That's it. izgledaj z nekaj, da smo vlasti v maju. Zdaj... Zdaj sem... Zdaj sem izgledaj, izgledaj, izgledaj, prantul system. V svoju dana, v svoju dana sem izgledaj prantul system in prantul system. Zdaj... Stacionalne prantorne system je prantorne, ali nekaj nekaj prantorne. Tako to je. Tako to je izgleda na u. In potem smo vse in kompressivnosti kondition, da imajo vse vse vse vse vse vse vse vse. potem včasno zelo je otekljamo vzal u boju tako že bomo vzal u boju, če je tako professionela vzal, napisak nekaj a polo na prostvenstven cans do, OK, primetno so bo vzal tko, More or less, we are interested in solving this in the domain x bigger than zero, y bigger than zero. So basically, the picture should have in mind, so this is x, you take some x equal zero here, you impose an initial condition, which is this u zero of y, and then this u zero of y is positive, right? u zero of y would be positive, so then this guy is positive, and then the equation itself, you will think about it as some evolution equation in x, okay? So x plays the role of time, right? So then you want to solve this till some time or some x x star, right? So this is where you are going to solve, right? So the idea is really we are thinking about this as some evolution equation in x, okay? So this is the stationary prank tool. So today the plan is to discuss three types of results. I'm not going to go into details of all these things, but there will be three types of result that I'll be mentioning, and more or less these three types, I will be discussing more or less exactly similar three types of result for the for the full system with the u t term on Wednesday. So the first result will be a result about blow-up for the blow-up result. I mean the term blow-up maybe is not the best term used, but it's more separation. That's the term that we use usually, the separation of boundary layer. So then there is another type of result which is long time asymptotics or large x asymptotics, but of course you'll be seeing that large x behavior. And then another type of result that, okay, I will not go too much into details about it, but validity of the prank tool system. Okay, so this is related to what we discussed last time. Validity of prank tool. Validity of prank tool means that if I take a solution of Navier's talks, it can be well approximated by prank tool. Right? So, and these are, I mean I'll be mentioning two results here. Okay, so these are more or less the plan for today. For Wednesday we'll be talking about a result about separation, but for the evolution problem and also some results about validity for the evolution problem also. Okay, any questions? No. Okay, so for one and two, I mean in one case we will only be solving till some value x star, in the other case we will be solving globally. What's the difference? The difference will always come from the pressure, whether the pressure is helping you or is acting against you. Right? So, more or less here this equation, you can solve it as long as you is positive. As long as you are positive, as long as you are positive, this looks like some evolution equation in x. Right? So your enemy is when you start become, tries to become negative, that's your enemy. So, so here that you are forcing you by this guy. Right? So if this guy is bigger equal than zero, maybe you expect that you can solve this globally. Whereas if this term is negative, that's where it's acting against you. So, that's what we call favorable or adverse pressure condition. Okay, so let me first recall a result from Olenik, a result where she proves local existence for the system. I think I may have mentioned this maybe around 64. So, I mean, I will try to state more or less how the result goes. Yeah, maybe just one remark, I didn't insist on this again. So, this guy is the Euler flow. Right? So, normally the Prandtl system that we are normally, I mean, we will be having Prandtl here, and here we have Euler. Right? So, the limit y goes to infinity, when y goes to infinity here, that corresponds to Euler on the boundary. Right? So, basically this guy, if you are in the Euler flow, this corresponds to the boundary value of Euler. And, of course, now it's not difficult to see that there is a compatibility between this guy and this guy, because when y goes to infinity, you expect this guy to go to zero, and you will be having only these two terms. So, normally what you, what we expect to have, what we should be having is that u Euler. Right? So, this equation, this is the x derivative of, this is the x derivative. This is, and basically this equation is the trace of the Euler equation on the boundary. So, if you take the Euler equation up and you take the trace on the boundary, that's what you get.  I mean, we are just looking at Prandtl. But, I mean, if you are trying to understand the coupling between the two, that's where it takes place. The other thing here in this model is that the Prandtl system does not affect the Euler flow. Right? We are only seeing, we are only seeing the effect of the Euler flow on the Prandtl. I mean, I was thinking about adding a fourth point, but I don't think I will have time. But let me just put it, but I will not talk about it. But there are, related to this question, I mean, there are slightly more involved model than Prandtl, that somehow try, I mean, like people saw that the Prandtl system is not always the good thing. So, they wanted to find models where what happens here can affect what happens here. Right? So, they wanted to see some feedback of the Prandtl on the Euler. So, then there are models, there are a few, a lot of physics papers dealing with a few models. The mathematics about it are still, still, I mean, there are starting, there are a few results now, that I can mention by Tali Var, David Gerard Vary, and then there is also some recent work of Samir Iyer and Vikol about trying to understand what is called triple deck models, where there are three layers here, or interactive boundary layer models. I mean, all these are models where you try to see a feedback from the Prandtl on the Euler. Ok, I will not be talking about them here. Ok, now, if I go back now to the result I want to mention about from Olenik, so you take some u0, which is, let's say, c2α bounded, some c2α function, which is bounded, u0 of 0 is 0, and u0 prime of 0 is positive, the limit of u0 is, and u0 of y, ok. So, basically, I mean, you have some profile, you have a profile of this nature, like it, your profile is more or less something like that, ok. This is y, this is u0 of y, ok, so it's positive. Ok, now, in this assumption, in this assumption, this is not necessarily monotone, so it can be, it can even do this, right, in this assumption it can do this, for instance, for here, this is also valid, ok. So, the other assumption, so this is first assumption on u0, we have another assumption that, let's say, this is c1, there is a compatibility condition, there is a compatibility condition that states that is a big O of y2, when y goes to 0, which is more or less, you can understand that, because we, oh, maybe I didn't write it down, right, I didn't write down the fact that u at y equal 0 equal v at y equal 0 equal 0, right, so we have also this boundary condition.  Okay, so then, these are the assumption of Olinik, and then the conclusion is that, if we have these, then there exists x star positive, sej that the Prandtl system has a local solution, which is, let's say, c1 in 0 x star, open, I mean, it can blow up at x star, times r plus, okay? So, so basically is that you can solve, you can solve from x equal 0 till x equal x star, and at x star something bad can happen, or some separation happens, okay?  So, let's say u is bounded, continuous, derivatives in y, derivatives in y actually are bounded and continuous till including, I mean, till x star, but derivatives in x, dxu, v, or dxu and v are locally bounded, are locally bounded, which is more or less understood. I mean, the reason is that, when you approach the separation, I mean, the separation will be the point x star where u, where u will be, can't touch 0 in a sense, but then you lose control on this kind of derivatives. So, so this is the, some of the property that we have, okay, now this is regularity, the other property is some sort of non-degeneracy, which says that u of x, y is positive for, okay, and so that's the, and then the other important remark is that if the pressure is favorable, if the pressure is favorable, meaning that if, if the pressure is, if the pressure is negative, dp, if this term is negative, then x star equal infinity, okay, if the pressure is not acting against u, then you can prove, you can do it globally. Now, what happens if we have separation? What is separation here? Now, if x star, if x star is finite, so meaning if the pressure is acting against u, so there are two possible scenarios, there are two possible scenarios. The first one is that the y derivative at zero of, the y derivative at zero at the point x star is zero. Remember, like when we started, we started like this, right, right, so we, our starting, our starting slope is strictly positive, or there exists y star positive, such that u of x star, y star equals zero, okay. Let me try to explain these two things. Usually, like here, okay, here I'm putting y like this and u like that. Sometimes when we draw these profiles, we like drawing them slightly the other way around, like I'll be drawing them like, like you see them in some of the pictures. So, so, so the profile we started with, the profile we start with is let's say profile like that. So, again, here I'm more or less changing the, this is my u, u is this, right. Now, now I change, this is u, this is u of y. So, y is like this and this is my u of y. Okay, so I just change the axis. This is how usually we draw them and I'll explain why. I mean the idea is that you think about this as being x and you are trying to see how your u of y evolves with x. Okay, so that's why we like, many times you like drawing it this way. So, this