 I think at the end at the end I'm going to do much more of 2d than 3d but I think maybe that's that's already good. So let me spend again like maybe five minutes to give another small exercise. Unfortunately I'm not going through the exercises but for students who they can still see me this week or next week if they want to we can discuss the exercises I gave. So yesterday if you remember I ended up with talking about the problem of this type. I mean we start we looked at an equation of like this and I explained that the main growth was coming from high frequencies here and low frequencies here and then I went through like some reductions to get to some sort of ODE system that mimic what is the maximum growth of this ODE of this equation. And I wanted to explain at least for those who want to do the calculation how you get really the jovree like what will be really the simple ODE systems you have to solve to see the growth. So that's what I want to explain now very rapidly. I mean if you remember I did I said okay let me look at high high low and then you write this as a convolution and at some we ended up with so I go a little bit fast but it's more or less the calculation we did I did a little bit last time maybe giving slightly more. So I wrote something of this type L t square f hat of c and this comes from the low frequency let me not write down precisely what that is. So I said okay this this will be a good candidate to mimic the growth and I'll be putting absolute values everywhere. The first thing I explained is to remove this replace this like also by one replace this by eta so this was the first thing we did okay put absolute values think about there there are absolute values everywhere and then I said only the modes k less than square root of eta these are the modes that are important but then you look here and you start seeing that it's really if you think about this as being time and I explained this I think last time it's if I write k zero to be like square root of eta at each I mean if you look at times between square root of eta and eta so I'm only looking at times here I mean that's where that's where this denominator can start having problems actually so the whole I mean I have to spend some time looking at this ODE and see what is really the path where you can have the maximum growth and try to estimate it and it turns out that path is more or less the following one let me put I mean at the beginning the times are much more closer very close to each other but at later time they they get I mean here close to square root of eta there are many many resonances at later time there are more spread basically okay so the worst the worst actually scenario is when the mode is the following is the mode k zero talks to the mode k zero minus one talks to the mode k zero minus two till you get to talking to the mode one right because that's where you will be getting the maximum growth at each time this was already like in the paper of or of 07 1907 he more or less says something about that sort of transient growth I mean all these pictures related to that transient growth and he mentions that okay he mentioned in his paper that it's possible that this transient growth gives problems because of like transient growth I mean the idea here is exactly the fact that you see like the mode let's say the mode k zero will grow at this time but then at this time it can talk to the mode k zero minus one and the mode k zero minus one will grow later right that's more or less so the mode k zero will talk to the mode k zero minus one and the mode k zero minus one grows later and so on so that's really more or less the the sort of possible cascade that happen it is it is not it is like a non-linear effect because there is this term that may remove I mean there should be some low frequency there that makes this energy cascade possible okay so it is a non-linear effect okay so now how how can we how can we get to exactly that exponential of square root of eta the way you do it is so you split these intervals so there will be at I split this the whole thing into small intervals centered that around these numbers and on each one of these intervals I will solve some sort of ODE that I mean like that may make that may make the the maximum growth basically it's not a difficult I mean like the the ODE one can solve it I mean it's very simple it's not difficult so what what is this so FR will be will correspond to the mode k in which you are here let's say you are in this interval FR if you are in that interval FR will be fk0 minus 2 if you are in this interval right and f and r will correspond to all the other modes and you can say okay this is what is the growth that this system of ODE can encounter along this interval okay so you you plug in you do some calculations and you end up and I leave this as an exercise for those who want to do it you end up here with exactly the growth that you get is eta over k square to some small power okay the small power is related to kappa the kappa here is related to the smallness of my data so the kappa is related to the size of this guy I mean this guy has think of this as small it's of size kappa okay so it's not difficult like you take this as a system and you try to solve it between actually between times let's say okay I'll make a change of variable I replace this by tau now if I replace this by tau okay now I'm going to replace t I'm going to write tau equal t minus eta over k and then this becomes one plus tau square okay so then you look at this ODE and you look at it for tau in the interval minus eta over k square eta over k square that correspond exactly to this interval here okay and then you you assume that your f r and f n r are of size one at this time