Stability of the Couette flow

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Stability of the Couette flow
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I will present new results on asymptotic stability of the Couette flow in dimension 3. These results give an estimate for the size of the basin of attraction, depending on the viscosity, in Gevrey and Sobolev topology. This is joint work with J. Bedrossian and N. Masmoudi.
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as the and you have to and that's you and delivers thank the organizers for the opportunity to speeches today and whole months and spending here which is great as long as I'm going to talk about the stability of the quake In the 1st should introduced and that is it's a question of the Saudi aggression is as follows and New is a positive coefficient which is called the viscosity and he knew it was 0 the questions could on and there is a diversion the conditions to their versions of you is 0 thanks you is valued in Oct 3 and it's a function of the x y but the eggs is in the chorus the Y is in the winter space and he was also so I agree there is the few fluids that live in such a geometry nevertheless it has the advantage of the getting rid of boundaries so what I wanted just today is the stability of sheer flows and that's already a complicated question and you add down there is it adds the numbers of dissident is which I do want to consider today so this particular geometry allows 1 to have a share flows without boundaries so what is the sheer the shear flow is the flow of the tide said F of y 0 0 an end but fearful that I want to look at is the question where F is a deserving of functions if you have 1 0 0 0 consistent with and this is solution of the Navier-Stokes for any new budget so the stability of Sheriff frozen in general is a a dissidents questions and what we discussed today is the stability of this particular shear flow which is simpler for 4 reasons that we see so maybe I should draw a picture so this is why this is a vexed and this is the and so the velocity of the fluid at a point on the Y axis glistening that but the symbols of the Union our are share so at home you get a sense for the geometry that good so the question that we want to understands is the stability of this flow which is stationary solution of and maybe even the asymptotic stability please NOTE :colon it does not have fired energy some OK so what I was there was stability so the the 1st point is that you need to take a positive viscosity questioned otherwise you pretty much have no chance to new has to be by so next thing you do is try and look at the linear problems so informally stating just dropped the connection during in which a cheaper centuries the the heat equation so it's a bit more complicated because you need to have eyes around the question but essentially get the union integration with something more complicated terms which are your order but what you see that the forum is stable and isn't very stable in that you get respect your debt so what's Massachusetts the stronger spectra information it's not hard to reduce the knowing our stability the question is of the spectrum you the home states and definitely walls so Maryland this requirement in the revision and that he was a year so let me remind you of it In forward at this point but significant reference all however there is something which in hegemonic stability is known as the summer fare paradigm so it's true for the quick flow and for many other shear flows so what happened is that the flow is Of the unstable as a result but experimentally unstable living close to some of the the problem our stability but the experimental instability partly because yes yes but if so of course it's not really a paradox in the woods it's underlines is that due the important notion is not so much money our stability as the size of the basin of attraction of the quit focuses on the resolution of the paradox is that just saying that that important the idea saying it is too early to quantify known in our stability in finding the size of the the basin of attraction and that this was already underlined by day and finally there is the last idea which maybe when that so used to winning the pole for differently questions which is that the the size of the basin of attractions really depends strongly on the topology that you choose OK and this idea goes back to Reynolds goal phrased it in more physical terms so as for the this together you see that the right question is the following the so many lives Nicole but it the change in addition delivered at could you not the Federation of adequate initially the deletion of the question and you had to be done the division of the quit fuel at a later time I think he's doing and I just serving the Navier-Stokes equations 42 and so I hope I convinced you that the right question is the following is a Viennese at the banner space X with associated to it then what you want to find is what is the smallest we remember Dharma such that if the initial motivation is less than in New
To the gamma to remember New is
the size of the viscosity and the more viscosity you add more stable the fuel so it it it makes sense that the dependencies is like that so that might ask why should the dependence power lights when it's somebody With experience teaches us so what you want is that the the the was numerous series it it it would use this sort of depend for more sworn in and this implies 1st that you of In smaller and 2nd that you goes to 0 as he goes to infinity so in other words which were asking is to really quantify the size of the vision of attraction of quacks I and what we are asking for is a that the stability of is sufficiently close to To quit emissions so there has been a lot of work on this question so I'm just gonna mention of few a few names in the history of some um yes that's right to destroy him and on of course you could ask you know you could ask you for the bitch remain closed to quit for lobster but I don't think this has been too much look at rights would have been a lot of the heuristic or American works that's so committed just mentioned your name and his Driscoll and Clinton there is no magic and Jefferson there is no violence this gentleman book and many others and that would actually do you agree to find that gamma lies between 1 and 2 several quarters depending on who you you are but OK so that's I wanted the work to where the authors did not free care about the importance of the topology rights just considered that emissions with be mentioning what sort of perturbations and then there is another word by ready Schmidt
