically given by this Fourier coefficient of the. Communication kernel the radio communication cannon and it has to do with the minimal value of the density and the maximum value of the density, and I still do not like that this is not symmetric,
36:25
but I cannot prove it without the two is apparently true, but I was unable to. This is not the point. The point is that as a consequence of this theorem, if we can bind the second eigenvalue, and if we know already that the internal energy has to do with the minimal density, so we can see that in both cases the minimal density is a critical component. And 5th immediately corollary is that as long as the maximal variation of the density is not too large in a way specified here,
36:54
then this ratio of the densities is bounded from below and we get exponential decay. So here are a few examples. If we have one dimensional dynamics in a which is supported on an interval, then we get exponential decay and the same thing happens. We stood dimensional dynamics. These are results which I was looking for for a long time and I'm pretty pleased with this result. Except. Because they are independent of the support of five, they are independent of the pressure.
37:27
They independent of the internal energy. However, the key issue is what is the behavior of the density? So at this stage we see that the density, which is usually a silent variable in many other anomic equations. Here is the most important is much more important than velocity because the density being away from vacuum is the thing that enables connectivity. It enables propagation of information and the lower bound of density it has to be assumed. So far I was unable to to cover that,
37:56
so this is an example. One example for short range kernel. The other example for short range interactions where we can prove things. For example, when we have short singular kernels. So here's an example for a singular kernels which is completely supported, and in this case this is an older result with Roman. Tried calling if the density is bounded from below with a certain rate than smooth solution must look, so again we see that we have to control in
38:26
some manner the density from below and it's left open. There is an easier which were unable to prove. In this case we get unconditional flocking for short range, non vacuous interactions. There are cases where these things can be removed, and I mentioned here few results in the 1 dimensional case, but in general the connectivity to non vectors density is still open. Let me go to 1/3 option example for short range interaction which has to do with topological kernels.
38:57
I mentioned them before where we take the distance between two positions at X&Y, not to be the metric distance, but the distance which is weighted by the density enclosed between them in some domain which is denoted here by Omega. So this is is the dictator to be original communication and then we look at the short range protocol. It showed range because it's cut off. Within a bowl of radius R0 and it takes into account the density so that the communication is inversely proportional
39:29
to the crowdedness the crowded it is there is less communication less crowded. Then there is increasing communication and this has to do with the topological description. I do not have time now to explore why it is topological, so we now look at this kind of model. There is no pressure and then we showed that in this case smooth non vacuum solution. Assuming that it's bounded away from the vacuum mass flow. So this is the third example. And finally, let me go to the first example for
40:04
multispecies.
40:08
In this case, there is not only once one equation, but there are collection of equations. These are multi species and you can see that now the velocities depend on a index Alpha which dictates what the species is. What is the difference between once PC and another species? The way they communicate in between. So if all the files are the same, you get one species. If the different fire AA betas then you get different species. So in this case you get flocking.
40:39
And again I have to skip lot of details, but the key element here is that this matrix of communication has to be connected exactly along the same lines as before. Connectivity is measured by the 2nd eigenvalue of a weighted Laplacian and which is quantified. Here the details are not important, but the message is important that connectivity is the key aspect for this large time behavior. In fact, what we happen where we have here is a sort of a game. For example, if you have two groups.
41:10
One talks with the other, but we not visits on kind. Nevertheless, in the long range in the in for long time they will coincide in the same direction. OK, I will skip the other talk. What is the origin of this multi species and I would like to talk also about the existence of strong solutions. How many minutes do I have? Some we have 5 minutes. OK, that's fine. We have 7 minutes. OK, so this is the second part and it's important part because all the statement that I said before they
41:45
they they have the following flavor. If Roy you is a strong solution then so and so. What about this? If that's the question that I would like to ask, so we look at the system and I would like to know whether or not it admits global small solution.
