 أعرف. أعرف. أطلع. أصدقائك أعرف أسدقائك. أطلع. أعرف. أعرف بس묘 بالدرجة. أعرف. بالدرجة. أعرف. تصدقائك. أعرف. لذلك. منتي. سأحاول أن لا أذهب إلى too much detail and maybe give some perspectives about other problems or other related problems. ما نحن نتحدث is again the parental system. سأحاول أن نتحدث about other problems and maybe give some perspectives about other problems. سأحاول أن نتحدث about some u Euler of x when y e goes to infinity. سأحاول أن نتحدث about some u Euler of x when y e goes to infinity. سأحاول أن نتحدث about some u Euler of x. ي lucky this as just the trace of the oil equation that normally you are solving it's again like here the ideas that we are solving now your stocks in the inside of the plane here we haveерь and here we have وي is an infinity here goes to the trace of the oiler on the boundary, okay? Okay. So this is the system. And normally you, you will have to take an initial data some u0 of xy now just let me recall what are the what are the questions that we discussed about this system so some of the questions we discussed was the derivation so how you derive this from Navier Stokes right the derivation from from Navier Stokes somehow the idea behind the system itself is that you consider Navier Stokes here you consider Navier Stokes in the half plane so Navier Stokes so I'll be writing everything in 2D so Navier Stokes I will write the velocity as being uv okay u will be the first component so the u you see here is a scalar that corresponds to the first component of the Navier Stokes so then if you look at Navier Stokes okay so this will be this will be your Navier Stokes system okay because I mean it's it's it's solving Navier Stokes in the half space yeah so the domain omega is the set so then when the viscosity goes to zero when the viscosity goes to zero we expect solutions of the Navier Stokes to converge to converge to a solution of Euler when the viscosity goes to zero so Navier Stokes becomes Euler of course now these conversions I would say in this framework is one of I mean it's a major open problem in fluid mechanics as of now formally formally this becomes this but proving that solutions of Navier Stokes converge to solution of Euler we discussed this already this is this is still an open problem so so the problem is that for Euler you cannot impose that the whole velocity vanished on the boundary you can only impose you can only impose that the normal velocity is zero so the normal velocity means u.n equal zero so u.n here the normal is what is the normal the normal is this right that's your normal so this will be v okay so my notation I mean my notation there is some sort of inconsistency in the notation because the y here and the y here they are not the same right so I'll try to explain this but we discussed this already now now the fact that these two equations they they don't have exactly the same boundary condition meaning here that only the normal component of the velocity vanished whereas here all the all the velocity vanishes so then if you try to do energy estimate we see that things do not work and Prentall and that's Prentall in 1904 proposed the following idea that there is a small layer of size square root of mu where here you will have Euler but here you will have Prentall okay now to derive Prentall you have to make you have to rescale so so instead of working with y you introduce capital y equal y over square root of mu okay so if you want to reduce this scale so you change you go from x y to x capital y this is the scale where you are seeing where you have the Euler or the Navier-Stokes equation here where you have the Prentall system so then you make this change of coordinate you plug it in in that equation and you just keep the leading order terms so if you only keep the leading order terms then you got this system okay and then what happens when I mean if I want if I if I try not to confuse you really to be precise then I will put capital Y then I will remove this capital Y but just for the sake of being precise and being consistent with the notation okay so I think we I spent some time explaining how the derivation takes place formally like formally again like you you look at the Navier-Stokes equation you make this change of coordinate right you plug it in so for instance okay so just the new Laplacian xy this guy becomes new dxx plus dyy okay so then when you do your formal asymptotics this term you drop it because this is small and you keep this one so that's why only the dyy remains okay that's clear for everyone sorry what about the second term the second term you mean what the y component of the second the V right so the V term okay the V term so the V term so here the equation for the Navier-Stokes is a vector whereas here this is an equation for a scalar right so it's an equation for a scalar so where is so something was lost in the in this process but that's that's just coming from the second equation the second equation of Navier-Stokes when you rescale to leading order term the leading order term will be the fact that dp over dy is zero right so so basically the the second equation you replace it by this one the leading order term the second equation becomes this one the fact that P doesn't depend on y and that's why the P e there you see is just a function of x doesn't depend on y right so that that's coming from that's coming from the second equation and it is also consistent with with physics the fact that we don't expect the pressure to move fast in the boundary layer so there is also some physical طوش behind this okay and other questions okay so let me give you now today a plan about a plan about what we are going to what I'm intending to do so I see many new faces so that's why I'll try to go slightly slower but so Monday Monday what we did Monday we looked at the stationally stationally prentor today we'll do the evolution so Monday we looked at the problem without this term and sorry I mean like the prentor equation you get rid of the ut and then you look at the stationally problem today we'll be looking at the evolution but Monday the two questions we looked at were mostly two the singularity formation or like separation like usually for prentor we don't use the word blow up but the word we use is separation which I think makes sense because like that that's what we discussed Monday what happens really at the time of separation everything is still well defined and there's no no I mean nothing going to infinity but it's more like a derivative which is going to zero so and it's called separation separation of boundary layer so this was one of the result