the Baggett Henningson where the authors find them like it would be happy for 1 because the rough the divisions look at you perturbed requests by something which is like white noise whereas here there were more perturbing made junctions and OK and there is also a vigorous results so it started with a Roman who looked at the case of the channel rather than the more space and then there is the prize room Henning sons book and once again there is many references think everybody but do do world record was gonna less frequent and 4 by rigorous means in dimension fleet so everything's admission 3 and in some the college so that was the state of the art noted me describe our results so this is so limited 1st at the start of the 3 three-day and and then I'm going to say a word about what happened in 2 days so in this way days there is a total qualities that you can consider that either direct apology or somewhere the provision of about so in the Jurek apology he said we found cannot be close 1 and in support of the budget we fell down much less than or equal to 3 have so this is a joy to work with Jacob across I was in Maryland and in smoothly with it anyway as woods recognizes that it said that the debates right we have clearance which countries say that indirect apology demise 1 and so is less than or equal to 3 where furthermore able to say that this result is all the and that's why I stated as again that it was 1 this 1 we suspect it is popular because it agrees very much with what these guys found for beat up and so if the provision certainly rougher than traveling so we suspect is optimal but we could not really prove it he has you gain anything from was yes or even if you will further to undertake but none of you when you would get To be saying the same thing so many of the others for years so he wouldn't measure it is unsafe the 2 would be the 1 hand breasts too and where inmates were left it is not the way to the final to the 5 had pressed you know in it OK so that's that's what we did with Jacob and later but but there is also a reserve on the details of which will be given for picture so In July topology so this is where everything started as you find gonna input 0 and this is due to be president and this would be that's for the day's new accords 0 and then it was extended by the trust him next Monday and the gun to the days of the positive viscosity and recently in the case of someone eventually has been treated Syria finding gamma less than would ask and this is due to the trust him because and waned OK so it's really nice because for this particular flow it's possible to get a pretty complete picture of what happens in terms of stability and there is this the state's future that you have dependence on the on this interesting future that it's dependent on the topology which this is not so common in the interest of all of its consulate in most of the people 1 of used we did so at anything above 5 hats we we we believe should give the hats and pursuant to a very low overnight and then there is something else happened but all and the many stick it into a city that has with the tribute under the tubing that change would treatment so this is tuition-free right this is emission-free and into any of those arrangements is the same and the Sobolev exponent is 3 hats think if fled fear OK so that's that's the reserves of snow and rain to explain why would the picture is unique in the region I so well so the 1st thing to do is to consider the but the question around quipped please you each of these things what is so you want to avoid something later as you pretty much since you have viscosity if it's possible that maybe you stuff very close to quit because of a lot of completely insane things and comes back to quick right so this we don't count as a stability we counted the stability if remain or was reckless to quit this is the views them of amounts of water flows remorse transients always so I'm going to explain that it is indeed this the fact that the Yaris problem is that not since the joint there that gives trends in growth and which is ultimately a think responsible for this difference depending on the topology Yukon today so I'm going to try and explain again so that the part of the question I'm so as I was saying would consider a solution of Navier-Stokes the former quipped that is why 0 0 press you now you is the perturbation of where so if you do this more competition you realize that you saw this inquisition OK and so did this this spot is not for drawing In and of course that the divergence of view is still there OK so as it turns out that it's a good idea to try and remove this conviction durum and this can be done easily by changing the independent viable Though it comes to the expense of modifying the differential operators so let's do that so is used to remove why the Exuma and what you do is you said to that the likes of which explains why the Capital One was completed the