you run this and you want to see how large they become at this time it turns out interestingly enough very interesting thing happens that they first of all first of all this guy grows faster in the first part of the interval and in the second part this catches up and it turns out at the end they they grow exactly of similar way and they grow exactly of of this amount okay I mean simple exercise just just ODE so then okay you put all this growth together so you put all this growth together so starting from the growth coming from k zero remember k zero was this and then you continue till you get to one to the power c okay so it's the product product of all these numbers then you use your sterling formula and you see that this is exponential some constant related to the constant you have here square root of eta so using sterling formula so let me leave this as exercise so there are two parts in the exercise first of all you look at this ODE and you try to find out that this is the growth that you get and then you put all these growth together and you see that it is some exponential square root of eta so then it means that if you I mean if you if you think that this sort of growth is possible in your equation it means that you have really to be assuming jeuvre regularity and I think we have we are almost now confident that it's possible to get this growth right you can you can cook up some way of taking your data so that you see this growth okay so let me now state a theorem at least I can state a theorem and then maybe I'll say two words about proofs but but then I want to talk more about the 3d case maybe insist more on differences so what is what would be the theorem so the theorem so this is a theorem with I forgot the year I think the public published last year that was the publication so we start with s here and we take some lambda 0 lambda prime so this will be I mean s will be fixed this is like the amount like as the radius of analyticity that you have initially and this is what you get at the at the final time so then there exists an epsilon 0 such that for all epsilon less than epsilon 0 if your omega initial so omega initial is a function of x and y satisfies so I think there was any question about it like the the integral dx d omega this is 0 I mean it's always you can always take that to be 0 and there is a reason I mean that I got the question about it so I think you can remember why we need that we need also some sort of what this is less than yeah less than epsilon or so so you need like some weight but just one weight will be enough and your omega initial in the jeuvre lambda 0 I mean the jeuvre of class 1 over s but with the radius lambda 0 let me recall what this is this will be the sum of a k the integral omega initial hat k eta to the power s I put a 2 here because I'm squaring things d eta so here I'm integrating in eta summing in k and this is the weight I'm putting okay s s is strictly bigger than one half so remember because this is the possible growth I'm I may be getting from this sort of toy model so at the end of the day I will still keep a little bit of of this okay yes I demand that this is less than epsilon square so then the conclusion is what so these are the assumptions on my initial data then there exists some f infinity of x y yeah I mean I think it's better that you I write it as capital X because yes okay so the integral of this f infinity is also 0 but now the f infinity in the lambda prime norm is now instead of being less than epsilon it will be less than some 10 epsilon maybe okay so around to grow a little bit but you see I'm not allowing it to grow too much but I'm losing regularity so again I mean the point here is that all this growth that you see here instead of seeing it as time growth there I'm seeing it as loss of regularity so that's I think that's really the main thing and then the conclusion is that my my omega so my vorticity will be converging to this f infinity so that will be my final profile in a sense but then I have to I have to make to write down the right change of variable so x remember like if you remember the x was my x minus ty in the linear case but then in the non-linear case I added some other stuff but then I see how I'm going to write down this my omega omega is a function of small x so then I have to write down the small x as function of capital X plus ty plus all those other partial things so that's why it's written this way plus some phi of ty this is coming from the non-linear term y minus f infinity of capital X y in the jeuvre norm this will be decaying like epsilon square over t okay I mean here somehow I I mean the statement I'm writing it in terms of the original omega but the proof is really done with what what I think last time I called f f was written in the new coordinate system so basically the the way the proof goes is really that you prove that your f converges to this okay and phi phi of ty is is really the piece that was here yesterday right so it's it's really the non-linear the non-linear part of the change of variable so it's somehow something like this um the integral between zero and t and then it is the the zero mode of the u1 which was a function of s and y ds okay and actually one can prove that this is equal to some u infinity of y t plus some b o of epsilon square okay um the u infinity the u infinity of y is the velocity the first component of the velocity related to this guy so um let me finish writing the result here so u infinity of y is like dy and then it's dy y minus one so of course then it becomes dy minus one of the integral of f infinity of x y dx okay so it's really the velocity associated I mean if you think of this as your vorticity um what will be the first component of