will be your initial data. Okay, so this will be your initial profile, for instance. So, then what can happen? Either this happens. So, this happens means u, y equal zero, u, y equal zero means that your profile becomes something like that, right. Because remember this is my y, this is my u, u of y. At the point x star. So, this will be the point x equal zero. Right, so my profile that was like this, due to the pressure term, the unfavorable pressure term, it is pushing. So, then you get this. So, this will correspond to one. That's the first, this is the first scenario. The second scenario, the second scenario, the second scenario is that something like this happens. It touches here. So, the second scenario can only happen if your profile is not monotony in y, right? Your profile, so this is y, this is u of y. The second scenario can only happen if you are not monotony in y. Is this clear? Of course, now, this is also, this is theorem, this goes back to Olenik. This is, this is, goes back to Olenik, yeah. Yeah, so this is a theorem, like what can, the two cases where you lose, where you cannot solve globally is either this one or this one. Of course, now, if this happens, and this is what we usually expect, there are a lot of question about how can you extend the solution afterwards. Physically, what happens here, this is the point of separation, that we call separation of the boundary layer. And then what you expect afterwards is that you get a flow, which looks like that, right? So, we have a flow that goes backward. Now, in terms of well posedness of that sort of transport equation, this is bad, I mean, like, so one has to understand this really differently. Like, you cannot just think about, just thinking about access being time will not be working very well, I mean, you have to think about that. And that's, as of now, it's not very clear how one can attack these problems mathematically. Okay, now, there's one important case, there's one important case where we can actually rule out this. If our origin of, if our starting profile u0 is monotone, monotone means it has to be increasing, right? If it is monotone, so it's like the picture I draw here. Then only one can happen, then two cannot happen. The reason that two cannot happen is that this monotonicity you keep it. You can prove that if u0 of y is monotone, then u of x, ux of y as a function of y stays monotone, right? So, if you start, if you start with a profile like this, which is monotone, you keep it monotone. So, if you keep it monotone, you can never have this. You can only get this. Proving this is not completely trivial. If you want to go through the whole proof, it uses some sophisticated version of maximum principle. It's one level, it's more than just maximum principle. You use something called comparison principle, which is like maximum principle, but it allows you to compare. OK, any questions? OK, now the first result I want to mention, as I said, is to describe the separation, right? So, an important question is to try to understand what happens here exactly, right? How does this separation happen, OK? So, that's the first result I want to describe. So, let me write down this theorem. So, this is a theorem that... So, it's with Anlorda Livar. OK, so, I mean, there are a lot of assumptions for this theorem. I'm not going to give all the assumptions, but maybe some of the important ones. But, more or less, the conclusion of the theorem is that it gives us... It tells us exactly how this guy... How this guy goes to zero as a function of x. So, if I define... So, let me start by a short version. If I define lambda of x to be Uy of x0, U of x0, OK? So, that's the derivative in y, right? Then, we know that lambda of x will go to zero when x goes to x star. That's really the definition of x star being the separation point. But we can say more. I mean, we can say that lambda of x behaves like some constant square root of x star minus x. And we can be more precise about this. But this is really the short version of the result. OK? OK, so, what are the assumptions? Yeah, so, I'll be giving the assumptions. So, this is what we call, at least what we think, to be the stable separation process. OK? And this confirms some of the formal asymptotics done more or less in the 40s, 50s, and 60s. So, at least some of the Goldstein has some formal asymptotics, Stewart, so on. They have some formal asymptotics. There is also an interesting, Landau himself also has some very interesting heuristic about why it should be like that. OK. OK, now, what is the, what are the assumptions? So, let's say we need some regularity. I think seven derivatives are OK. U0 increasing. We start with some lambda 0, which is the U0 prime, at 0, which is positive. OK. We have to take this to be already small. So, our lambda 0 will be small. So, the second type of assumptions are assumptions that are the second types. These are assumptions that will be used in maximum principal arguments. Sorry, so you are saying that lambda 0 is small compared to what? OK. All right, so it's good that your question is good. It will be small compared to 1, and the reason is that I'm going to take the pressure term to be 1, right? So, here in my result, this is equal to minus 1, OK? So, I already, I already, I already, I already, I'm already choosing a size here, which is 1. So, then lambda 0 will be small compared to that, OK? So, then U0 double prime minus 1 be infimum between y square and 1. OK. So, and then we have also an assumption on the fourth derivative. The fourth derivative is of U0 is bounded from above and below. OK. I mean, as of now, it's not clear where these things are used. I mean, all these assumptions are not completely necessary, and you can relax them if you want. Modulo, making the proof more complicated, right? But, but what is behind, behind it at the end, and the end of the day, is that what we really need, which is now 0.3, right here, now 0.3 is that our U0, we can write it as some approximate U0 plus V0. So, the U0 approximate is some sort of approximate profile that we can construct by hand, and then we try to close some energy estimate on this V0, right? So, that, that will be the idea. Now, our U0 app will be what? Lambda zero Y, OK, so that's the derivative, plus Y squared over 2. The Y squared over 2, that's, that's related to the Y squared over 2, that's related to the fact that the pressure is minus one. The pressure term is minus one, so it's related exactly to this condition, right? That condition tells you that this term, this term has to be, this term has to be Y squared over 2, plus, OK, also there's no Y3 term, also because of that. Y to the 3 term, there's no Y to the 3 term because of that. So, so, so you see, the, this guy, this guy is like the compatibility condition at the first order, like you just take the equation and that's the first compatibility condition. The next term that you are seeing here, they are coming from higher order compatibility condition that your solution should satisfy. So, I'm not going to write everything, but we need some other terms. But you mean at zero. Yeah, yeah, this is at zero. Yeah, at zero. OK, so I'm not, I'm not writing everything, but we need this till maybe the order Y to the 10, for instance. We need to know the terms. I mean, you can compute them, right? You can compute them just from, from the equation. OK. And then our V0, what we, what we can say about V0, after we remove all those terms, V0 is less than some constant, plus some, another constant, Y8. Yeah, maybe I don't want to go into some unnecessary details, but let me here write down everything, so that at least you get the sense about how things are scaling. So, right, so maybe this doesn't make a lot of sense, because I didn't write all the terms here. I mean, there are some other terms, and you have to push this till, like, Y to the 11, more or less. You need some sort of good compatibility condition at the origin. So, but then after you remove those terms, you want your rest, the error here, to be of this size. OK. I mean, basically, you need it to vanish till the order Y to the 7, right? So, the other terms, they come just from scaling. Whenever you make the scaling, that's the natural thing to put. OK. One important, I mean, the important thing in all this, even though, I mean, this is really the one of the technical parts that I will not be discussing too much, but if I put remarks whenever, if one looks carefully, is that V0 is much smaller than the approximate solution, and this happens if lambda 0 is small. Under that condition, oh, yeah, I should mention that this we need this for Y small, and more precisely, we need it in this region. I'm just writing down what is needed in the proof, but at this level, you just think about it as Taylor expansion at 0, right? So, you just do the Taylor expansion at 0, use the equation, and then you can, if your solution has enough regularity, then all these terms are completely necessary. So, the second remark, which is related to what I mentioned there, that the monotonicity of U0, the monotonicity of U0 is propagated, again, I insist, like, we are prescribing some sort of Taylor expansion till the order 7, and this is needed to derive our estimates, because we'll be taking higher derivatives, so that's why we need a Taylor expansion at a higher level. So then, the conclusion is what? The conclusion is that there exists some c0, positive, there exists epsilon0 positive, such that if our lambda0 is less than epsilon0, then there exists x star, phi star, phi star, phi nit. Actually, more precisely, x star is some v go of lambda0 square. So, it's really your x star here is not that big, such that lambda of x, lambda of x being the number that they define there, lambda of x will behave like some constant square root of x star minus x, as x goes to x star. So, it's more or less, it gives us exactly what is the rate of, what is the rate of separation. OK, now c0, c0 is the number that appears here, there. The c0 is the