42:02
So here there are two competing mechanisms. First of all, day is the usual stiffening of local fluctuations where the graduates want to tend to minus Infinity and the same time we have alignment and it tamed the fluctuations and then it induces the divisions to be bounded from below. The question is who wins? Of course I cannot answer this question now without the closure on the pressure P. So I will take the mono kinetic closure and I will assume that the P is different from zero.
42:34
P is different from zero and then we have a balance between the older dynamics or learning dynamics and the alignment. But before I'm doing that he was something very interesting. When you look at the system behind the system, there is another equation for the internal energy. The internal energy is the intelligent defined before. In fact the internal energy users usually satisfy this equation equal to 0. But in case of the alignment on the right hand side we get a new term.
43:05
And the term is to do with the trace of their normalized. Pressure matrix and the Alignment tab, which is given here as we know before Iftar equal to 0, so these are the usual equations. Now it turns out that this is a nice exercise that if you ever matrix whose traces one and this is true for people, if you take the trace of a normalized matrix P bar times the gradient of you another matrix, the grand interview is the usual stress tensor.
43:34
It's always bounded by the minimal eigenvalue of the symmetric part of this tensor. In this statement is independent of the pressure on the internal energy. So we start with something in the determined dynamic level. We end up with something which has to do with the dynamics and using this simple exercise. We look for a threshold which is independent, the thermodynamics. Namely, we want this scalar multiplier J to be positive bounded away from below bounded away from zero.
44:05
Why? Because if it's bounded away from zero, then the total amount of the internal energy by integration will decay. So we impose this kind of decay as a motivation. Well, we would like to prove it is very nice, but the question is, is it true? In fact the question is assumed that this ETA indicator is initially bounded from below, which we can assume there's this persistent time. That's the general question that imposing and I would like to address it in the case that there is a
44:43
monochromatic closure. Of course, in the general case it's it's one has to be specific. Now this condition is interesting because in the 1 dimensional case it is true, infecting the 1 dimensional case. If we take the system that we had before in the wonderments case, it is well known that there is a global solution if and only if the initial conditions satisfied this threshold condition, which means that the minimal value of the derivative classify convolution rose 0.
45:10
These ones from below by zero and the same thing is true for the two dimensional problem. It's equivalent to what I'm showing here, and this led me to the following general theorem, so we consider now and later general 11 dynamics. There is no pressure we would like to know about the global existence in any number of dimensions, so this is an older alignment system subject to non vacuous initial data which are sufficiently smooth, and we assume a threshold condition which reads the following
45:44
the initial eigenvalue of the symmetric part of the Grand Interview plus style times 5 convolution row is bounded by a constant. Given here. The most important thing is that this statement allows these numbers to be negative. Not just positive becausw, this quantity is positive in this quantity is positive and together we allow these eigenvalues to run into negative values. Then the minimal value. I mean this quantity remains bounded by this threshold and the equation that made global small
46:16
solution. So this is a global existence under this threshold. And another important question is to develop a theory for weak solution for this system. Another question of existence applied to singular kernels. And we know that in the case of singular kernels, which involves which involve to political distance, they exist. Global solution. This result goes to show White Boy and myself. I believe that I I stop here about the question of existence. I would like to thank you for your attention and
46:49
I'll take the advantage that the topic and that enables me to put very nice video. Thank you. OK, thank you.
47:00
OK, so we'll have a.
47:05
Any questions and comments?
47:34
OK, I think can I ask one question. Just give in your in your one of the theorem that you show us. If law minus is bounded by one over skirt to T, then you have a frocking. Could you explain to me why how one overskirt OT appear naturally there? It's a yes so you can see that OK. Let me say you see that I I I. There is one theorem. When I have 1 / T, which is apparently the right decay. The point is that here is we have the one over square root Y that you are referring to.
48:18
This is the case that we have singular kernels. OK, in the case of a singular kernels, if the density is bungalow by one over square root T, it looks like a very ideas theorem, but I'm not proud of it because I know that this is not a sharp. This condition is too is too strong. Then smooth solution must look now. So first of all I would like to take. And to criticize this, not to Christmas this theorem, but to put it in the appropriately in the proper light.