we discussed and then I discussed few other result about the validity of prentor validity of prentor meaning the fact that navier-stokes converge to prentor in the sense that navier-stokes converge to prentor of course now normally what you would like to say it's more like navier-stokes converge to Euler and the prentor yes so that's that's those are the type of result we are interested in but somehow usually whenever we look at these kind of problems we'll be mostly interested in rescaling space somehow we are rescaling space so that we are mostly trying to focus on what happened okay but I mean there are also results that do there are results that do that navier-stokes go to Euler plus prentor and depends how you state it okay so these are the type of things we tried to discuss last time I mostly spent time talking about this problem I mostly discussed this one last time and this was mostly a result recent result I have with and lore delibar we talked about this one today the the plan is to do more or less the same thing but for the evolution prentor but I will I will try to discuss exactly more this point there are few words few words that we have on this so I can tell you so okay so here we have one and two here we have result with with with with and I will discuss a little bit but I'll talk a little bit more also about the validity of the prentor so validity or and validity like whether it is valid or not valid okay so that will be the plan any questions so Monday when when when we discussed the validity of prentor I mean I mentioned these all I mean the point that all these results are very very recent I mean most of them are just preprints not not even yet published but for the validity of prentor one of the main I will say geometric condition one of the condition on the profile on you is that you is positive like in this type of result I mean the I would say the the main assumption or the main thing that that is needed is that you is positive and of course there are many other things I mean for instance I mentioned like for this result of Samir Iyer and Yungu or another result of David Gerard Varey with Yesonuri and Mayakawa another assumption is that X is small so X has to be in a small interval but but the main the main thing is that you is positive and that they work in sobof sobof regularity right so so it's sobof regularity you is positive if you are in this kind of condition you expect that solution of Navier-Stokes will converge to solution of prentor now it turns out that similar result I mean like similar logic or similar similar results for the evolution prentor do not hold and I'll try to explain this a little bit meaning that if if if we follow so I'm trying to make this parallel because I think this is interesting and I mean for me this is a problem like even physically it means it poses some problems so again if we look at I like to make this comparison between the two problems if we look at so if I if we look at the stationary problem the stationary prentor this problem I mean like if you want to prove existence for this equation like local local existence meaning small X again here when we look at the stationary problem we are we are thinking of access being playing the role of time and the main assumption that is is used is that you is positive you is positive except of course at the boundary where it has to vanish here but inside it is positive so then when you is positive this behaves like this like an evolution problem and then you can solve and the results of go higher and David Girard Vary with Maya Kawa yeah so they more or less tell you that Navier-Stokes converges to this on a short time on a short X so for some short interval in X you can prove conversions and all what you need is sobole irregularity so these are very interesting results of course I mean I think that there is a lot of improvement for this result yet so but but they are very I find them very very important results I mean they show like that we can justify we can do we can justify the derivation in sobole irregularity now if we try to make a parallel with the evolution problem if we try to make a parallel with the evolution problem prentor the prentor system we have we can also solve prentor in sobole irregularity if we have so so the evolution prentor evolution prentor is well posed in sobole if my profile is monotone so if it is increasing then I can solve in I can solve in sobole right so again what correspond in the stationary case in the stationary case somehow the assumption that you need to get local existence is positivity of you it is understood by the fact that you want to see it as some evolution in X that same assumption here in the evolution problem correspond to the fact that you need you zero P to be monotone monotone in Y right you need it to be monotone in Y okay so then you can you can solve prentor in sobole then a natural question is to say okay since here we have this convergence result in sobole do we get the same convergence result here in sobole or not and the answer is no the answer is no let me try to explain this and it turns out that somehow the result that correspond to this will be more like some result in in jeuvre irregularity right so we even though even though here we can solve even though here we can solve for monotone profile we can solve in in sobole but the convergence requires jeuvre irregularity okay let me ask you one question okay here you concentrate to prove the profile and so on when you were positive no derivative in you you are the monotone property okay no the right Monday I talked about I talked about the case where this is has a monotone right here here okay so the monotone the monotone the right so in the paper with allure about separation so that's another in the paper with allure about the separation we also assume that we are monotone and the reason the reason there is that the reason there I can re-explain it so if it's okay it's just no no but okay so just to re-explain it so this for instance this is a profile where you is positive so think about this as this is why and this is you right so you start like that you can go now if you start with the profile like this it's maybe as you are solving this touches here right so so then the separation is not happening at this point but it's going to happen here okay I don't think I don't think there are okay I'm not sure about that I don't think there are but but but but somehow right so I think this is kind of things where I mean I will compare to some of the results on water waves where people prove the splash singularity yeah I mean