 Thank you, and let me first thank the organizers for the opportunity to speak here today and the whole Month that I'm spending here, which is very nice so I'm going to talk about the stability of the quite flow and first of all I should introduce the Navier-Stokes equation So the equation is as follows Where nu is a positive coefficient, which is called the viscosity And if nu equals zero the equation is called Euler equation And there is a divergence free condition the divergence of u is Zero okay, so u is valued in r3 and It's a function of t x y z x is in the torus y is in the real space and z is also on the torus So I agree there is a few fluids that live in such a geometry nevertheless it has the advantage of Getting rid of boundaries so what I want to discuss today is the stability of shear flows and And So it's already a complicated question and if you add boundaries it adds a number of difficulties Which I don't want to consider today So this particular geometry allows one to have shear flows without boundaries So what is a shear flow? A shear flow is a flow of the type say f of y zero zero And the particular flow that I want to look at is the Couette flow where f is a Just a linear function So you have y Zero zero, this is the Couette flow and this is a solution of Navier-Stokes for any new positive so the stability of shear flows in general is a difficult question and What we discussed today is the stability of this particular shear flow, which is simpler for reasons that we will see So maybe I should draw a picture So this is y this is x and this is z and So the velocity of the fluid at the point on the y-axis Goes like that. So you see there is a linear shear So I hope you get a sense for the geometry that we are looking at So the question that we want to understand is the stability of this flow which is a Stationary solution of Navier-Stokes and maybe even the asymptotic stability. It does not have finite energy Okay, so what can we say about stability so The first point is that you need to take a positive viscosity coefficient, otherwise you pretty much have No chance. Okay, so you have to be positive So next thing you do is try and look at the linearized problem so informally speaking you just drop the Convection term and what you keep essentially is the heat equation So it's a bit more complicated because you need to linearize around the the quiet flow But essentially you get a linear heat equation With some more complicated terms which are lower order But what you see immediately is that the linearized problem is stable And it's even very stable in that you get a spectral gap So once you have such a strong spectral information, it's not hard to reduce Nonlinear stability. Can I ask a question? Yes Spectrum depends on the space And we have an infinite energy flow. So now we need a that's to a finite energy Protubation and just L2. Is that what you want? Yeah, so let me remain a little bit informal at this point, but say yeah in L2 for instance Okay However, there is something which in hydrodynamic stability is known as the sommerfeld paradox so It's it's true for the quiet flow and for many other Shear flows. So what happens is that the flow is Linearly stable as we saw but experimentally unstable Okay, so it's called sommerfeld paradox So linear stability But experimental Instability yes, yes Okay, so of course, it's it's not really a paradox really what it underlines is that The the important notion is not so much nonlinear stability as the size of the basin of attraction of the quiet flow Okay, so if you want the resolution of the paradox is just saying that the important Idea say is to sort of quantify Nonlinear stability in finding the size of the basin of attraction and This was already underlined by Kelvin And finally there is a last idea which maybe we're not so used to in Partial differential equations, which is that the the size of the basin of attraction Really depends strongly on the topology that you choose. Okay, and this idea goes back to Reynolds Though he phrased it in more physical terms so If you put all of this together, you see that the right question is the following so maybe let me call Let me change notations a little bit. Let me now call you not the perturbation of the quiet floor initial perturbation of the quiet floor and you To be the perturbation of the quiet floor at later times as it is given by just solving the logistics equation For later times. So I hope I convinced you that the right question is the following If you give me say a banner space X with the norm associated to it Then what you want to find is what is the smallest? real number gamma Such that if the initial perturbation is Less than new to the gamma so remember new is The size of the viscosity and the more viscosity you add the more stable the flow So it it makes sense that the dependencies is like that so then you might ask Why should the dependence be? power like Well, it's it's simply what the experience teaches us so what you want is that if the Look numeric Theory it all gives this sort of dependence small new yes This implies first that you of T In X remains small and second that you goes to zero as T Goes to infinity. Okay. So in other words what we're asking is to really quantify the size of the basin of attraction of quets What we're asking for is asymptotic stability if you're sufficient sufficiently close to To quit initially Okay So there has been a lot of work on on this question So I'm just gonna mention a few a few names Yes, that's right, that's true. Yeah I mean of course you could ask You know, you could ask her for that you remain close to quit for a large time, but I don't think this has been too much looked at Right, so there has been a lot of heuristic or numerical works So, okay, let me just mention a few names. There is baguette Driscoll Traffitin There is baguette and traffitin There is valet there's Chapman okay, and many other and Well, they actually don't really agree they find that gamma lies between one and Seven quarters depending