cuisine and you will be right the question in this new virus it has the effect of turning gradients into what he called Grand which is the white line CDX the banks and like into Laplacian L which is great and square to this new coordinates and Nicholas capital you can t completely recorded an international treaty with the Clintons new coordinates the question reads but the bulk of the notes OK it's so pretty much we just remove the widely excuse that and now the different operators had this more complicated structure "quotation mark King has so if attended linear eyes problem maybe it's best to view it in this new coordinates when it's pretty clear that teaches you win and you will and do the pressure to win 3 again expressed the pressure has to tread on inflation in the UK and OK so that's the nearest problem and that's the food question so what I'm going to try and explain in detail that remains is a few hours and striking properties of this 39 as a problem which has a lot of very you striking features now and then and then we're mentioned a few important features of the of the of the normal the and the it will so the first year heuristic 10 it is the so-called lift effects destination that and understand scholars in the physics literature so this is the domain of destabilizing effects in the dimension 3 and is absent in dimension to this explains why the exponent In dimension 3 on his good than in dimension accused pottery the focus of others it worked well a very important idea is that the frequencies which shocked 0 you need capital X behave or the next behave very differently from the frequencies which are not 0 in years and this is because of the sheer here so the if you look at the U.S. program this year will only affect the non-zero frequencies and X so the picture is very different depending on whether you look at 0 frequencies or 0 frequencies so here we look at 0 frequency is an excess and we call on it's not the integral of of X the X which is the same as interval of capital due to the fact that it doesn't just 0 frequencies in the X viable so it turns out the unionized problem is extremely simple for frequencies and because then the the ECU to concerns that the right-hand side Kansas and the belt that Ellen becomes a regular diet of so doing a thing that remains is the keel -minus the apprehend you fill up at 2 0 2 indicated 0 frequency X press you 2 0 0 the cost 0 that's very simple and you can actually explicitly sorrow and you find that you it is so 1st and that is the nudity that clashes you want to appear even In initial to distinguish from the 0 which your frequency -minus the Mutual Insurance Co the new chief of the National Avenue to Commissioner of Insurance Co 0 right time 0 frequenting the nutrients professional you and nature that India and so obviously the 2nd and 3rd coordinates or will behave but this 1 is the grows linearly until the he currently takes in but this happens only at the time to like 1 of new so this gives me a she likes 1 Over the end state intervention to and this does not soccer because you too uh 0 has to be 0 due to the divergence free conditions so that's the main source of instability and it's really a 3 D effect OK so the 2nd piece of years heuristics that to presented is the so-called enhanced dissipation at the time back OK so let me try and explain the idea in physical terms so if you look at this picture is if you have a good physical intuition that if you have something that depends on X due to the transport in this direction for this direction is like that Torres direction if you have the day that actually does depend on acts of this leads to the creation of a very high frequency is very fast and is get then even up by the viscosity already heated wish so it's easier to see it through the questions so it's actually very easy but if we just look at the year the question in the new coordinates consumers the inclusion in the new coordinates and it's just forget about the rest of the of the Yaris problem adjusting this piece and remember that that that is the X squared because the wife minus TD Bank square present easy square and so is becomes very big of course what matters is this guy here give access to essentially what remains is the who you she square the bank's squared that's something that that citizens needing lots if he goes to infinity the cost 0 so you see that the viscosity becomes a man
as he goes to infinity and if you switch to fully viable this becomes the key hats -minus new chief squared case where in fact equals 0 and it's easy to start as part of key chain you something next year minus annuity include case squared initial so here you have secured instead of a team because we did mixes huge difference and it is viscosity act extremely fast once you have to share and so it is that the disability landscape there is no good 1 of renewed the one-third rights instead of 1 avenue for regular and he did look as I should say this sort of idea has been exploited
1st by the Dresden promiscuity and OK and then there is the last of linear effect which is the famous In this impending which is the wire version of Arredondo damping so in New York heuristic 3 In addition the thing which is the once again the pointer version of longer than and of course this phenomenon received a lot of attention after the work over the wall and the lid on London meeting for his of whistles so it's actually very easy to proceed in this context there is just a trick and the trick is to switch to the new arrival here too which is a lot national airline Utah good so it it looks a bit magic which the standard change right but that this is due for instance and it was very nice is that in this new environment but the equation aside by Tutu is indeed the future -minus New Left left and a chance to put the cost is 0 To that end the year rising levels of this sort the as you will have to work the other laborers don't believe that simply so I that's where just single out this so for simplicity if we just look at the case and you request 0 so which is rooted in this case of the oddities what happens is that here too it is constant right could you just did the teacher to equal and that's the reality so it's very simple to recover you to U 2 is simply 1 of the passion and kitchen so now think they'd 248 France forms and switch from xyz landfall To take 8 . edu and you see what comes out is you to act the heaquarters 1 over key square note given the definition of foreign aggression and a dump -minus Keats's squared off and square thank you too initial and this formula is extremely instructive because from this fall I in read 2 things 1st what happened as he goes to infinity so as the goes to infinity