same in the equivalent, so it's the same, because it's the same. Yes. OK, so this is the statement. This is the statement. I'll just give you ideas about the proof without going into all details. Of course, I mean, this is really just the statement about the lambda of x. I mean, the proof has in it the fact that v is much, I mean, is normally what you are proving is that v is much smaller than the u app. OK, now how are we going to prove this result? So, we have this lambda of x, lambda of x is the, I mean, that's what we are interested in. We are trying to, we are trying to find how lambda of x goes to zero, right? I mean, more precisely, we are trying to see what is the stable, what is the stable behavior of lambda of x. I mean, we are not, we are not necessarily saying that all separation should happen with that lambda of x, and as of now, we don't know. I mean, maybe there may be other ways, but we are trying to identify the stable behavior. OK, so let's start by asking a question. What are the, what are the, what are the scalings that keep the equation invariant? OK, so what are the invariant scaling of the equation? So, the fact that I put a pressure, the fact that I put a pressure reduces a little bit the allowed scalings. Right, so if I don't put a pressure, if I say I have terms like this, if I don't put a pressure, there are more scalings in these equations. And actually, I will be discussing one later on, which is related to the so-called Blasius profile. For the Blasius profile, we scale x like y square. For the Blasius profile. But here we are putting a pressure, so this is minus one. So, here the, if you solve the equation, if you solve p, then if I look at u lambda, then I need to put here lambda square. OK, lambda is not necessarily the one there, but afterwards it will be more or less. Right, so if you solve this equation and you do this scaling, then this will also solve the equation, the same one, right? It's not difficult to see. With minus one? Yeah, to keep it minus one. I mean, if you don't put this, if you put, no, if you don't put this, then you have more scalings. You can, putting this reduces the scaling. OK, so this is the natural scaling that one would like to do. And so then, so I think, so all this is inspired, of course, from works on singularity formation for Schrodinger, like Frank and Pierre. So, inspired by those works, like what we introduce, the following change of coordinate, we look at u of s y to be lambda minus two. OK, where s ds over dx. Yeah, maybe in my notation, I should have. OK, so maybe it's confusing the lambda here and there, but that's more or less.  You are extracting the profile from. OK, now this change of so this is now the object we are interested in. This is the object we are interested in. Again, the way we are scaling, the way we are scaling things, the reason we are putting lambda, the reason we are putting four, the reason we are putting two here, I think these are the things that one should ask, why we are putting it like that, is really motivated by the fact that the equation is invariant under this scale. OK, so now the nice thing that happens is that this profile u of s y satisfies now a nice equation. Nice equation. So capital u solves the following. So it's just change of coordinate. So OK, I have to write down what is b. So b is equal to minus 2 lambda x lambda 3 which is also minus minus 2 lambda s over lambda. OK, so that's really the equation in the new coordinate system. OK, so basically all what we did is just to explain how that comes. It's not difficult. OK, so we started from the x y coordinate. We make the change of coordinate to s capital y where ds over dx is 1 over lambda 4 and capital y is small y over lambda. Right? So we just make that change of coordinate. So then just you rewrite the equation and that's what you... Oh, yeah. And also we made the u of s y exactly that guy. 2 u of... The other thing is that in this formulation I'm not using v. I mean you may ask why what's the meaning of this guy, for instance. But this is just v. But written in the new coordinate system. Right? So basically like whenever you take derivatives you have... these are like the transport term. The reason you get these extra two terms that's coming from when the derivatives is hitting the scaling that when you are differentiating the lambda. Right? Right? So when you are differentiating the lambda you are getting these two extra terms. And this is coming from the u y y in the new coordinate system it becomes this guy. Is that clear? You want me to go... You can spend like 5-10 minutes to do this but maybe you can skip that. Okay. So the other thing the other thing is that the limit when y goes to infinity of it becomes some u infinity of s. So the u infinity of s is corresponding... it corresponds to this guy. To the u Euler. As I mentioned the u Euler satisfies normally the u Euler satisfies u Euler prime u Euler prime okay. So and you can you can see what this means means that you can take your u Euler to be the square root of some x zero minus x. Maybe there is a tool. Maybe there is a tool. Okay. You can always write it down something like that because the u Euler square over 2 prime equal minus 1. So you can solve it exactly. The derivative is in x. This is minus 1 so you can it should be of that form. Now when you change this u Euler into the s variable it becomes something like that. But then this guy now will solve I'm sorry but so if you solve that so u Euler if you take this okay. So the x star is before x zero Yeah the x star will be before x zero. x zero is can be large. x zero is he has to be a theorem here okay so u zero is defined I mean okay I mean here I don't need to I don't need to tell you that x zero is coming from here I can tell you from the beginning I tell you okay this is my profile I'm choosing some u Euler of this nature and then you make all those assumption but all those assumption will imply that x zero has to be slightly large cannot be too small exactly but it doesn't have to be very large like x zero is yeah okay so this equation this equation that happens at infinity for that system translates here to something in the s variable that has this extra b okay I mean when you do the change of coordinate you get this extra term okay it's a good exercise I mean I really encourage you to do all these small calculations so that you see how the change of coordinate is happening okay so the last thing about this system that another important fact about it is the fact that I mean and that was the reason behind all this change of coordinate the change of coordinate will also imply that I write it down here because I think this is an important thing is that why at zero what should this be when we do the change of coordinate so we started lambda lambda was u lambda x zero okay and we made this change of coordinate right so now if I look at capital U lambda right capital U lambda capital U y when I take the y derivative here there is a lambda that comes out right so this will be what lambda minus 1 U U y but U y U y was lambda this is lambda so this is 1 okay so the scaling was done so that U lambda is identically 1 make sense now the next term the y square over 2 term by the way the y square over 2 term is something that stays all the time this expansion at zero in kondition at zero that derivative like the second derivative at zero has to stay always like that and of course I mean that will like this equation will satisfy it also because of the scaling I mean that the scaling really keeps the the y square term you can see it like if you take the second derivative here the this so now now you can just look at this you can forget about the original system you can forget completely about the original system I mean the the condition we have on the on the initial data you can translate it into the new on capital U like at the end we'll be doing estimate in this in this coordinate system so you can completely forget about the original system the only thing that you have to keep in mind is also lambda also lambda you can forget about it because lambda doesn't doesn't really appear here so the fact that B is this guy this I will only need this I will only need to go back but as of now I can forget about this guy so then all I'm having I need to solve this equation I need to solve this equation but globally in S maybe I didn't insist on this but when when X goes to X star we expect S to go to infinity and then the question is ok can we can we solve this and what is the natural behavior of B what I mean how should we choose B of course now when you get to this point I don't know whether like saying that B is like 1 over S is ok we will be proving that B is 1 over S is the stable behavior but I don't know whether there is a clear precise intuition why why it has to be 1 over S it's not clear right of course it's not always the case yeah but I mean at least for us it was motivated by and also the use of the letter B and you can explain why so we will prove that B of S behaving like 1 over S is some sort of stable behavior right so again I want to insist on how now we are going to try to solve the original problem first I forget about the original problem I only look at this equation I can forget about this one I look at this equation with this prescribed behavior with this behavior at infinity with this compatibility condition at infinity and with the original profile of course now the my original solution U0 I can translate it and now my goal is to find the goal is to find so the goal is to find some and B I want to solve globally I want to find U of Sy and some B that solve that equation so honestly I don't know if you get to this point and you don't have a precise intuition about what should be B if you don't say okay I want B to be one of S I don't know how to proceed I find that at some point there is something you can get you need some intuition from somewhere to say okay let me let me assume that B to be like that you do some calculations and then