who you believe Okay, so that's one set of work where The authors did not really care about the importance of the topology, right? So they just considered perturbations without really mentioning what sort of perturbations And then there is another work by ready Schmidt Baguette Henningson Where the authors find gamma equals three half for what they call rough perturbations Okay, meaning you perturb quit by something which is like white noise Whereas here they were more perturbing by Gaussian say Okay, and there is also vigorous results So it started with Romanov who looked at the case of a channel rather than the whole space And then there is Christ Loonblad Henningson, okay, and once again, there is many references. I don't cite everybody But the the world record was gamma less or equal than four by rigorous means in dimension three so everything is in dimension three and in Subol-F topology Okay, so that that was the state of the art Now let me describe our results So this is So maybe let me first But the three decays and then I'm gonna say word about what happens in 2d. So in the 3d case There is two topologies that you can consider either Jeuvret topology or Subol-F Topology so in Jeuvret topology we found gamma equals one and In Subol-F topology we found gamma less than or equal to three half. So this is a joint work with Jacob Bedrosian Who's in Maryland and Who's at NYU So what's very nice is that it settles the debates, right? We have theorems, which actually say that in Jeuvret topology gamma is one and in Subol-F Gamma is less than or equal to three halfs. We are furthermore able to say that this result is optimal and That's why I Stated it as gamma equals one this one. We suspect it is optimal because it agrees very much with What these guys found for rough data and Subol-F topology is certainly rougher than Jeuvret So we suspect it is optimal, but we could not really prove it Yeah, do you gain anything from an authenticity? Yes. Oh, you mean if you go further to analytic No, no, you would Keep the same the same exponent so Maybe it doesn't depend on Jeuvret's exponent. Yes. So here what I mean by Jeuvret is Say e to the d to the one half plus F in M2 And what I mean by Subol-F is D to the 5 sorry to be you not 5 half plus you not In L2, okay So that's that's what we did with Jacob and Nader But there is also results on the 2d case. So which really give a full picture so in Jeuvret Topology so This is where everything started you find gamma equals zero and this is due to Bedrosian and my smoothie So that's for the case new equals zero and then it was extended by Bedrosian my smoothie and The call to the case of positive viscosity and Recently the case of Subol-F in 2d has been treated Here you find gamma less than or equal to a half and this is due to Bedrosian recall and Wang Okay, so it's it's really nice because for this particular flow it's possible to get a pretty complete picture of what happens in terms of stability and There is this this nice feature that you have a Dependence on the I mean this interesting feature that it's dependent on the topology, which maybe It's not so common in in PD What about us exponents or exponents he doesn't doesn't matter if you go Yeah, yeah, so anything above five halves we believe should give three halves and then if you go to very low regularity Then maybe something else happens if you go to very low, but oh and say my mistake I think I said five halves, but it should be So this is dimension three, right? This is dimension three and in 2d The Gevry exponent remains the same and the Subol-F exponent is three halves. I think if Vlad is here Okay, so that's that's The results so now I'm going to explain what the picture is in in 3d So Right, so the first thing to do is to consider the perturbed Equation around quit Okay, so you want to avoid something like you know Pretty much since you have viscosity It's possible that maybe you start very close to quit does a lot of completely insane things and comes back to quit, right? So this we don't count as a stability, right? We count it as stability if you remain always very close to to quit You see what I mean Straight to me these are these ideas of Trapehton of a non-self-adjoint Flows where they're large transients. Oh, yeah, so I'm going to explain that It's indeed this the fact that the linearized problem is not self-adjoint That gives transient growth and which is ultimately I think responsible for this difference depending on the topology you consider So I'm going to try and explain that Okay, so the perturbed equation So as I was saying we consider a solution of Navier-Stokes of The form quit that is y zero zero Plus you so now you is the perturbation of Navier-Stokes Okay, so if you do a small computation You realize that you solve this equation Okay, and so this this this part is is not a self-adjoint And of course the divergence of you is still zero, okay, so It turns out that it's a good idea to try and remove this Convection term and this can be done easily by changing the Independent variables though it comes to the expense of modifying the differential operators. So let's do that So if you to remove y dx u What you do is you set capital X equals X minus T y capital Y equals Y capital Z equals Z and You rewrite the equation in this new variables It has the effect of turning gradient into what we call grad L which is dy minus T dx dx dz and Laplacian into Laplacian L which is grad L square Okay, so in this new coordinates, let me call also capital U of T Equals little u of T x y z so in this new coordinates the equation reads Okay, so pretty much we just remove the y dx u And now the differential operators have this more complicated structure Okay, so if you look at the linearized problem Maybe it's best to view it in this new coordinates. Well, it's pretty clear And the pressure term so you can express the pressure term as to grad L Laplacian L inverse dx u okay, so that's the linearized problem and That's the full nonlinear equation So what I'm going to try and explain