it's more like it's more like in the spirit yeah yeah yeah I think yeah yeah those those type of results so yeah I think I think this is very similar to those I mean one can but but but like in terms of physics physically okay not necessary no the the the splash singularity itself doesn't require analytic but like in the course of the proof they use some somehow but but the result at the end doesn't I mean I think that's how but but but but here here like for for this problem I don't think you will need anything like analytic because like whenever you talk about the whenever we talk about the when ever we talk about I don't think the analyticity is used all always used is really this so that you think about it as an evolution problem but it's true that this kind of question are very natural but but again here the the the monotonicity in the result with and lower is more or less to avoid this kind of things so that the separation only happen here but do you think the picture will be the same like if you if you signify the things you know what do you mean you have to touch down the derivative zero here right when it touches it touches like that the derivative zero so do you think okay so it looks a bit similar above or maybe different different style or different singularity you I'm not sure that there are I can have a look and tell you it's okay so go ahead no but okay I mean that's that's a good but okay but you are going to speak about the yeah yeah that we still don't we still don't have it yet we still don't have it so I'll talk about related questions like on burgers or on but okay so and no but today I wanted more to talk about the validity of the because I mean I think maybe that's even more interest I mean more important okay now if one if one looks at this says okay positivity give us validity in the stationary problem why we don't have it here so it's true that okay for for the evolution parental for evolution parental and I mentioned these kind of things last time or maybe like in May for evolution parental we don't have only one type of existence result we have I would say three or four types let me recall them very rapidly so there is so type of existence types of existence results so we have results about monotone profiles monotone in Y but then here in sobolev we have another type which are analytic result so analytic result these are results of كفليش سامارتينو these go back to 98 and here analytic result كفليش and سامارتينو they prove existence for parental and also convergence of navier stocks to parental in the analytic setup so here we have everything is done you have existence and the fact that navier stocks goes to Euler plus parental everything here this goes back to كفليش سامارتينو of course I mean their result has been improved later on because in their paper they were having analyticity in both X and Y and then people realize that actually the analyticity in Y is not necessary so you can get rid a little bit of some of the analyticity in Y okay but let me not go to that so then there is another type which is جولوري ريجولاري تي which is maybe more interesting and there were few results about جولوري ريجولاري the first result is a result I had with David Gerard Vare and there were few improvements afterwards I think the best result now we have is a result of David with Helger and then there is another type which there is another type in a paper we wrote with actually with four authors كوكاوكا myself and وانك so I mean in this result we are able to prove also local well-posedness in a case where we can split in X we have regions where we are monotone in Y monotone in Y and regions where you are analytic right so you can divide your domain into places where you are monotone some place where you are analytic and you can still solve okay so I mean if you compare the type of existence result to the station in case I think here you have much more type of results right now now if we are interested if we are interested in the validity now so if we are interested in the validity so trying to go from this to this so there are results that there is a result that goes back to around 2000 by Emmanuel Grignet but I am going to talk more about some of the some work that I find interesting by Grignet, Gu and NGUN first to motivate to motivate another result that I have with David Gerard Vare and Yesru Nuri Mayakawa so it's a very natural question to ask in sobolef regularity we have in sobolef regularity we have existence for Prentor and again like the idea behind that existence I think since we have a lot of new faces let me re-explain it very fast because I find it a very interesting consolation that is easy to remember what is behind the sobolef regularity the local well-posedness in sobolef in the monotone case so this result by itself goes back to Olenik Olenik proved this result but she proved this result by not using so called Croco Transform which is some complicated transformation that you do here which is not very adapted to the Naviestox I mean if you do that transformation the Croco Transform you don't understand what it means on the Naviestox precisely so so the idea that I had with find a good cancellation in the monotone case and this cancellation comes from very simple observation by the way so the observation is the following if you write the Prentor system if you look at this equation now the difficulty if you try to do energy estimates the difficulty if you try to do energy estimates is coming from this term because this term loses derivatives in X so let's say if you want to do HS estimate so if you do HS estimate then you want to solve you do this in HS so so if I do HS estimate the S derivatives they can hit here they can hit here those are the two terms this one and this one right or they can hit some of them here some of them here or all of them here some of them here or all of them here but all those other things are okay they are not losing derivatives in X now this guy is good because it has some energy structure and you can integrate it by part so this guy is problematic okay now if you take the same equation but you go to the vorticity formulation so if you take this you look at the vorticity vorticity for the Prentor means DYU so it turns out that there is a cancellation here so that's why you get these terms okay if you do the same thing plus okay okay means that you are not having S plus 1 derivatives in X on one term you don't have S plus 1 derivatives in X on one term so then you end up with this now the idea that comes from that paper is the following cancellation is that you say okay I have