48:49
Now where does where is the source of the square root, you ask? Well, it's the same point where this square root appears.
49:02
Just a second. Is the same source. I'll explain in a second.
49:10
I'll explain in a second.
49:19
Just one second. I have to find it.
49:27
Yes, it's the same source for this role. So what happens in the singular case? So let's let's look at this theorem. You just have to believe me that in the case that you have single occurrence, this is not this case. In the case of a singular kernel, we do know that the maximum of the density is bounded from above, so that's not an issue. So the real question now is when can we say that this quantity is intact? The integral tends to Infinity, so you can see that the borderline is exactly when
49:56
the minimal of the density is like one of the square root. The.
50:03
So they're all tide together to the same thing. Can we improve, uh, this this square root, and in fact, what can we say so the message take home message is really what can we say about these models? They had a dynamic swarming. What can we say about the behavior of the density? Becausw this framework the spectral analysis and so forth tell us that if we know something about the density, the density is the medium which carries the information of connectivity.
50:35
And then this large literature that you're familiar, of course. About what happened with global interaction kernels applied to local interaction terms. Now local interaction skills. Of course. I'm much more realistic models. In the to address this question, it reduces now the problem to the behavior of the density. So to your question, is the same problem, why heavier the square and is the same problem now? What can I say about a lower bound the density?
51:08
OK, so thank you any any other questions and comments?
51:21
I have question. Yes, yes please can we go back to page 23. Wow.
51:37
Yes. Uh, yeah, um so here, Phi is scalability, right? Here find scalar yes so then can you obtain some kind of this kind of global strong solution result for metrics valued kernel?
52:00
I mean the as you said in the previous stages for like Yes, yes exactly understand exactly. Question. Uh.
52:15
I mean that if this kernel is, yeah, yeah yeah, I understand. I. Um?
52:29
I do not know. I mean, it's a very interesting question. It's not only I do not know, not that I do not know if it's true. I do not know even if it can be done because what is behind this theorem. Is there a lot of technical data which is?
52:48
So if Lisa matrix. Very good question. That's a very good question and I I cannot even. I have to talk to go back in to see I don't know if I say I ask the question to prove global existence when there is a 5. Uh.
53:13
Ah, I see so actually I see. Yeah, I don't know. That's very interesting because in the 1 dimensional so far, if you ever matrixes dimension D by D and I see in the scalar in the 1 dimensional case it must be scalar. So the question already appears in the two dimensional case and I never thought about it. I don't know. That's a very good question. In fact it is a good question becausw. As they try to hit here, the original cookie smell goes with scalar interaction kernel whereas.
53:51
This one derivation of the Cucos mill leads actually to a matrix interaction in a very natural manner. Yes. Yes, yes there are more than one works about matrix interactions, but it's not clear. I mean, besides the matrix form. So I tried to explain why it follows from higher principle. Would be very interesting to talk to raise this theory into the Matrix case. I see thank you very much. OK, other questions. And comments. OK uh.
54:29
I. Any any other questions?
54:38
OK, so if not, then let's thank our speaker again.
54:47
Thank you yeah yeah. Do you have something to say? Yeah, just a few words. We got yeah your your speaker is alright I'm sorry bout that So what I was saying is basically what I just wrote in the chat. So thanks first of all to both of you to our Chairman, speaker today. It was very nice session I think, and also thanks to everybody for joining us today next week. We have another talk at the same place at the same time. You can use the same link again next week. Talk about yoga Gomez from coast.
55:25
And sorry to everybody for the confusion in the reminder email that was. Not working link. Actually I think for many people the zoom link was wrong. In any case, thanks a lot to all of you, goodnight. Thank you for staying up so late, I think. Have a nice day. And everybody else have a nice day as well. Goodnight or whatever the time is. Thanks a lot. OK bye bye bye.
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