the the first component of the velocity which is the zero mode the zero mode of the first component of the velocity okay um so that's that's really the main statement about the convergence of the vorticity and then okay I mean there are other statements about um the decay of the of the velocity really um so then I mean these are not very difficult to get I mean that's really the main part of the proof that the the rest comes for free so now if I take this minus u infinity of y so it's really like the error between this and the limit guy so I think I mentioned this last time normally in in problems like this you expect this to be one over t right usually but it turns out here it's one over t square there is some subtle some subtle cancellations that okay I mean like and it's really the reason why this is t and then constant is no log term there's a no log term I mean the first version of the paper we had to log there but okay we were we observed that there is some sort of nice cancellation that makes this one over t square but that's really for the zero mode so the zero mode converges faster than maybe what you expect okay I mean this is again I mean think of this as a function of t and y okay and then the non-zero modes the non-zero modes if I write down if you remember I used this notation for the non-zero modes I mean this happens in Juvre the non-zero modes if I write them down in the original variable okay I'm mixing variables but if I write down the things in the original variables for the non-zero modes I can only get convergence in L2 I will not get convergence in high space because like derivative will cost me t of course if you write them down in the rescaled in the in this coordinate in the capital X coordinate you get convergence in Juvre okay but that's I think I don't want to insist on these things because that's not really how the main part of the proof goes okay so that's more or less what will be a statement I think you can ignore these parts of the statements but just remember this part yes yes here zero I don't know you said the non-zero the zero mode yeah yeah okay I mean I wrote the integral so I wrote the average here so but I think I can if I if I am if I no no but I think it's better I mean if I am if I am more I will use this notation I mean that that would be the better way of writing so then yes yeah yeah so right yes the integral yes so the integral is exactly like to avoid to avoid when you invert the Laplacian when k is when k is zero when you invert the Laplacian so it's it's really inverting this guy so then you don't want when you invert this you want you don't want to be dealing with low frequencies and so on so it's just problem of low frequencies in y and also the second one is is is also the condition so both are really for which I think I mean like in the Navier stocks community it's always whenever you are in the torus it's it's really the the natural assumption that you make like you have average zero I think you can even recover I mean if you just take yeah I mean like if you take the velocity you take that the vorticity is the gradient of the velocity then you'll have it yeah I mean it is it is I mean otherwise you are in energy not infinite energy right so f infinity basically norm I mean here it is like a perturbative result so basically like your f infinity is like equal to your omega initial plus big o of epsilon square so it's like scatter I mean it's like in scattering theory like no no I took the translation out I mean this is taking the translation out right so I mean I think one can think of this I mean it has a lot of similarity with scattering but of course like it's it's not just I mean the linear thing you have to filter with depends on your final thing I mean that's one another difficulty also in the proof like you don't know this state initially so but then you filter with this and then you prove that this behaves like that at the end right but of course you can write a form I mean you can write an infinite integral to to write down what is this right it's possible okay so this is the statement um I'm not going to talk about proof because otherwise I mean I will not have time to talk about the 2d the 3d but let me just mention one thing since I thought I told you a little bit about this toy model and so on it's so I mean in all the in proofs whenever people use proofs using these jewelry norms the typical the way usually these proofs work is that you will you will try you will try to um you will try to um so as I said like f I explained what was f last time f is after I apply the change of variable so my f my f of t x v I mean yesterday I also introduced v but in the statement you don't need the v is my omega of t x y I'm not going to recall the change of variable just so usually proofs um in analytic or jewelry regularity at least um will try to estimate f of t in some lambda of t right and you will allow your lambda of t to to go from um from your lambda zero to your lambda prime right and this is really what is behind the idea of Koshy Kowalewski like when time goes you your radius of analyticity let's say decays and I think I think this type of norms in the fluid community was used by first I think the moment voyage like the paper of end of the 70s they they use this type of norms but they used it for in the analytic case okay it turns out that I mean this was our first guess I mean let's try to use norms like that that's not enough that's not enough because this will be losing the jewelry regularity for all frequency in the same way however if you remember this toy model like for the frequency eta the loss of regularity happens exactly in this interval right so we cannot we cannot use this let's say democratic way of losing