these two equations this one and this one and they have a problematic term in each one of them so I mean the idea is really to kill the two problematic terms but you make them fight together and they kill themselves together right so the idea is really to say that in the monotone case this guy has a sign UI has a sign so then if you want to reduce GS equal DS omega minus DSU so somehow you take this equation and you multiply it by this coefficient you multiply it by that coefficient and you add them together then these two terms they cancel so then what you end up with is plus many other plus commutators so you end up with something like that so that's really the idea behind the existence in the in the Sobolev case so I mentioned this in May a little bit but I like it I wanted to say it again I have many new faces today but somehow it makes a lot of sense to ask yourself I mean this is a nice structure can we use it in the Navier stocks can we use it to prove that Navier stocks converges to Prandtl in the Sobolev in Sobolev regularity so if I take an initial data for Navier stocks that that more or less is coming from a monotone profile so just take a monotone profile of Prandtl rescale it put it in the Navier stocks solve Navier stocks and try to prove that that solution of Navier stocks will converge to Prandtl can one do this in Sobolev regularity so this is a very natural question so it turns out that this is wrong right this is wrong in Sobolev regularity even though we have a good a good local well posed result in Sobolev but we don't have convergence in Sobolev like in the stationary case and basically like the the enemy was already known in the physics literature from the 40s I mean this I mean this is some sort of instability called Tormey and Chishtien instability coming from Navier stocks it's really Navier stocks it's really a Navier stocks problem and it turns out it's what I'm going to explain a little bit is like a sort of destabilizing effect of the viscosity so the reason behind the fact that these Navier stocks will not converge to the even though we have a very nice local existence for the Prandtl in the monotone case is coming from a destabilizing effect coming from the viscosity so how to understand this okay so I mean these are papers that go back to Grignet and then Grignet and NGN I mean like the ideas so when we do the derivation of Prandtl when we do Prandtl when we do Prandtl we make the following change of coordinate txy we replace it by tx this is for Prandtl one can actually introduce another change of coordinate which is slightly more motivated by I mean the physics behind it some explanation about the physics behind it is that we expect some creation of vorticity at the boundary and if you have vorticity being created at the boundary maybe you should treat x and y in the same way right so there's another change of coordinate that one can do which is slightly more democratic you scale everything same so what's the meaning of this it means that in x you are looking in small x you are looking at high frequencies right so you are looking at frequencies like maybe which are like one over square root of nu not so okay if one does this if one does this if one does this change of coordinate and then you rewrite you rewrite your Navier-Stokes equation you rewrite your Navier-Stokes equation actually all what happens is that so I'm looking at Navier-Stokes but now I'm making this change of coordinate so then what we end up with so I'm not changing notations I'm not changing notations but you end up with this you end up with a viscosity that now became square root of the viscosity becomes square root of nu but besides that everything is the same okay you also change time everything is okay just you change time right you can convince yourself it's very straight forward and of course you rescale the pressure but the pressure is just the Lagrange multiplier so we don't care too much now okay now let's let's try to ask the simplest question I mean let's think that we are taking the profile for Prandtl so a very nice profile it can be we can even take we can even take a solution of Prandtl which is independent of x right so we talked about the fact that if you take Prandtl with that is independent of x I mean we can even take the pressure to be zero okay so I can I can take the simplest what will be the simplest solution of Prandtl the simplest solution of Prandtl it can be u t minus u y y let me be consistent with my oh okay never mind so let me keep capital Y so this will be the simplest solution of Prandtl I can take I can take a solution of Prandtl of this type okay so this will I will call it up of t y this is a solution of Prandtl so this solves the heat equation but with some funny boundary condition right so it's a profile that it's really like it has some sort of self similar behavior but it's really a profile that goes from zero to some constant which is this u Euler I mean if I don't have a pressure it means that my u Euler is a constant independent of x and t so it can be just a constant number u bar right so so I have a up that it goes from zero to some u bar right so this will be a up at y equals zero is zero and up goes to some u bar when y goes to infinity right so this will be like the simplest type of profile that you can write for Prandtl and okay I mean you can write it down explicitly I mean the goal my goal here is not to write it down explicitly but this is this is a solution for Prandtl this is a solution for Prandtl that is it has even like you can write it down in a self similar way I mean you can so okay so usually this is what we call like a shea profile because it only depends on why you can choose it so that your up is monotone and it stays monotone and so on so so then it's very natural question to ask yourself take this guy take this guy plug it in in navier stocks now this guy solves also navier stocks by the way because it's independent of x it's independent of x so you can so the viscosity the the viscosity term in x will will vanish there I mean so but if you perturb it what happens right so you want to perturb this guy and try to see whether this still solves yeah what happens if you perturb it okay so very natural question perturb this in sobol f does it converge does it converge okay so very natural question take navier stocks okay so will be more working with rescaled navier stocks so this is I will call this rescaled navier stocks so so I want to take u bar solution of rescaled navier stocks with the initial data u bar at time capital T which is this guy plus some small perturbation