you close it at the end of course there will be a place where no, maybe I am slightly wrong so while constructing the approximate solution while constructing the approximate solution there is a place where you want BS plus B2 to be more or less 0 but of course at the beginning to start your expansion you should have in mind that B should be small otherwise you will not even start so okay now let me just give you without going into the whole detail the steps of the proof okay so I will just give the steps of the proof so step one is to construct an approximate solution right so we want to construct an approximate solution in what it turns out that there are two ways of thinking about the problem and more or less they give you both work the first way is to say okay B is small let me expand in B you can try to expand in B you can look at you can say okay my starting point is I will call it I think I am calling it u0 or u1 let me be consistent so u1 u1 will be y plus y2 over 2 okay I mean you can check it is not difficult to check if that if you put that into the equation if you put that into the equation it solves the equation you put B equal 0 you put B equal 0 you plug that in so it is something that is independent of S so the first two terms they cancel I am taking B equal 0 and then this solves the equation with B equal 0 solves P with B equal 0 and in the language of the Schrodinger equation this will correspond to a soliton for the ground state for the problem of course it doesn't solve it doesn't satisfy the good boundary conditions because it's unbounded I mean whereas we expect this guy when y goes to infinity to go to something finite of course I mean this guy can grow with S because of the rescaling okay so you have u1 and then and then you can keep solving you can keep expanding in B so I will write it down as plus Bt1 plus B2t2 so on I am expanding in B for instance I mean if the Bt1 term the Bt1 term should be and if you think a little bit about it you need dyy of t1 dyy of t1 this has to cancel the term coming from exactly this one I just show you the first guy I will not go into the okay so the Bt1 term the Bt1 term the Bt1 term here the Bt1 term here will cancel the term coming from BU1 these two terms as of now I don't need to know as of now I don't need to know what is the law of B how B behaves as a function of S so this you can calculate and then you can calculate t1 which is some minus a4 so this gives us t1 equal minus a4 y4 a4 is 1 over 48 you just compute you can see all these guys here leading order you have a term like this will be like some constant times y2 so then what happens is that and this is where where there is the choice of B will come let me just explain here and then maybe we will so now if I look at if I call u2 equal u1 plus Bt1 and then I plug u2 in the equation I plug u2 in the equation so as I will write down equation applied to u2 so equation I mean by that equation of u means this guy plus 1 so normally what I am trying to do is to write equation of u equal 0 so if I write down if I write down the equation applied to u2 what is the error that I find I will find so there are some calculations that one has to do so basically u2 is an approximate solution it's not a solution because when you put equation of u2 you don't find 0 but you find something ok you find something and you find a term which is growing like y to the 5 term y to the 6 term like y to the 8 ok now comes the place where I want to tell you what is Bs how I should choose B right and again I mean this is also a little bit inspired by works of Frank and Pierre so you look here and you see the term which has y to the 8 has B times B2 so this is B3 so this coefficient is smaller than this one or this one right this coefficient is smaller than this one ok so you can convince yourself that you can forget this one as this level so you have this one and this one now this guy grows like y to the 5, this guy grows like y to the 6 and it's better to make this 0 because then this growth will be weaker so that's how you choose your B you choose it so that this is like 0 right so you say ok I want Bs plus B2 to be 0 at least whenever when you are constructing your approximate solution like afterwards it will be it will not be exactly 0 but it will be lower order terms but in the construction of the approximate solution in the construction of the approximate solution we can just replace B by so then I'm going to choose Bs plus B2 to be 0 which means like B equal 1 over s right so whenever I'm going to construct my approximate solution each time I see Bs each time I see B I can replace it by 1 over s ok so then you can keep constructing your approximate solution to any order right and so then I can say that I'm going to construct t so u3 will be my u2 plus B t2 and then I can write down the equation on Bt2 so basically the main term that you have to cancel is this guy and that's why your t2 your t2 is some sort of minus a7 sorry so there is B2 here ok and a7 is also explicit a7 is 1 over a4 ok ok so so then what happens let me just show this I was not really planning to go into this detail but I just want to show you at the end how the law closes but when you do it this way you are looking at your solution as what I'm calling the approximate solution plus some error and the whole thing is to close some estimate on this guy now since we expanded till this level so since we removed this guy this guy it will start with terms like y to the 7 which is like consistent with this kind of with this guy so so basically our v3 sy we can see what are the first terms in the expansion you can compute them of course now when I write down the precise thing I put back vS and actually the control of v3 will allow me to control this guy this is how the control of this guy will be coming it will come from controlling v3 over y to the 7 plus other terms let me not write them down but the term like y to the 8 and so on but this is more or less at the end of the day this is how the how the law on vS plus v2 will be derived it will be derived from getting some good estimate on v3 ok ok, maybe I should almost finish with this let I have a question about the formal approach if he decided to cancel the other guy in YouTube this one? I don't know, I see that you have odd power of power of y and do you think that this will make up here something like even power of y? no, no, no actually they are here there are also some power 8 so it's not really so the next term has a power 8 and this coefficient is explicit so it's not question about even or odd but for me it's more like a question about stability if you try to if you try to cancel this guy I think my thinking that if you cancel this guy you get a better behavior at zero but it's bad at infinity so in particula if you cancel this guy then this term here instead of being 7 it will be 8 so you are losing at 2 places actually you are losing in the fact that you have a worse behavior at infinity first but the other thing which is bad is that you need to control v3 over y to the 8 so that you control this I mean you have a different coefficient of course I mean you get that coefficient so it will I mean I don't think it works and maybe that corresponds to some unstable behaviors but I have no idea in the case which I know it doesn't work in some case for example there is no solution of such it's a prototype where the approximate solution has it's more the fact that you have an approximate solution doesn't mean automatically that you have a solution which looks like there is no so what are the other steps of the proof what are the other steps of the proof so the real approximate solution here we talked about the approximate solution but for y small actually why not very small like why here can be what we discussed is why like s to some power alpha it can go to ok at some point we have to choose alpha I mean it's there is room in choosing alpha more or less we are choosing at some point we are choosing alpha equal to over 7 I mean it's related also to the powers we had there but there is freedom so what we want to do is to take a cutoff right and to plug in u3 actually it turns out even though at this level normally u3 is enough because when we go to u3 we have v3 and v3 controls everything but I mean it's usual thing we push it even to u4 ok so this is the approximate solution for y smaller than as to the alpha but then what happens afterwards afterwards I have to I want to reach this guy I want to reach this guy now my b is 1 over s my b is 1 over s so b is 1 over s so basically this guy u infinity should be like s plus 1 right so u infinity is like s plus 1 ok ok you can check that will more or less do the drop so it's like s let's say it's like s like the difference between not important because all the norms we are going to use they they are weighted norms so what happens far away is not that important ok I mean you cannot really make it y square but it's like constant that infinity is fine because of the weights we are going to use so then what we do more or less here actually what I'm going to do is subtracting y square over 2 ok for some small reason it's just a choice but plus 1 over b some function theta because the y square will be coming from here so the theta behaves like y square over 2 for y small and it behaves like 1 for y large ok so basically you just take that profile that we constructed we only keep it when y is less than s to the alpha then we want our solution to behave like there is a region where it will behave more or less like y square over 2 and then it will behave like 1 over b which is s so in a sense there are more or less like 3 regions so there is this region s alpha there is a region which is more or less s to the 1 half and then there is a region at infinity so here it's really the approximate solution here it's more or less y2 over 2 and here it's like 1 over s sorry s so that's how your approximate solution looks like ok so this is the approximate solution then then the next