regularity for everyone the same time so each one will be losing each frequency loses its regularity at a particular time so that's why we need a much more complicated norm that that actually uses a lot this toy model so it is based on the toy model precisely you mean that you have a weight function which depends on eta in a more complicated way exactly exactly exactly I mean I can write it down but I think I think I get I think I will not do it because I mean you get the idea basically like we will write a weight function that mimic this growth of the ODE and we will put it here as a weight in front right it is a little bit also in the spirit of this I always mentioned like Alina like this ghost the ghost energy of Alina it is in that spirit also I think okay so any questions about this 2d case on otherwise I'll move to the 3d okay yeah I think I said more or less what I wanted to say about this okay now 2d so let's let's go again back to simpler thing or 3d so let's go back to some simple things and what I want to insist upon is differences more than doing the the 3d precisely oh let me leave this because this is part yeah so I mentioned that this was the result with Jacob and then we had another result with also in collaboration with V call where we extend this to the case where we put a viscosity of course in this proof with with we call the viscosity didn't help us at all like we had really more I mean the proof was more to just make sure that the viscosity doesn't come and destroy all the nice properties we had here and one of the other things about it is the fact that at the end of the day we have to go back to quit like this will go to zero basically because in in the in the I mean there is a visc there is a viscosity that will kill this guy also okay so now 3d okay so I'm not going to rewrite the the equation I wrote about you I'm more or less going to assume that you you have it or maybe I should rewrite it okay let's assume that you remember it or maybe I can write down the equation that was so I had the V which I wrote something like this plus U and then U is the small perturbation and I can write down an equation on this small u equal minus u 2 0 0 so I will try also to go to follow the same the same way that I talked about the 2d and mention first like linear effects and then try to talk a little bit about non-linear effects so what is the first linear effect the first linear effect is to say okay let's let's try to get rid of this large transporter and then the change of variable the natural change of variable is the same as the same as in 2d so at the linear level it is the same the change of variable the natural change of variable is the same and if you do that okay so right so here I'm working with with the velocity equation rather than the vorticity that's a big difference there I mean like you can try to write down the vorticity here but it's not useful at all so right so I'm going to use the notation capital U capital U in these variables is the small u in these variables so then I can write an equation on this capital U and then I get the following equation of course now capital U is not um capital U is is not necessary divergence free but it is it is divergence free you know I mean it is this divergence which is free okay and I recall which I explained last time this is what this is dx dy minus t dx dz and this Laplacian is just um the gradient l square okay so few observations here um okay I kept the non-linear term even though this is I call this like a linear change of variable of course there will be another non-linear change of variable to take into account this guy but if you remember last time when I wrote the equation on the vorticity when I wrote the equation on the vorticity there was a nice property is that there was some if you remember there was some nice cancellation and here I didn't have this gradient l it was just a gradient this is a one one difficulty that comes in 3d that we didn't have in 2d the fact I mean remember that you want the non-linear term to be the smallest possible so that it doesn't perturb your linear dynamic so that was one observation okay so so the p we have here I mean we can compute it actually explicitly I mean you can write down an equation about it so it's I mean you can take the divergence you can take this divergence term like this will disappear and um actually the pressure as I wrote it here the pressure has two pieces into it so there is a linear part of the pressure p l and p n l you can you can observe that you have two pieces in the pressure it's a simple calculation let me do okay now I want to I want to look at the if you remember last like when we talked about the 2d there was two important linear effect inviscid dumping and enhanced dissipation right so remember those two um so inviscid dumping is the decay that comes without the viscosity help okay and um and then enhanced dissipation was the decay that comes with the viscosity but it is enhanced by the help of by the help of the of the shearing in a sense um okay here here it's not clear um what what is the how how we can do inviscid dumping here so so so let me let me for now forget about the non-linear term let me for now forget about the non-linear term okay so just explain the difficulty that comes from the change of variable about it but just now think about this equation so there is it turns out that there is a good quantity that replaces the vorticity there's one good quantity that replaces the vorticity which is so called q2 I mean q q2 which is Laplacian l of u2 okay I mean you can also write it down in the original variable in the original variable it is just the Laplacian of u2 right in original variable it is q2 equal Laplacian of u2 and um I