perturbation can be like new to some large power new to the power n and you ask yourself when u goes to zero does this guy converge to this guy okay okay the question is more or less so let's so the interesting thing is that this question itself is related to a very important question in stability in stability of oiler right so I mean one of the main one of the main questions that people were looking at in the 19th century mostly physicists but also like arguments were mixed between physics and math like is the stability of simple profiles and that's how for instance we have the very famous really criteria for stability in oiler and it turns out that this particular question is related to this to the stability in in oiler okay yeah share flow yeah share flow okay now um so so here I mean the natural thing to do is to linearize of course I mean like or all what I'm saying here tells us that we have to linearize so I'm asking the following question I'm looking at this you navier stocks my you navier stocks I want to write it as this now the is written in the with the time so if I go to the time then this will be square root of sorry right so as I said this is an exact solution for navier stocks and you want to perturb it right and that I want to write down the perturbation of this guy okay so then this will be let me call it plus some V maybe the notation is not very good because sometimes I'm using it there so let me call it not V but let me call it so so then we have to linearize so H I will write it down so now we have to linearize so if we linearize our H will solve what H will solve plus so my profile so so for this analysis actually for this analysis I can replace this by I can replace this by zero so I can think about this UP as a fixed profile because as of now I should put capital T okay and all this is really in capital T capital X capital Y so so then now we are back into very interesting questions about oiler itself we start by looking at problems related to oiler so why is that I mean the whole thing is really in time scales so when you look at this problem first you can right so maybe I should insist on exactly what we want to do precisely so we would like to prove again we would like to get the fact that Navier-Stokes converges to Prandtl so then it means that I want T to be of order one T of order one means capital or maybe small order one or even small can be one of a million even if I prove some result convergence of Navier-Stokes to Prandtl on a very small time I'll be happy so then I want capital T to be one over square root of nu okay so that's justify why I'm writing this like zero I want to take T to be large but large of this size I want capital T to be large but of this size so then it still makes sense to replace this guy by zero right I mean one has to work a little bit but so the whole thing in the new capital T capital X capital Y is to try to understand this linearized problem over times of this of this size so it's true that over times of this size you expect this term maybe to be at some a little bit negligible at first approximation it's not completely correct but first it makes sense to try to first look at the problem without this guy so then if you look at the problem without this guy that's a very classical problem which is the stability of Euler profiles right so that's how we'll that's how it goes okay now I'm going to derive so again so we have that linearized equation on H okay so let me do it step by step I'm not going to get rid of the viscosity from now but we'll do it in first step so when you look at this now UP only depends on Y and basically the question you want to ask is do you have growing modes for that linearized problem do you have growing modes for this linearized problem so the idea is to do some Fourier analysis in X and time so you do Laplace in time Laplace in time and Fourier in X because like UP only depends on Y right so that's why I wanted to replace square root of new t by 0 so that there's no dependence in T and no dependence in X so then it's very natural to try to find solutions which are some sort of exponential um I alpha X minus CT so what we'll be trying to do of course now I should recall that H is divergence free okay H is divergence free in our capital X capital Y so so then I have to look at H I want to look at H of the form DY minus DX of some stream function Psi okay and my Psi of T my Psi of TXY my Psi I want to look at it in the form some profile some function of Y times okay so this is the form um this is the form of the solution we want to to find right so alpha is is coming from the Fourier in X and alpha C it's a sort of sort of Laplace in time okay so the question is um can we find solutions um where C has some imaginary part right and of course if you have one you get the other actually but but yeah so with the with the right sign so that you get some growing mode so you want to avoid that huh normally you want to avoid that you are looking for it depending on whether you are trying to prove the so at the end you are looking for it yeah at the end because like what but you are the example where you don't have that I mean yeah but we'll see the whole discussion now is that you will always for the Navier stocks part you will always get something here for the even if the Euler doesn't have it right so that's the idea at the end so the right so it depends whether you want to prove the stability or the instability so the result of uh they were uh looking for I mean they were trying to prove the instability so they were trying to find this guy which has some imaginary part now what is the equation satisfied now by five right so we get some equation satisfied by five and um so you plug in everything you plug in everything you get rid of the pressure and you get rid of the pressure by looking at the curve right so you take the curve so take the curve so when you take the curve the equation will be written in terms of the vorticity and the vorticity the vorticity is is the laplacian of I mean what you are seeing is the laplacian of five um so curl of h I can write down curl of h curl of h is the laplacian of psi is what we are the way we write it down is the second derivative minus square five times the exponential that's that's the vorticity now the um the equation the the equation that you get the equation that you get is the following um so I'm going to use the notation u yeah okay you can write up minus c r r is one over the viscosity okay so okay maybe as of now let me not use r let me put it here as viscosity so um let me just to explain how the