in the time that remains is a few Striking properties of this linearized problem, which has a lot of very You know striking features And then I'm going to mention a few Important features of the of the nonlinear Okay, so the first linear heuristic is the so-called lift-up effect. So that's the name that This thing is called in the physics literature So this is the the main destabilizing effect in dimension three and it is absent in dimension two So this explains why the exponents in dimension three are less good than in dimension two at least partially Okay, so how does it work? Well A very important idea is that the frequencies which are zero in Capital X behave or little X behave very differently from the frequencies which are not zero in little X and This is because of this shear here So the if you look at the linearized problem the shear will only affect the nonzero frequencies in X So the picture is very different depending on whether you look at zero frequencies or nonzero frequencies in X So here we look at zero frequencies in X and we call F sub not The integral of F DX Which is the same as the integral of capital F D capital X. It doesn't matter just the Zero frequencies in the X variable. So it turns out the linearized problem is extremely simple for zero frequencies because then the DX U2 cancels so the right-hand side cancels and the Delta L Becomes a regular Delta. So the only thing that remains is Dtu Minus new Laplacian U. So I put a zero to indicate that it's the zero frequency in X plus U2 zero zero equals zero So that's very simple and you can actually explicitly solve and you find that you not is So the first coordinate is E new T Laplacian U1 so here I put in for Initial to distinguish from the zero, which is the zero frequency minus T U2 initial E new T Laplacian U2 initial Initial zero right everything is at zero frequency E new T Laplacian U3 initial at time zero So obviously the second and third coordinates are well behaved but this one you see it grows linearly until the Heat coronal kicks in but this happens only at a time T like one over new. Okay, so this gives linear growth until T like one over new And you see in dimension two this does not occur because U2 Zero has to be zero due to the divergence free condition So that's the main source of instability and it's really a 3d effect Okay, so the second piece of Linear heuristics that I want to present is the so-called enhanced dissipation Okay, so let me try and explain the idea in physical terms. So if you look at this picture You see if you have a good physical intuition that if you have something that depends on x Due to the transport in this direction. So this direction is like a tourist direction, right? If you have a data that actually does depend on x This leads to the creation of very high frequencies very fast and these get then eaten up by the viscosity or by the heat equation. Okay, so Maybe it's easier to see it through the equations. So it's actually very easy if we just look at The heat equation in the new coordinates, okay, so that's the heat equation In the new coordinates and let's just forget about the rest of the of the linearist problem. Let's just keep this piece and Remember that delta L. It's dx squared plus dy minus tdx squared plus dz squared So if t becomes very big, of course what matters is this guy here tdx. So essentially what remains is dtf minus new t squared dx squared f Okay, something like that. That's the leading part if t goes to infinity equals zero so you see that the viscosity becomes immense as t goes to infinity and if you switch to a Fourier variable This becomes dtf hat Minus new t squared k squared f hat equals zero and it's easy to solve F hat of tk is something like e minus new t cubed k squared initial So here you have a t cubed instead of a t Okay, so it makes a huge difference and it means viscosity acts extremely fast once you have the shear and so what it means is that the dissipative time scale is Now one over new to the one-third right instead of one over new for regular Heat equations. Okay, so I should say this sort of idea has been exploited first by Bedrosian, Masmoudi and Nikol. Okay, and then there is the last linear effect, which is the famous Invisit damping, which is the Euler version of London damping So linear heuristic three Invisit damping, which is the once again the Euler version of London damping and of course this phenomenon received a lot of attention after the work of Mouot and Villani on London damping for Vlaise of Poisson So it's actually very easy to see in this context There is just a little trick and the little trick is to switch to the new variable q2 Which is Laplacian l u2. Okay? So it looks a bit magic, but it's the standard change of variable that Physicists do for instance and what's very nice is that in this new variable The the equation satisfied by q2 is dt q2 Minus new Laplacian l q2 equals 0 so that's at the Linearized level Sorry Right, so the other variables don't behave that simply so that's why I just single out this one So for simplicity if we just look at the case and u equals 0 So which is really the inviscid case of the Euler case What happens is that q2 is? constant right Because you just get dt q2 and that's at the linearized level. Yeah so it's very simple to Recover u2 u2 is simply 1 over Laplacian l q2 So now if I take the Fourier transform and switch from x yz My Fourier To k eta l You see what comes out is u2 hat Equals 1 over k squared Now given the definition of Laplacian l eta minus kt squared plus L squared times u2 Initial and This formula is extremely instructive Because from this formula you can read two things first What happens as t goes to infinity so as t goes to infinity this guy becomes huge so that u2 hat Goes to zero and then u2 goes to zero as t goes to infinity So this is why it's called inviscid damping So for u equals zero you have a very nice Hamiltonian system no dissipative effect whatsoever Never the less You get this decay of u2 as t goes to infinity and I should say it's