designate becomes huge so that you 2 hats goes to 0 and then you have to go through 0 as the goes to infinity so this is it's could envisage them being so freely because you really have a very nice having been insistent no dissipative effect whatsoever nevertheless you get his decay of you 2 as the and I should say it's not an effect which is like scattering 4 the way the questions in indoor space because you don't have space to scattering the exterior then OK as a next to the good side of the picture but on the other hand this factory here vanishes the years was 8 over food despite the the vanishes and then you get the growth compared to what happens at things look at this is to say the fact that this 1 is quoted in this campaign and this is rather helping us and to prove that stability and this 1 is good or mechanism and of course it's rather annoying because it means that you have growth at various moments in time and of course the fact that you have these growth at various moments in dying is affinity linked to the nonsense of a joint director Of those in eyes operating since its months of giant it's well known that you should expect growth for the evolutionary problem OK so that those of you can't put it evidence good if you look at that it's so the very nice all the time but there is 1 particular moment rate rose and they did again so 1 might that intermittent so now the is just not enough problem so far the only heuristics at 1 into so-called streak solutions so thought I should call the I should write again the known in Europe the nonlinear for defending the original violence and important observation is that there is a large set of explicit solutions to these problems which consist of a U 2 the depending only on y and z new 3 depending only on y and z as well I just serving the 20 minute Stokes equations why you want is driven to it you bring that in use ceded it works and that you want is simply driven by U 2 and you 3 in the following way all right so that someone who is to usually gets if you will 1 you truly each 1 them you could see why pursue through you go on the quarters minors you because it want as a slave to the others and is to solve the problem is that an exact set of solutions and has said Dr string solutions because supposedly did look like streets in experiments and you see that the role of an attractive because when the and hence dissipation goes is that very fast it's killing any dependence on right to here to get out of this range
fast decay you need that day non-zero offices that the
days none 0 you get very fast then things which remains does not depend on eggs and that is our the streets could be are the tractors at least on the bench scale of 1 of renewed to the 1 that OK ended last thing that I would like to emphasize that at the include various forms which are found in this equation soul because there isn't it there is a lot but it may just mentioned to if you look at the most threatening interactions when 1st the most threatening interaction is due to the growth of you 1 of them you see that you want is growing Louisiana so the worst that could happen is if you want it's an 0 frequencies talking to you 1 had 0 frequency maybe with derivatives 1 for this would be very bad because both grueling only so it would be completely to the structure unfortunately this does not happen there is no such interaction another problematic term is the bread and for each remember agreed that you don't red and its you won the ax arrest you too the association with Chiquita deal what you want when you make this Q 3 and so of course this key here is very threatening but luckily it comes standard with a you too and we feel that you choose as decays thanks to invest that for those of cancelation between these 2 key factors here and the U to which dictates what that's so the the prove consists in putting all of this together so as century the picture that emerges is that but 1st on advanced is 1 of many of the one-third you did this and hence dissipation which gives water None 0 X modes then the solution looks like a stream a 40 and larger than 1 of and one-third Street press something very small and then what happens is the lift up effects of Dixon and did growth until you reach I'm wondering and new pray you can keep within a control order and you can't do it thanks to reduce cancelations which are present in the equation and that's what I wanted to thank thank you for your attention we have have to do is make it useful you workers the incidents of the last month of the year on its it's the big worry here the unit to so that for this reason that um the joints and would introduced their time within 2 days I need to do that's really the main problem but because they're able to the case Newark was 0 In through the it's not me it's 1 of the problems but it's not the only 1 anymore In of Newfoundland explains the collection Group and the you can call all right yes or OK so what would this uh or mechanism is doing Is that so it's Sayer if you think of 8 out being big look at high frequencies no sir we think of it that being small which 1 would think the good news right so what you want to avoid is high to low right so that's what or a mechanism could do and that the best way to avoid high to look at state is just too start with very few high frequency and and that's why it had so that's that's how it's going to hit mechanism of to do this the university Japan has to move his unending 20 years into the trickier into India things are messier and it's I'm is not the only effect that 1 has to In particular the list of effect is move more threatening and that's from we Yemen it's 1 if the perfected through the witches services that will hear the explicit love through the high name you have fought and the In Jevric apology do 1 of a new newspaper is easy to explain that the look the Ministry of those have it's listed there is no simple I can point to a single mechanism but the military is on ultimately right with the use of force because we tried hard to do better and that makes you the same thing really I agree it's few him