calculation goes like first you do it I do it like this let me put i alpha here and then you divide by that so so I I mean the way I wrote what I wrote here is just exactly what you get so for instance the time derivative the dt um the dt applied to the vorticity so so of course now when you take the curl when you take the curl you get that you get that the Laplacian of psi t plus ut dx Laplacian of psi um now you apply the curl there one second so the curl is um okay I mean you have to write it down right but let me keep it like here more or less that's what you get I'm just trying to explain the the different terms but the term the the guy with c that's coming from the time derivative the guy with up that's the term there with the up and the term with the uw prime that's the term coming from from there right the the i alpha the i alpha that's the x derivative that's this guy that's coming from the second component of the velocity so you get this equation and then what classically we do is that we divide by i alpha here and we replace we introduce the Reynolds number and that's how we write it 1 over i alpha r so this equation is very well known in in the study of stability of Navier-Stokes so as of now these are questions not necessarily related to Prentor the problem as I'm writing it here is more like stability of Navier-Stokes I mean you can just ignore Prentor completely and this equation has the name of or somerfield and this equation if you take if you take the limit r goes to infinity and more precisely it's alpha r goes to infinity so if you look at what happens when alpha r goes to infinity when alpha r goes to infinity this term disappears and then you end up with this equation which has another name which is the Rayleigh equation this is Rayleigh when you add this guy it becomes or somerfield or somerfield is a fourth order equation in y Rayleigh is second order but it is singular it is singular when this guy vanish and depending on where is c so if c is in the range of up then this becomes this becomes singular of course now this the question the study of the Rayleigh operator the or somerfield there is a huge literature on this mostly like in physics and so the question is do we have solutions do we have solutions with do we have solution with imaginary part of c positive because so assume assume we have a solution with imaginary part of c positive let's assume that for the let me first do it for the Rayleigh so assume the Rayleigh assume Rayleigh has a solution assume so basically assume that up is an unstable profile for Euler assume up is an unstable profile for Euler actually what I'm saying here is goes back really to the work of Grenier I mean then there were a lot of so assume that up is an unstable profile for Euler what does that imply that implies that the equation this equation find a c which imaginary part positive and a phi that will solve the Rayleigh equation so then what we get we have a phi with some value alpha and with c we have a phi and alpha and c with imaginary part of c positive that solve solves Rayleigh Rayleigh meaning the problem without this guy then it's not very difficult I mean it's not completely trivial but it is not very difficult to do some perturbative argument to introduce a so called sub layer some viscous sub layer actually of size new to the power 3 quarter but somehow it's not difficult to construct something similar but for Navier's talks our mean depends on the Reynolds number so you can more or less construct things like this where c r is more or less same as c right you move you move a little bit this c moves a little bit by something which is of size 1 over r right you can make it like if you take r very large if you take r very large this is perturbative this term is perturbative it's true it's a singular it's a fourth order so it's a singular perturbation but somehow you start from this from what we call an unstable profile for Euler to perturb the unstable profile for Euler you get some sort of unstable profile for the or somerville and there is some work to be done but it's not very difficult so but then your c r is more or less like c I mean it's more or less c plus 1 over r but then you keep the imaginary part away from 0 now what's the conclusion of this so here I'm starting with some unstable profile for Euler so what's the conclusion so if we try to go back here the conclusion is the following you come back and what do we have we have a solution for the linearized equation we have a solution for the linearized equation that looks like I mean the phi depends on r but we have this sort of profile we have exponential i alpha x and then I have exponential minus i alpha c r t so now if I go back to the tx y if I go back to if I go back to txy what do I get I get phi r y is what but we said the imaginary part the imaginary part of c r is bounded from below I mean like it's a number so basically what we have here we have a frequency in x so the frequency in x so the frequency in x is like alpha over square root of u and the growth in time the growth in time is like alpha imaginary part of c r one over square root of u and it's exponential it's exponential of this times t so if you are in situation like this the only way you can get estimates that you are in analytic regularity if you want to prove some sort of convergence in some analytic and in some functional space this will require this requires analytic right so this requires okay so this will require analytic regularity I mean like if you are not analytic if you are not analytic I mean you have like these are going when u goes to zero these are becoming higher and higher frequency and the growth is I mean it's like you have something it's like you have something like this exponential c t right if you have if you have something behaving like this with some constant in front I mean with this sort of constant if you have something behaving like this the only way the only way you can control such kind of things that you need to be in analytic regularity right because you have this growth so your initial data should have should have the same decay initially so you should be like some constant times c so if your initial data has this behavior on the Fourier side even though you get this growth but for short time for short time this will will kill this growth so that's why you need to be in analytic regularity okay so so this is if up is unstable for oiler now what if up is stable so if up is stable then you don't have these guys so all solutions of this type will only happen if solution of this type only happen if c is