part of the proof is to perform some good energy estimate on the rest which is v right so then we write our solution as u u up plus v and then we need some estimate on v now there are 2 types of estimate so we will do some maximum principle type estimate and the second type are energy energy type with weight ok so and it's very important that I need whatever norm I'm going to use on v so I need a norm on v but I need this norm to control the trace of v divided by y7 so I need this to control v so v is what I call v v3 so I need this to control I need a norm that controls that and that one thing and also I need to make sure that this whenever I'm estimating this I want this to be better than 1 over s2 plus some eta so those are the two things the requirements is that I need a norm to be strong enough so that it controls this guy then it controls bs plus b2 and I need it to be smaller than this because if I am smaller than this then I can deduce that my b is like 1 over s plus something smaller ok so that's at the end the whole game so so any questions about this ok so can you be more precise about n ok, so n n can be for instance an example of n is that if you can take something like this if you take something like that that will at least it will do this of course now in our choice we take something more complicated with weights and so on right but that's really the point that whenever you take trace whenever you apply trace theorems you can control it at the end we are using something adapted to the linearized operator right ok maybe yeah, I can say a few words about that nice I don't think it's technical I think there is a good idea there so maybe I can explain it so you see like the way the operator here is given the way the operator is given it's not really an evolution in s you have us here but you have also us here so this is not very nice if you want to do energy estimates and you are slightly you don't know what to use so it turns out there is an idea you have to put these two together and ok I mean these difficulties also slightly related to what we see whenever we do Rayleigh operators when one looks at linearization of Euler we get sometimes things like that or also linearization of Prentor so so an idea that we introduced is to introduce operators of this nature so this L so if I do this if I introduce this then actually my equation my equation will look like this my equation can look like LU US minus BU2 so just introduce this this notation so somehow you combine these two terms together then what you do is that you apply LU minus 1 to everything and you can check that LU minus 1 makes sense and you can apply it and then it turns out that LU minus 1 applied to this you can compute it, it's simple LU minus 1 applied to this is simple you can compute and then what you end up with at the end and this is one of the version of the equation that we use so let me write it down so this is a I mean there are some interesting cancellations that happened that minus LU so this is your new equation I mean here we use the fact that LU minus 1 applied to U2 give you this LU minus 1 applied to this give you this I mean there are identity that you so then the good thing is that it only appears here now so this is again this sort of things I think Frank is more seems looks like type of things that also you you see like this sort of transport and the fact that we put the fact that you differentiate more you get more dumping also right so now you differentiate more you get more dumping so then you differentiate till you get this kind of dumping that would be the idea of the energy estimates now we will not be using this kind of things because this kind of things do not compute very well but it's more it's more like applying this kind of operator many times in this kind of operator related to this guy and you apply it so that's any questions about this so I think we spent most of the time talking about one result so let me mention I want to finish by talking about few other papers by other people but that I found interesting in that are recent papers about results related to printer and it turns out like the interesting things that some of the difficulties that we saw here like for instance this kind of this kind of object they are also hidden in all those works like at some point you have to deal with the fact that your equation if you want to think about it as an evolution you need to do something to this U.S. because the U.S. appears in two terms so ok, so I'm going to mention two one result that I found interesting so let me mention this result by some year I heard about long time behavior of it's about long time behavior of printer but when the pressure is 0 ok, so of course now the separation that we have here was really driven by the pressure it's really the pressure that is pushing you to become negative in a sense so if you don't put a pressure if you look at printer with zero pressure so if you look at this equation ok same initial data everything is same so in this case the result of Olenik the result of Olenik gives you global existence ok, so Olenik gives you global existence so so then so Prentol had a student Prentol in 1904 wrote this equation, had a student his name was Blasius and maybe few years after Prentol, maybe in 1908 or he looked at self-similar solution of this so as I said when you put zero somehow you have much more scalings right and one scaling which is maybe more natural is like some sort of parabolic scaling where you scale y like square root of x you can scale y like square root of x and if you do that so if you introduce eta equal y over square root of x and you can add plus x zero if you want and that's how it appears afterwards but if you look at this guy and you try to find solution of this equation then you have profiles which are these Blasius profiles and which are like in application turns out to be really very important like all physical, I mean many of the profiles that we see as solution for Prentol I mean when you don't have a pressure are coming from this because that's really the long time the large x behavior of this Prentol with zero pressure so then you can look at solutions it's not difficult to see that you can look at a solution that behaves like this so you can find the solution that that behaves like that I mean because this will satisfy the incompressibility this satisfies the incompressibility and then you plug it in and you derive an equation on F so there is an equation on F you get that F should satisfy this so it becomes just an ODE and then you fix I mean you need this to be zero because that's the initial condition and then you also fix F prime at infinity so you can fix F prime at infinity and the fact that also F of eta over eta goes to 1 at infinity so you fix all this you fix all this then you can solve that ODE you get an F you scale it like this and that gives you a solution so these are solutions of of course here we are fixing some behavior at infinity right so again, since you don't have a pressure since you don't have a pressure it means that it means that u Euler has to be a constant because remember like the u Euler and the pressure are are related so this is the case where u at infinity goes to 1 so this is the case that u of y goes to 1 when y goes to infinity ok so Blazius constructed this guy this solution, 1908 now what this guy Samir did like recently is to prove that solutions of this equation they relax to they converge to the Blazius profile so this is a long time behavior type of result so I should mention that his result was already proved actually but what I like about his result is more like the method used and which has some potential for other things so so as I said so Olenik gives global existence Blazius this is 1908 constructed this sort of particular solution now there is a result from 66 by Seren who proved using maximum principle using maximum principle techniques but not only maximum principle but also you need to go to the so called von Meiss coordinate we mention this last time von Meiss change of coordinate so you make this change of coordinate and then use maximum principle technique to prove that the u if you take a u here you start with an initial data which is not the Blazius profile you can prove that u will relax to some Blazius profile so the so you get the fact that you relaxes to Blazius so there is some sort of dumping some sort of dumping that pushes you to look like Blazius as x goes to infinity so it's more like a relaxation result so I mean I found this interesting because even that proof it's not completely okay so what what this guy did is to re-study that but more using energy estimates finding the right energies and so on which is I think very important in this business to be able to do things by energy estimates now this is one type of result that are recent that I found interesting so there is another two other results that I want to mention but maybe don't have too much time but the two other results are one by by go and same guy and the other one is by David Gerard Veret and and Yesunuri Mayakawa more or less around the same time more or less about the same time which is about the validity of the parental equation validity validity of the stationary parental both are from so somehow the parental system we derived it based on some asymptotics nothing I mean the result of conversions to parental are very few and like in the evolution case some of these results use analyticity and ok, Wednesday I will mention another result that I have with Gerard Veret and Mayakawa where we have to use we need jevre regularity but it turns out in the stationary case it can be done in sobolev so that's the interesting thing in sobolev so they can prove the validity of the parental in sobolev regularity so it turns out like the setup they have are different for Gerard for David and Yesunuri they look at the problem in the periodic setting in X so you look at something periodic in X you take some solution of Navier stocks with