not an effect Which is like scattering for say wave equations in the whole space because You don't have space to scatter in the x direction Okay So that's sort of the good side of the picture, but on the other hand if This factor here vanishes T equals eta over k so it means this factor here vanishes and then you get a growth Compared to what happens at time zero. Okay, so there is this two sided effect This one is called inviscid damping and this is rather helping us To prove stability and this one is called the or mechanism And of course it's rather annoying because it means that You have growth At various moments in in time And of course the fact that you have this growth at various moments in time is ultimately Linked to the non-self-adjoint character Of the Linearized operator right since it's non-self-adjoint. It's well known that you should expect growth for the evolution problem Okay, so that's the You can call it intermittency, right because If you look at that it's sort of very nice all the time, but there is One particular moment where it grows and then it decays again So one might call that intermittency Okay So now let me discuss the non-linear problem non-linear heuristics one It's a so-called streak solutions So I should call the I should write again the non-linear the non-linear problem This time in the original variables and an important observation is that there is A large set of explicit solutions to this problem which consists of U2 Depending only on y and z And u3 depending only on y and z as well Just solving the 2d navier-stokes equation While u1 is driven So if you plug that in you see that it works and that u1 is simply driven by u2 and u3 In the following way. Oh, I forgot someone here. There is minus u2 So here you get u1 So u2 the u2 dy plus u3 dz U1 equals minus u2 Okay, so u1 is just Slave to the others and these two solve the 2d navier-stokes. So that's an exact set of solutions And you see So these are called streak solutions because supposedly they look like streaks in experiments And you see they play the role of an attractor Because what the enhanced dissipation does is that very fast it's killing any dependence on x Right, so here to get this very Fast decay you need that kb non-zero. I should have said that So if k is non-zero you get very fast damping what remains does not depend on x And these are streaks. Okay, so these are attractors At least on the time scale one over new to the one-third. Okay, and the Last thing that I would like to emphasize Is the various null forms which are found in this equation So Okay, so there is it's There is a lot, but let me just mention two If you look at the most threatening interactions Well, first the most threatening interaction is due to the growth of u1 up there You see that u1 is growing linearly. So really the worst that could happen is u1 And it's at zero frequency Talking to u1 at zero frequency maybe with derivatives So this would be very bad because both grow linearly. So it would Completely kill the structure. Unfortunately, this does not happen. There is no such interaction another problematic term is the the grad l So if you remember grad u dot grad l It's u1 dx plus u2 Sorry, I should put capital u dy minus t dx Plus u3 dz And so of course this t here is very threatening But luckily it comes paired with u2 and we saw that u2 decays Thanks to inviscid damping. So there is sort of a cancellation Between this t factor here and the u2 which decays Uh, so The proof consists in putting all of this together. So essentially the picture that emerges is that First on a timescale one of a new to the one-third you get this enhanced dissipation which kills all non-zero x modes Okay, then The solution looks like a streak For t larger than one of a new to the one-third like a streak plus something very small And then what happens is the lift-up effect kicks in And you get growth until you reach time one over new And then you pray that you can keep everybody under control And you can do it thanks to these cancellations Which are present in the equation And that's all I wanted to say. Thank you for your attention We've answered So these mechanisms we described that here and where does this And it must there must be a big head Oh, yeah, it's it's a big worry. Yeah, so you need to so that's for this reason that Bejoison and mass moody introduced their toy model in 2d And in 2d that's really the main problem Uh, because they're able to do the case new equals zero in 3d It's not I mean, it's one of the problems, but it's not the only one anymore Yes, so you didn't quite explain the connection between the proof and the fact that you have this difference between between topologies All right. Yeah, so Okay, so what this All mechanism is doing Is that so if say, uh, if you think of eta being big so you look at high frequencies No, sorry, we think of uh, eta being small So what you want to avoid is a high too low cascade, right So that's what our mechanism could do and The best way to avoid a high too low cascade is just to start with very few high frequencies And and that's why It helps so that's that's how it's related That mechanism that at the end of the day It's at the end of the day Jeffrey classes to do that I mean in 2d. Yes in 2d. It's very clear in 3d things are messier and it's Not the only effect that one has to take into account In particular the lift up effect is is is more threatening And and it's yeah, maybe it's it's more the lift up effect in 3d, which is annoying So is there a simple heuristic explanation of the number three over two? No No In in jeuvrette apology the one of a new scale it's easy to explain That's lift up effect, right? Yes, exactly But hadn't you three halves, what is it? So three halves it's less clear. There is no simple Like I cannot point to a single mechanism that that does it All right, no two mechanisms together But why do you think it's it's Optimal Right, we think it's optimal because we tried hard to do better and the numerics give the same I agree, it's uh, you know