imaginary part of c is 0 right now now if we think that now the whole discussion the whole discussion is what okay so if we are stable then if we have if we have what we call like embedded eigenvalues we have imaginary part of c 0 so then formally if I if I think that cr is like c plus 1 over r of course I mean this is to be checked if I think that that happens then I come here then my cr will be this cr will be let's say the imaginary part will be then it may make sense to think that this will work in sobolev regularity right if I am able to prove if I am able to prove that this logic is correct like cr is c plus 1 over r then we we can expect that we have result in sobolev but it's really that conclusion which is wrong this is not true this is not true meaning that it's a little bit it's in a sense counterintuitive it's very counterintuitive because what you are saying here what we are saying here is that you take this equation which is the inviscid guy and of course there are boundary conditions that I didn't write the two equations do not have the same boundary conditions but you take this guy this is stable so you only have you only have solution with imaginary part of c0 you perturb it you perturb it and then you start getting growing mode even with the cr which is much larger than 1 over r okay any questions about this so let me try to explain a little bit but maybe maybe I will not go into the whole I will not go into the detail because I want to state few other things but there is a whole construction that is I will not give the whole construction but I want to it's more like an ODE construction and if we think about it I mean it's really we are talking about ODE's at the end singular ODE's so the question you start with some and actually the whole construction uses uses three small parameters uses three small parameters which are alpha c and the viscosity okay and it uses a lot of property of every functions but at the end at the end without really trying to go into the whole detail at the end we end up with some profile phi r alpha r cr where cr is some sort of c infinity c infinity is the guy which has imaginary part zero plus some big O of alpha r minus one third I mean as I said the right the right parameter that you look at is look at alpha r going to infinity so that's really the guy that you get at the end after this ODE construction so this work here is going from a profile like this for Euler to this one this is singular ODE and it has to do with every every functions it's not a trivial thing at the formal level it was done since like the 40s by Thornman and Chishtine but then at the rigorous at the rigorous level that's the work of Groney, Go and NGN to construct really a growing mode for the Navier stocks now what does that imply of course then there is also right so now there is also a question about what is alpha as a function of r and that depends on the type of profile that you are looking at so there is another discussion about what is alpha of r I mean this depends on the type of profile that you are looking at for Euler on UP but somehow somehow there construction at the end it shows that it shows that what does it show if we go back here if we go back here then this guy is alpha r minus one third I mean one can work out this and somehow this will require some jeuvre irregularity I mean one can work out what is jeuvre regularity I mean basically their work is to prove that there exists one guy that one growing mode in some jeuvre irregularity now on the positive side we have a paper with David Gerard Vare and Yasunuri Mayakawa where I mean I don't want to go into the detail about what are the real assumptions that we have to impose on the profile so there are assumptions that one has to impose on the profile but we have few different result depending on assumptions on the profile we can prove so this is with David with Gerard Vare Mayakawa so depending on assumptions on the profile we prove that Navier-Stokes converges to Prentor in jeuvre irregularity so this is like the first result that do not require analyticity right so that's the first convergence result that doesn't require analyticity and in I mean like here what I mean depending on the profile I mean we can also allow the profile to depend on T, X I mean there are few X X I mean that's we hope to be able to do it that's not yet done completely but but in the best case we can really match exactly the growth of Gruny-Egu and Guyen so we can really get the result which is really sharp so the jeuvre space is sharp with sharp with sharp space with sharp regularity or with sharp jeuvre class which is like three halves and in the best case we get the sharp jeuvre class three halves okay so this is a type of result that I wanted to mention about the validity the validity of Prentor and again for me the important thing here the important thing is somehow this sort of destabilizing effect of the viscosity but that was already like understood physically like since the 40s more or less any questions about this and I will not go into how these results are proved but but somehow somehow in this result we use a lot of we use a lot of the work of I mean here I'm maybe I'm saying okay this is a singular ODE but I mean here there is a lot of work by the way like it's it's a very precise it's a very precise construction based on what they call really airy iteration so they have to construct a very precise object I mean these guys they have to construct a very precise object based on some what they call really airy iteration but I mean more or less I mean the idea here is that they have to choose in a very particular way some alpha and some c right alpha and c again alpha is like the Fourier in x c is like Laplace or Fourier in time and then they have to choose a very particular value so they choose a very particular value for which they can make a construction and they prove that the element they construct here the element they construct here has the growth of that jewelry 3 halves now our work is more or less to say we have to look at everything and prove that this guy is the worst guy right so that's more or less how our proof also inspired from their construction but it's slightly more general because you have to you have to understand what happens everywhere in the Fourier Laplace space so that's really the idea of the two proofs in few sentences but each one of the papers is like 70 or 80 pages so it's always like whole constructions and so on but at the end at the end it's like some ODE techniques like it's really some interesting ODE techniques so this is about