periodic data so there is no boundary condition to be imposed here and here because it's periodic and they put also some assumption about the profile for parental which is independent of X so they are taking some solution which is independent of X of parental and they can prove that solution of Navier stocks will converge to the solution of parental I mean the important thing for the days that you have a way of deriving parental rigorously from Navier stocks so that's the setup here the setup here actually is more interesting like more general because they are working in in here like between they have a flow that goes in so they have to impose some boundary condition but the type of boundary condition they have to impose are quite complicated like you cannot I mean it's similar to the type of approximate thing that you impose so they have an approximate solution that they compute to very high order and they need to be compatible with the boundary condition ok so they impose that they have to choose L small same here they need there is a condition about smallness in L and same thing they can prove that solution of Navier stocks will converge to the solution of parental I mean both proofs are really long this one has like 100 pages both proofs are really not very trivial one of the interesting fact about this even though this kind of things seem different from what I was talking about here but let's say an object like this guy also has to be used in both proofs like you need to understand how you can extract UX from that transport to structure ok let me stop here


The following is a presentation of TFNN. The morning market kickoff with your host, Tommy O'Brien. Good Monday morning everybody. I'm Tommy O'Brien, coming to you live from TFNN. A little bit stuffed up this morning, feeling good though. So we'll get through it, but I am a little stuffed up, you can hear me. Maybe I got a little something from the kids in the house, whatever it is, I'm feeling good. Just a little stuffed up. Maybe it's a little allergies. We got some cool weather in the air here in Florida, especially by our standards. But boy, these markets, what's going on over there in Israel over the weekend. Sure, we're all aware, but at this point can't overstate the seriousness and the headlines just keep coming up this morning. I was watching an interview this morning on Bloomberg, they were talking about the den as well. Boy, former ambassador to the US from Israel, member of their security council, and boy some strong words, talking about crushing the will of their enemies for generations to come. If you think about what that really means, things escalating dramatically. You have nine Americans killed over there. One of the headlines I was reading this morning, talking about a spokesman saying the US is working with Israel to locate others. The death toll, I think somewhere near 1100 at this point. Trying to just pick up some headlines as even they come out this morning. But boy, it is a dire situation. And we know one of the things that the gentleman that they were interviewing on Bloomberg this morning, who I believe I'm saying it correctly is, was one of Israel's former ambassadors to the US. Now a member of their security council, an aide to Netanyahu, a close ally to Netanyahu. Nonetheless, the verminage is pretty strong and you should probably take it for what it's worth. And I don't know how that abates anytime soon. And yeah, you go from there, man. Surprise markets actually knocked down a little bit more with the potential for how this ties into Iran and things could escalate from there. But nonetheless, you get the markets down about 25 points right now. You were down at about 4,300 and the S&Ps, we clawed that back a bit. Oil, all the focus potentially rightfully so. We get a lift in oil from 83 to about 86 this morning for some context here. Still well off the highs in 95 bucks, right? If we're talking about a Fibonacci retracement in that oil contract, you hit the 3A2 pretty much on the head last night. There, excuse me, 86, 68. If you're talking about the 618, you're talking about about $90 in the price accrued. We'll see where we go from there. The day is young, as our man Basil Chapman says. Gold contract catches a bid right now up about $15 to 1860. We jump over to notes and bonds. We catch a lift as well. You got the 10-year, up 23 ticks, the 30-year, up 24 ticks right now. We jump over, you're talking about a 10-year yield currently sitting. Excuse me. Come on, where are we? There we go. 4.8%, 4.79, almost 4.795, to be exact. So we'll call it 4.8% the yield on that 10-year. We're up to 107.16. We almost got a 105 handle last Wednesday. So you're talking about a full point and a half above where we're at right now, right then. But again, let's add some context. You can barely see the uptick when you consider the move that this thing's been in. And that's only going back five months. 117 was almost, oh no, exactly. That's easy to remember, okay. May the high was 117. We're sitting at 107 and change. Right now in the 10-year, the 30-year right now. 111.05, same conversation back in May. Look at that, nice round numbers, huh? 117 is the high on the 10-year and 133 is the high on the 30-year. We're at 111, 22 points we lost. From 133, 22 points you lost from May to October. Talk about being exposed to duration as you've had rates rip in the opposite direction than many had expected. All right, we jump over to the dollar index. We talk a little bit of currencies as we get some pretty substantial geopolitical news over the weekend, just from a humanity perspective, man. You're talking about nine Americans.  He was talking about, if you make the comparison, Israel only has 9.4 million people, about 9.5, maybe it's up to 10 million people, okay? I have a number of 9.36 million people as of 2021, coming into almost 2024. So let's call it 10 million people. And you got a thousand deaths, right? The US, let's get it exactly. About 335 million, 340, something like that. The number I got with them is 332 for the 2021 year. So you're talking about 33, 34, maybe 40 times the size of the country. It would be akin on a population basis when you lose a thousand people and you only have 10 million people, it's a much smaller population. And so we lost a lot of people on 9.11. We have a lot more people. I mean, it's just a huge event going on over there, man. You can't overstate it. I know we all know it. And this is not gonna get into a discussion of war tactics or what's appropriate because that's not what this program is. Oh boy, that's a very opinionated conversation that everybody has their opinions on. And everybody gets very heated on this one. But I think we all feel like if that was America and that was happening, trying to consider how we would react if we had a ally on our border doing stuff like that with a thousand people ending up dead. You know, the conversations, they're 9.11. And boy, that persisted for a while. So I was listening to Bloomberg earlier this morning and I said, day three of a war that's continuing or something like that. I said, day three, this is gonna go on forever, man. This is not gonna stop anytime soon. So day three, and you can keep counting those days because this is not stopping anytime soon, man. It's a sad deal for the families over there. No matter what, and it is still persisting and the visuals are shocking to put it lightly, man. Waking up. I mean, we're waking up drinking coffee on Saturday morning, right? Watching the news in all hell's breaking loose in other parts of the world. So be thankful for what you have, folks, every single day. My dad says it, right? Tomorrow's not here. Yesterday's gone, tomorrow's not here. What are you doing right now? It's a perfect example. I mean, there's so many videos out there. Israel, I saw one of like a dance rave, dance concert, just living good life, man. And all of a sudden, just like that, the world changes can happen to any of us. So wish for the best over there, as always. But it's a dire situation. And the verbiage coming out of the gentleman this morning on Bloomberg, and probably rightfully so, you know? That's not how you gotta live, man. Where just all of a sudden there's a surprise attack and a thousand people are dead and people are kidnapped and the number's probably gonna go up from there. So I'm surprised the market's only down 23 points, man. Because this thing ties into Iran, it ties into Saudi Arabia, the normalization that's going on there. So it is more than just a Hamas-Israel attack. And that's really the potential here, when you look at things. Crude oil, Neil Dutta, he was out on from Renaissance, out on Bloomberg as well, somewhat before the program. And he's talking about crude, he's making a great argument though, in terms of the gas pump. Man, wholesale gas prices, you're gonna actually see a decline. And it has to do, I'm sure, with the 95 to the 81 drop that we saw, gas has gotta catch up a bit, but you're gonna see gas prices drop maybe 50 cents. So quite a dichotomy going on, right? We got the markets down, we got crude spiking, we have a war, literally breaking out that probably is gonna persist and get much worse in the near term and maybe potentially the longer term, hopefully not, but in the near term, man. You know, I'll have to get that guy's name that was out on Bloomberg. Maybe somebody has, and I saw you talking about it in there at some point, because boy, his verbiage out there, talking about destroying the will of our enemies for generations to come. That's for generations to come. Can't blame him, but boy. Stay tuned folks, we got a lot to market to talk about. October 9th, we'll be back. Three minutes, stay tuned. If you're looking for potential trading setups in the stock market, then Rocket Equities and Options Report is a newsletter you should try. Tommy O'Brien delivers options and equity trades when the markets present them using a combination of fundamentals and technicals. Sign up for Rocket Equities and Options Report today with a 30-day money-back guarantee so you have nothing to risk. For all the details and to start your subscription today, visit the front page of TFNN.com. TFNN, educating investors. Everything in the universe is governed by the Fibonacci sequence. This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market. To stay on top of stock patterns you can take advantage of, sign up for the Fibonacci 24-7 newsletter at TFNN.com. When you subscribe, you'll get a weekly report from Veteran Day Trader Larry Pezzavento on stocks you need to pay attention to and you can trust Larry's analysis. After all, he's got 45 years' experience as a day trader. Larry will also provide daily charts, videos and data on the key markets that he's tracking. Expect notifications from Larry on market movement you need to act on at any time. 