mean this goes back actually to Kelvin Kelvin already already noticed that this was a good quantity actually the enhanced dissipation I mentioned last time the calculation like that sort of decay mu to the minus one-third was already computed by Kelvin if you go you read the paper of Kelvin you see that calculation precisely so uh there's this nice cancellation that happens basically I mean it's not it's not like something you can see immediately why that's a good that's a good guy but you you do the calculation I mean you can do it you have the choice you can do it on the original variable or the or the second one and what you end up with at the at the linear level you end up with the following equation dt q2 equal new Laplacian l of q2 that's very good right I mean it's some nice cancellation happening and so on and then you you get that I mean if you write it if you write it on small q2 if you write it with the small q2 you will get the same but with Laplacian small q2 I mean you have to do the calculation to believe that it is correct but somehow there is a cancellation between these two guys basically there's some sort of some sort of cancellation between actually this and the linear part of the pressure right so you you see like the pressure has I mean you can compute it even though I didn't write it down but okay let me I think to help you do the calculation I can tell you I can tell you what is this guy so you right in the rescaled variable you can write down that the linear part of the pressure is minus 2 dx u2 so if you use that if you use that you can do the calculations okay so this is good because then what does this mean so if you are in the inviscid case q2 is nothing happened to it it is just conserved if you are in the viscous case you get the enhanced dissipation right so if mu equals 0 q2 is just q2 initial I will say and if mu is larger than 0 we have enhanced dissipation but I'm like in both cases your q2 is bounded let's say it's it's not going to grow or nothing bad is happening to it so then the good thing is that now u2 the way you recover u2 is that you invert this Laplacian l and okay Laplacian l gains you 1 over t square so from here we deduce that your u2 in sobolev norms your u2 in sobolev norms will will be like 1 over t square let's say your q2 initial okay these are this is for the non-zero modes I should say always like non-zero mode referred to the non-zero mode in the in the capital X variable okay so this is type of things you'll be getting this is if you are in the inviscid case if you if you add if you get the viscous effect I mean you can add some sort of exponential minus some constant mu t to the 3 if you add the viscous effect but this is more or less like the type of of course like it's more this decay which is more important than this one in when when we talk about u2 okay so if you remember in the in the 2d case in the 2d case this decay gave us also the decay of u1 for of course for a simple reason that like the incompressibility if you think about incompressibility in 2d I mean like you will write it down as ik if I am non-zero hat plus dy of course like the dy will lose me actually like if I if I write down if you remember the divergence free is like here it is I'm writing it in Fourier so it's like plus i eta minus kt u2 this was equal to zero so in 2d in in 2d if this decays like one over t square this will decay like one over t when k is different from zero okay it's not difficult to see that okay but now we are in 3d and I have to add il u hat 3 so these two guys will not decay okay there's no decay for the other components okay and it's not difficult like to look at I mean you can you can go back to the equation you look at the equation about u1 and u3 and you see that there's no decay mechanism in them I mean of course there is a combination of them there is a combination of them which is good but but there's another one which is not good and okay I mean one can think okay is there like some sort of null good null condition that helps with this particular non decay but no there isn't okay so so the only way you can get u1 and u3 to decay is by enhanced dissipation is really the this type of decay for u1 and u3 and that's one of the reasons why like in 3d we definitely need the viscosity otherwise the result will not be is not correct okay let me go now to the second the more problematic actually effect which is the so-called lift up effect I mean lift up effect this is I mean this is a physical terminology but it has it has it's it's also a simple linear effect that we can understand which concerns the zero mode actually so I mean for the zero mode I mean for the zero mode I mean whether you do this change of variable or not is the same I mean you can see the the calculation at this level or at this level it's exactly the same so you just average in x you take the equation and you average in x and you see what you get so if you average in x you get the following equation so I'm writing it in the new system of coordinate but it's it's not okay so now you remember you remember things from study of matrices I mean if you remember this this has this is like a Jordan block basically right so there is some sort of Jordan block here and I mean this you can solve it explicitly I mean like you have a formula so so think of these as functions of y and z right these are functions of y and z and you can solve it so u01 of t is exponential t mu Laplacian so here it's Laplacian if you think about it it's just in y and z but it doesn't matter u initial 0 1 minus t u20 why I'm not writing initially I should be an initial I'm getting confused what I should put the initial there or not yeah so the u2 and same for the u3 anyway so I mean the way I