the validity of Prentor so now let me talk about a little bit about blow up for Prentor I mean I don't have a lot to say about blow up for Prentor so that's why I'm not taking too much time for that but ok so these are whole program that we started with so I'm not going to give really precise results but I'm just going to say what we are able to do as of now so this is a program we started with شاركولو تجغول سليمي براهيم so the question we want to understand is can we describe the blow up for Prentor for the evolution Prentor so now I forget Naviestox and I want to understand the blow up for Prentor of course I mean if we are really interested in Naviestox itself then blow up for Prentor is not the goal because if you have blow up for Prentor it means that your approximation from Naviestox to Prentor is already has already a problem and it means that Prentor was not the good model to start with and that's how other more involved models were discussed in the physics literature as of now mathematically we still don't have a very good theory let me just mention names about models but there are at least there is so called triple deck model interaction boundary layer I mean these things here the idea about these things and it's related really also to the type of blow up we think to construct for Prentor itself the idea about either of these is that so here you have Prentor here you have Euler in the derivation you remember that Euler talks to Prentor through the boundary the boundary condition at infinity but somehow Prentor doesn't talk to Euler when we do the Prentor system interaction boundary the interaction boundary layer is like another slightly more sophisticated model where you allow the Prentor to talk to the Euler right in some not very complicated way but I mean there is a different model a different scaling that you use but this is still not very well justified and it turned out I mean for me that's interesting is that the type of blow up we are trying to prove for Prentor they more or less it's something that is shooting to infinity right so somehow the type of blow up we are even like numerically this is what is seen like in the blow ups for Prentor what is seen is that something like that like you get some singularity on the whole line that goes to infinity so it means that there is some information that is being ejected so it makes sense that one tries to look at this kind of models we still don't know how to do all this but at least the blow up we are proving for Prentor is of that nature do you mean no no I mean like let's say this is the point in X where you have so this is the point in X where you get the blow up but in Y the point in Y the point Y star where where you are reaching the maximum is a point that is going to infinity of course now the I mean there is like a sort of plateau here that forms where you are reaching your maximum which is going to infinity but that thing is going to infinity also so if you think about the problem locally in Y you are not really seeing a blow up it's bounded so I mean it's a blow up that is going to infinity but what I find interesting I didn't really see it precisely but I guess physicists they have it in mind I'm sure but that is what is behind this kind of models like the derivation of Prentor itself the derivation of Prentor itself is based on the fact that such kind of things are not happening so it's true that if this is happening that you expect to go to a model like that but now if we are just interested in Prentor we forget about Naviestox I mean we can just study it study what are the type of blow up and so on but for me like the next step is what does that mean for Naviestox how can we improve the the approximation I mean how can we improve the Prentor model itself but these are still I think we are still far from understanding this kind of things and this is even more complicated the triple deck has like different different sub layers okay now the results we have so I just take five minutes and they'll tell you the results we have as of now so um so we have the first result um so the first result is the first result is a result that only studies what happens um in the odd case if we are odd in x so if we are odd in x we can reduce the it turns out that if you are odd in x and you look at c of ty equal minus ux it's an interesting observation if you are odd in x and you look at the derivative at the vertical axis then you can write a closed equation you get a closed equation you get a closed equation on c so somehow instead now of having a problem in a two dimension it becomes a one dimension problem and it was already observed by e and inquist that this equation on c in some particular setup blows up but they got the blow up so this is a paper by e inquist this equation blows up the equation is more or less something like this dTx um something like this always forget the signs but they think and of course depending I mean if you put a pressure for oil you can get the pressure for oil but you see this is really in the spirit of uh parabolic equations but with this funny term and actually this funny term really completely changes the type of blow up you have really this funny term is like that non-local guy that is pushing things to infinity so here the blow up is not really as in the heat as in the semi linear heat equation where it is concentrated in one point but here it's really something going to infinity right so that really changes completely the picture so then we have a result about this so basically we in this paper we can justify the profile here so that's one first result so then we have two other results um with sharl and tej the first result is studying uh burgers equation so burgers equation means that you take print all but you you get rid of the viscosity and you get rid of the y variable right and then burgers and but then we also study the burgers with vertical we also study this model so this is burger with vertical viscosity with viscosity in y so we can study blow up here and uh um so whole constructions and so on so we are we are able to do that and and then the third thing we are able now to do is also to do the inviscid the inviscid print all and there is a whole literature about the I mean not whole literature but there is an important paper by hunter and uh and uh on the inviscid print all like the fact that this is you will post even in soba left turns out that these equations locally will post in soba left and we can study the blow up here now the next step is to put everything together but that's still under under work okay let me stop here