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At TFNN, all our newsletters come with a 30-day money-back guarantee, so you have absolutely nothing to worry about. Visit TFNN.com and try Mastering Probability 30 days risk-free today. TFNN, educating investors. TFNN has launched the Tiger's Den. Hosted at Discord, TFNN has been educating traders for more than 20 years with live programming hosted by a variety of professional traders during market hours, the Tiger's Den. Available to all Tigers and Tigresses for just $1 for the year. There's no catch or added costs when you join our community of traders. Sign up today and become a part of this educational community of traders. Just visit the front page of TFNN.com. Folks, we got the S&Ps off about 22 points right now. You got the Nasdaq 100 off 99, trading just about 15,000. Dow off 161. The Russell right now, off about 16 points. Yeah, jumping back to some of the headlines. So the journal out there, this morning, quite the visuals, right? Just nonstop, man. And this is, you know, you see some of those buildings in the Gaza Strip. You're talking about high-rise buildings, man. Civilians everywhere. And, you know, that's where nobody's gonna have the perfect answer on this, man. Not to get too philosophical for a whole hour. But what do you do? There is no perfect answer here. So we'll see where we go. But boy, the visuals pretty intense to put it lately. Yeah, and you just got more. It's just not gonna stop me in the visuals. So we could spend the whole hour looking at visuals out there, but we're gonna jump to some market action. And let's kick it off with Tesla, man. Tesla this morning. Tesla, down with the market from 260 to 256. Tesla prices now rival the average US cars after billions in cuts. The latest round of price reductions brings the EVs to a new level of price parity, and they're also driving a wedge between investors. That's the interesting part of this, right? The Model 3 is at $38,990. I think these are the cuts that happened Thursday or Friday. That's $8,700 less than the average amount paid for a car or truck in the US. It's almost nine grand less than the average vehicle price. Think about that, Tesla, an electric Model 3. The starting price for a Model Y is $3,700, below the average auto price of roughly $48,000. Tesla's prices now compete with the average new car, the Model Y and the Model 3 price differential. Look where they were. That's the interesting part, okay? In July of 2018, you were paying almost 14 grand above the US average price of a car. Now, what has happened over that time, though? I wish they had this somehow. What's the average price of the car done since then? It's going up dramatically, okay? So it's not all been Tesla cuts. You've had the average price of the US vehicle go up at the same time that you've had the Model 3, the Model Y and the Teslas, especially over the last year, just get cut in dramatic fashion. This is the starting prices for every model and they reach record lows against the US benchmark, the Model S and the Model X. And as you can see, I mean, they were lofty prices September of 2022. So you're talking about with 13 months into this. Some of Tesla's price cuts is the cost of propping up demand after he bought Twitter. Yeah, I mean, that's gonna matter. It's for a lot of people that are, it's just a fact of life, man. Elon doesn't care and kudos to him for speaking his mind. If that's the deal, there's a lot that I don't agree with what he does. There's a lot I agree with what he does. Everybody should just speak their mind and not be biased by profits, but everything has, you're always speaking your own book to a certain degree, especially the richest person in the world, okay? So he wasn't too worried and that's what always would shock me that somehow Tesla investors, a public company were cool because remember folks, where was the acquisition? Was it here? I think it was. Let's see, I mean, it was pretty lofty prices that he put it out there, that he was coming after Twitter and the market punished him thoroughly and not many other people would have been able to get away with that and still retain their position because of the negative consequences of that. If you're looking at Tesla, man, you're getting quite a bargain compared to where you were at. If you're talking about Tesla profits though, they're in a different game changing scenario in terms of they're going for the masses, that's gonna result in sacrificing profits. You look at this thing longer term, okay? We're in a different stratosphere from where this thing was trading at when we came into the pandemic. Different stratosphere folks. So don't think that there's not a room for some multiple generation changes if they decide to forego profits for an extended period of time to ramp up supply. They are well behind the type of vehicles you're talking about, four GM, et cetera. Look at this market. Look at this market.  Is the market gonna say that on Monday after war breaks out that stocks and equities are more attractive than they were on Friday? Maybe they are. We're about to find out, man. S&P's off 18 right now. And yeah, so some of the, it's the State Department saying nine Americans killed over there. You got USS Carrier being sent over there, right? So these headlines are not going away and can't overstate it.  And I hope this does accelerate the need to have a speaker of the house because everything gets so political when we're arguing over nonsense in many occasions, okay? And now we have real war breaking outs and the U.S. carriers overseas and we don't have a speaker of our house. So we need a speaker of the house, okay? In case things need to get done, they need to get done in a timely basis. Hopefully this provides an incentive to some of the politicians out there. They see a speaker vote on track as usual adds to urgency. I would put it lightly, man. Gonna be interesting to see how the Republican caucus with the majority plays out here and who's it gonna be? But we find out, I think they're gonna have a vote on Wednesday, right? They're gonna get together for a meeting on Tuesday. Interesting to see what they have going on in that meeting. You have Jim Jordan. You have Scalise. Yeah, no real news, man. But pressure is on and they should get it done because we need a government, especially right now, right? All right, we talked about Tesla. We talked about the markets. Let's jump around to some of the equities this morning. Let's take a look at some of the airlines. See how we're trading. Yeah, drop off there, right? From 128 to 1244. You have all the big airlines, Ceasing Operations and Tel Aviv, American back and off with the market. This isn't gonna crush them, but boy, you get a breakout of war that extends in areas. And yeah, I'm sure that could hurt travel, especially over Europe or something. Delta Airlines, we're gonna be down about a dollar. So they're off well more than the market, right? What's that? Almost a 3% drop for Delta shares on the open. You jump over to United. They're down a dollar as well, 2.5% drop right there. What are you doing, United? Yeah, an American off 30 cents, 2.5%. So you get the airlines off 2.5 to 3% right now. Crude spiking, that's not gonna help them in the same key. We jump over to the big dogs. Amazon shares are lower by about a dollar from 128 to 127, the big dog, Apple. Look at these markets clawing it back, man. Apple down only a dollar right now. We jump over to Microsoft shares. Off a bit to 3.25, 25. We jump over to NVIDIA. NVIDIA trading at 4.48. This thing's been so strong, man. Just pushing all-time highs, right?  We're up to 3.77. They had another remarkable quarter back in August.  I mean, just holding on to it. No, no, no pullback of that acceleration at all. All right, let's keep our eye on the dollar index and yield to get the dollar index right now. Shorter term basis, back to a 10-minute chart. Pulling back a bit at 106.33, we keep our eye on yields. Pulling back a bit off the highs we were up at almost 107.23, excuse me. We're backing off a bit at 107.15 right now. And what is that correlating to? As we got yields rising a bit, probably just over, yeah, we're sitting 4.8% the yield on the 10-year. Let's jump over to the VIX as we come into the opening bell. We're just right back to where we were on Monday morning. Imagine that, I mean Friday morning. Right back to where we were Friday morning. Remarkable. Stay tuned, folks. We're coming back for the open. Currencies, commodities, and bond markets are as important as ever right now with how they're driving the volatility in equity markets across the globe, which is why it's a great time to try out Teddy Kegstad's Tiger Forex report. 