think about it forget about the viscosity I don't want to think about the viscosity but if you forget about the viscosity which means let's say if you are four times less than mu to the minus one four times less than mu to the minus one you can forget about the viscosity but then you see that u01 will have a linear growth right you'll have a linear growth which we didn't have in the in the in the 2d case so so basically this sort of linear growth means that if you start with if you start with a perturbation of size epsilon and you wait till time so this is at time zero and you look at what happens at time mu minus one then you become of size epsilon over mu okay so this means that you cannot allow perturbations larger than mu right you cannot allow perturbations larger than mu because this simple this simple I mean just looking at the zero mode gives you this and then I mean this is really physically very relevant and it is observed it has has a name it's lift up effect and so if you remember in the 3d case I said in in the Gevry and Sobolev we had like two different kind of size of perturbations here we can allow epsilon actually to be that small constant times mu in the Sobolev we allow it to be of size mu to the three halves so the this sort of size of size mu is basically related to this in the Sobolev it is much more subtle why why the say the size should be this one I mean at least we I mean we don't have the the I said in Sobolev we don't we we don't have the instability type of things but in the in the Gevry we have the instability we can prove that your solution can really do this we can prove that your solution can really do this thing but we can follow this sort of zero mode and we can see it really grow from size epsilon or size mu till size one okay so any questions or before we move to non-linear effects yeah maybe there is another linear effect I want to talk about which which is yeah let let me talk about this even though I mean like it is it is related to the fact that you want it is related to the fact that you want and you three do not decay and we give it a name I mean I'm not hundred percent sure it is I mean at some point we were calling it vorticity stretching I mean it's more like a stretching behavior and it turns out like in the proof it is an important it is an important fact to take into account and even though I think I mean I will not go into the proof but it it shows an interesting thing so I mentioned that q2 was was already introduced by by Kelvin he didn't look at the whole Laplacian he only looked at Laplacian of u2 I mean he thought that Laplacian of u1 and u3 were not interesting and I mean at the beginning we we were trying to do estimates on u and q and q but then after some time you immediately see that since you are taking the Laplacian of u2 you have to take the Laplacian of all the other components so so that's how we introduced like q which was the Laplacian of u and capital q which was the Laplacian l of capital u so this is in the original in the new variables and then I mean you can write down equations on q2 and q1 and one one one nice fact so let me write down just the linear case so then I get some Laplacian l of q2 so this is the time derivative and there are bunch of terms sorry q1 so there are terms that come from the q2 and then there is another term which is something like this dx dx dy dx dy l if you remember the notation this means it is a dx dy l dy l is is this guy okay so there are other terms so you you just try to see how really um yeah so at the linear level this is what you get equal zero so why why I wanted to explain this sorry so this is u1 here so um so in a question like this like we said that q2 is not u2 is decaying so this is a good term q2 nothing is happening to it it's like a constant so this normally can give you some sort of linear growth of this guy but actually it turns out that this guy will grow quadratically it has to grow quadratically because um I mean I can recover I'm always talking about non-zero modes I can recover u1 I can recover u1 by inverting Laplacian l minus one of q2 q1 right and I said that u1 was not decaying so if I'm saying that this does not decay and I know that this gains me one over t square this has to grow quadratically and actually the quadratic growth is coming from this guy and you can see you can see that there is a quadratic growth because then this guy actually write it as Laplacian l minus one of q1 okay and um I mean if you if you forget about the viscous effect you can see I mean it's it's an ODE also an ODE and the factor two here is important it's actually the factor two that gives you the t square um the t square right so basically I mean um what you get you get I mean like to mimic the ODE you get you get something like this um plus or minus I think it's minus two q1 over t you get something let's say if you mimic it a little bit you get something like that and this gives you quadratic growth um one of the reasons I'm also mentioning this is that um so if you remember when I met when I talked about the toy model so we still have like 10 10 minutes Frank or 50 oh okay okay good so at least I can explain I'm going to explain few facts like that are new in this proof without really trying to go into um basically I mean if you remember in in the in in the 2D case I explained that the norm has to be built based on that sort of toy model here we also need to take a norm that is built on linear effects and non-linear effects so in particular um one part of the choice of the norm is based on trying to control this sort of growth by the by the Laplacian right um I mean again this I mean here I'm talking about something which is about the non-zero modes right it's about the non-zero mode and I explained that I don't want this non-zero mode to come and destroy this sort of growth that I'm having here I mean at the end of the day I want really to make sure that this is the main thing that is happening and that all the rest is under control so um so then we write we introduce a multiplier that more or less may make the growth that you can get from this and then there is a time where this kicks in and and this the decay coming from here will overcome the growth so so basically you can see that your Q1 start growing and then decaying but now the decay is coming from the viscosity not from some inviscid effect like um like in the 2d inviscid case okay so that was another linear effect um that I thought was interesting to to to to mention um okay so let me now talk about non-linear effects unfortunately I will I will not tell you where why the Jovray 2 here happens I mean here if you I said in this case we need Jovray class 2 and here we need like some sobolef but the Jovray class 2 here is coming from a toy model like the toy model I explained for 2d oiler but they have nothing to do I mean it's it's much bigger toy model that involves that involves um writing a system of equation on Q1 Q2 Q3 right so I showed you the linear part of Q1 it's already has many terms if now I write to the non-linear term and it has like maybe six terms like you have to now one of the difficulties here is that you have to keep track of Q1 Q2 Q3 separately because each one of them has different properties so I will not really try to give you the toy model that may make the worst growth that will tell you that you have to take that Jovray but instead I want to mention um some other thing about the change of variable actually so I want to more um yeah so I'm going actually to mention two things I want to mention two things one of them is the change of variable and another one is some sort of null conditions in the non-linear terms so these are the two non-linear effect I want to mention so the change of variable so let me put back the non-linear term so I have a change of variable and some sort of null conditions here I don't know whether this is right terminology but we are like borrowing terminology from the dispersive word just just to mention that the non-linear term has built into it some nice cancellations and that um I think for people familiar with Navier's talks like many of the results about Navier's talks can be proved by replacing u grad u by anything that looks like a derivative I mean quadratic term with a derivative it turns out I mean in this result we need a lot of the precise structure of course like here we are losing but we are not losing too much basically and um in this term let me mention this first because I think I I mean I think it's important to keep this in mind so in this term I mean you have growth but it turns out um there are two interesting facts that do not happen in this term and one interesting fact is that the u01 so the u01 is the worst guy I mean the the guy which is growing so there is no um self interaction of the u01 with itself because like you can see here the u01 comes into the first component so the first component comes with u1 dx of u so if I if I look at the first component the first component of this so then I will get u01 u1 let's say but then there is a dx here so if I put u01 and u01 this term will disappear so u01 does not um does not appear quadratically in the nonlinear term so that's a good effect that's that's one one thing that we call um I mean if we had a term like that all these exponents will be much worse okay so so this is what we call like a non-self interaction of the u01 okay so that's one was no self interaction remember that I want things to to look more like the linear thing so I want the non-linear term to be the weaker possible so that was one effect another effect I mentioned here that I have this gradient l and this gradient l is bothers me a little bit because there is a t in it but and a good thing is that this guy I mean if I write down the if I write down the convective term I will get something like this this is the convective term right okay so what what can we observe here this is bad this will lose me a t but the good thing is that this guy the guy that loses you a t is always paired with the guy that decays better right so I said that this guy decays like one over t square so the the bad growth that you have here is always comes always with the good decay of this guy versus like here this guy is not decaying but it comes with a derivative without a t so so at the end it balances well so somehow that that's another good effect from the non-linear term that um that was important also okay so um those are at least the two main things that you can really that I can really like explain in a simple way I mean there are many hidden other small things that you see in in the course of the proof but let me not talk about them okay um okay so let's let's let me now go to the non-linear non-linear change of variable or maybe before that I like to yeah since I will not have time to give theorem but um yeah maybe before yeah maybe I will not give you the change of variable but instead of that um what comes with lift up effect so in the physics literature you will see something called lift up effect but you see another terminology which is so called streak okay so what is a streak so um these are related to what is called 2.5 solution of navier stocks so it has to do so what does that mean these are solutions these are like solution that depends on two dimensions but they have three components in them um and the so the two the two dimensions here are actually y and z okay so so here I just explain the linear picture but the non-linear picture for the zero mode so if I if I look at the zero mode so if I call let's say v equal u0 of yz you can see that um so now I'm back to the non-linear equation for if I take that it's it doesn't matter what I do I work in the original variable or in this then you can see that actually v v2 3 actually solves two d navier stocks in yz and this is well known in the physical literature people are and then the the u0 1 or in my notation will be like v1 v1 the v1 here I just wrote the linear piece will be solving some sort of drift diffusion so it will be solving an equation that will have a v2 3 in it also v2 3 grad yc of v1 equal minus v2 okay it's a scalar minus v2 plus mu laplacian yz of v1 okay so that's really the the equation that you get and this sort of v will be called a streak because it is like a two dimension solution of navier stocks in yz now right so you have to change like the way you think about things and then you have some sort of linear growth right so so basically this is exactly this same picture but at the non-linear level so somehow it's really saying that the lift up effect which is like a sort of linear thing survives in a nice way in the non-linear case okay okay now the picture that we have now so I think I mentioned in the jeuvre case we have two results with Pierre Germain and Jacob one of them is to prove stability if I start with epsilon smaller than this and the other one is to prove sort of instability if I start exactly of this size so in the in the instability case I would like to show you I mean it's more like applied math type of way is to show you what's going on really what's happening why why we have these results so basically like if you start at t0 you can start with the data which is of size let's say epsilon epsilon less than some constant times mu at time zero so now if we put all these effects that I mentioned lift up effect enhance dissipation inviscid dumping if you put all these effects together what's what will be what will happen first of all if you go till if you start with if you go till four times larger than mu to the minus one third so the first times are these times so this is the time where enhance dissipation acts so what will be the effect of enhance dissipation it will kill all x dependence okay so no x dependence so so then what happens that your solution your solution will approach streak so that's in the physical literature you will see that as like what they call like metastable effect metastable state so you approach something which is not your the state to which you are converging but you'll be close to so we become close to a streak okay so that's the first thing that happens okay so then when you get till times like one of a new when you get till times like one of a new there are two things that's happening first of all the u01 u01 will reach more or less its maximum it will be like epsilon over new and then okay depending on depending on how I choose this constants so if I choose this constant really small very small so it can become it starts so then my your my u01 start like epsilon can become of size let's say one over million but then decays so you can follow all this in the good case so then that's where like for instance we get these two possible results either either we stop like either we say okay at this time if that c0 is large I get here till something of size one and then my proof stops I cannot control afterwards or if if I start with epsilon if I start with epsilon let's say which is smaller than very small constant times new for times larger than one over new so for times much larger than one over new then the viscous effect in the regular viscous effect will kill also the yz dependence and then that's how you go back to your quit I should say that this picture is correct in the jeuvre case in the sobolev case I don't think it is correct that there are many other non-linear effects that can start becoming important but we cannot really say precisely which ones maybe it's a good time to stop I guess so if c0 is big you say it goes out of the window but you don't know exactly what happened okay so actually I wanted to say that so so basically what we expect and I think if I'm not wrong this is what people doing I mean like in the physical literature trying to say so so this is a function of yz right so this is a function of yz so what happens is that you start seeing oscillations in z and now oscillations in z normally gives you growing modes right so for those who are familiar with shear flows if you take sine of y as a shear flow I mean there are cases where you can have growing modes so so what what is really expected is that this if this starts becoming of some critical size it will give you some instabilities in in in z direction so that's that is what is expected I'm hoping to see how I can prove results like that but it's really tough like I think a good result will be to start with the data start with the data like I mentioned here put the right amount of energy in a bad mode for this guy and hope that after you wait the right amount of time that guy that mode starts growing it's it's not clear whether one can prove such kind of things but but I think I think if I'm not wrong like some some experiments will see really like some sort of growth then you get some sort of turbulent behavior and then it becomes laminar again so somehow what you what is possible is that you reach a time let's say at this time some growth can come from z directions so then the Laplacian start killing things better and then you go back I mean I have to go back to some applied math talks where I saw this kind of things but trying to maybe now I understand the much better anyway yes no no more questions