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Currencies, commodities, and bond markets are as important as ever right now with how they're driving the volatility in equity markets across the globe, which is why it's a great time to try out Teddy Kegstad's Tiger Forex report. Teddy Kegstad breaks down the forex markets every Monday using his 30-plus years of experience as a trading veteran of futures, forex, stocks, and options. Teddy releases his weekly Tiger Forex report every Monday morning with coverage of all the major currency pairs, including the Dollar Index, the Euro Dollar, Pound Dollar, Dollar Swiss, Dollar Yen, as well as many more, and he also has weekly coverage of the crude oil market and the 30-year T-bonds as they both influence forex markets tremendously. When you sign up for the Tiger Forex report, you also gain instant access to Teddy's 60-minute webinar archive. He just hosted forex strategies and fundamentals, what is behind the Tiger Forex report. 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Watch online at TFNN.com or on TFNN's YouTube channel and become the investor you were born to be. TFNN, educating investors. Don't forget, you can listen to TFNN live on your mobile device 24 hours per day. Go to TFNN.com and hit Watch Tiger TV. That's TFNN.com and hit Watch Tiger TV. Welcome back, folks. We got markets open. You got the S&P off about 18 points right now. Drifting a little bit higher as we come into that opening bell. We're at 43.10. Basically, when I started the program, we come into the opening bell at 43.23. That's basically pre-market session highs. You open last down in the futures. At just under 43.20, Weber came into the opening bell at session highs and we catch a little bit of a lift right now. Remarkable to put it lightly. All right, let's talk a little bit of Fed. Let's talk about a little bit of Fed policy. You got Federal Reserve Bank of Dallas president, Lori Logan, said the recent surge in long-term yields may mean less need for the US central bank to raise its benchmark interest rate. Again, here's the quote, higher-term premiums results in higher-term interest rates for the same setting of the Fed funds rate, all else equal. That was in today in remarks prepared for a speech at the National Association of Business Economics meeting in Dallas. Thus, if term premiums rise, they could do some of the work of the cooling, the economy for us, leaving less need for additional monetary policy tightening. Now, my analysis and opinion would be that they're probably at a pretty restrictive rate and they're gonna try and keep them there if they can. If the data allows them to stay where they are, knowing it's restrictive, because they're at almost 5.5% and the R star is probably not 5.5% right now with the numbers that are coming out. Big jobs number on Friday, right? But what did the market do? The market hung on to and you can't overstate, man. You can't overstate. When I was doing the program, folks, 42.42 was where the 8.50 AM bar on Friday was at. The S&P traded up to almost 43.60. You're talking about almost 3% from where the first move was to where it was. I didn't see that one coming, man, on a hot jobs number like that. But guess what? This market I think is making the calculation that they're gonna try and stay where they are and the market can handle where they are right now and if they stay where they are and we march forward with those types of job gains and wages continue to abate, that's the kicker, wages continue to abate, which is what the market I think clawed into on Friday, then the equities will be okay where we are right now. If the Fed stays where they are and the economy's adding 350,000 jobs, 360, almost 400, 450 when you add in the net additions revisions I should say to the prior two months, you're talking about over 450,000 jobs added in the month and we had wages under what the market was looking for. So that's what the market clawed into. I think the market is taking that to mean that if we continue to see the wages declined, we continue to see inflation numbers declined. I think we have CPI out this Thursday, Wednesday or Thursday this week that the Fed won't have to hike again. And just the fact that maybe they're done hiking, even if we stay where we are for an extended period of time, the market may be okay with that, especially where equities are trading. Now, that's a little bit of a Goldilocks scenario, right? Because who's to say that if they keep rates where we are right now, then inflation will be under control. And that's the kicker and all that. So the data is gonna drive it. And I think on a longer-term basis, we're gonna be stuck at these rates for longer than people expected because the Fed is willing to wait it out at this interest rate policy that they're at. And Lori Logan, she's a voting member of the Fed this year. So, these are calculated remarks folks. This was not the off the cuff answer of a question. These were prepared remarks, okay? And the other thing I was talking about last week is they're talking about the markets doing the work for them a bit here, right? But the other thing to consider is that every time we have inflation waning a bit, rates where they are right now actually become more restrictive on the economy because as inflation cools and growth cools, which is what they're doing by hiking interest rates, right? They're cooling off the economy by making the cost of capital more expensive. Therefore, driving down growth and driving down inflation. So every time you have those numbers come down, the cost of capital is still at 5.5% the Fed rate. So what happens? Well, what happens is as that gap gets wider as inflation and growth comes down and the Fed stays where they are, it's actually more restrictive because the gap is larger from the natural growth rate to the interest rate that the Fed is charging for the overnight lending rate, okay? So as you've had that decline and it's kind of interesting that we first started with interest rates at zero and inflation at 14%, 9%, pick a number, right?  When you hit equal, that's where we keep it all the same. They had to get it where the interest rate was above the natural growth rate, but now as it continues, even keeping interest rates where they are becomes more restrictive.  Shouldn't use the word no, right? They believe that they're in a restrictive policy right now, which they probably are in my opinion. So I imagine they're gonna try and stay right where they are right now. We'll see if war complicates things as it possibly could. We'll see where crude goes as that could complicate things a bit as well, but no real huge reaction, no real huge reaction when you consider the potential here. Crude, just back to where we were last Wednesday at 85, 88 right now. Yields, right? Not spiking above where we're at. You jump over to the dollar index. No, we're chopping around at 106 and changed. Basically been where we at right now. The VIX, what did I say? The VIX, just back to where we were at 11 o'clock in the morning on Friday, just back to where we were in the middle of the day of Thursday. So no spikes yet in this market as the S&P claws back a majority of those losses overnight. We're at 43, 24 right now. And to put this another way, we're 80 points off of the lows of Friday right now. But think about that, right? That's context for you, okay? We're 80 points off of the lows of Friday right now. We are 32 points above where we came into the jobs number on Friday. When we found out that we added basically 450,000 jobs. When you add, what was the headline number? 360 something, net revisions were over 100,000. So you're talking about 460,000 jobs overall, net added. 360 something of those added in the prior month. Market's like, we got no problem with that, man. War in the Middle East. So we'll see where we go, but it is relentless to say the least. All right, what else we got? We got another strike out there. Yeah, Mack truck drivers. They're joining the Detroit Auto Workers. So about 3,900 UAW members with Mack trucks will go on strike Monday after a majority of members rejected a tentative deal reached last week by the union and the company. I mean, that's always the kicker, right? They got a deal between the union and the company and the members were like, nope, voted down by 73% of UAW members who voted significantly shorter with the unions demanding and negotiations currently being held with the Detroit automakers leading some workers to vote against the potential contract. So it's on, man. Employees feeling empowered. We'll see where that goes. But yeah, every time you hear about, you know, these automakers have been making a lot of money over the last five years, man. Even putting out less cars, making a lot of money. All right, let's jump around a bit as we come into this break. Markets behaving pretty well. All things considered. You get the S&Ps down 410% and Hasdeck 100. You're off by about 810% right now. Look at the Dow. Look at this. Pushing positive gets it all back. Dow spikes 200 points from where we were trading when we came on the program. Dow only down by four. Russell off by seven right now. Crew pushing higher to 86, 20. You got the gold contract. Up $13 at 1858. Stay tuned folks, we'll be right back. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life, before you decide it's impossible, get some advice from the experts. You might find that it's not so impossible after all. For daily market overviews that give you direction on the key indices, selective stocks and commodities, subscribe to the opening call newsletter at tfnn.com. The opening call newsletter is written by Basil Chapman, creator of the trading methodology known as the Chapman Wave. The Chapman Wave up-down sequence gives you an edge in identifying price turns, finding the peaks and valleys in stock prices.  First-time subscribers also get a 30-day money-back guarantee. 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To sign up today and become a part of this educational community of traders, just visit the front page of tfnn.com. This program is brought to you by Vista Gold, traded on the NYSE American and TSX under the symbol VGZ. Back folks, we got the SMPs continuing to claw back the losses. We're only off by 11 points right now. Dow pulls back a bit, they almost got positive. Dow off 21, NASDAQ 100 leading the way in losses off about six tenths percent. Right now you have the Russell, about three tenths percent in the red. Over to a Bloomberg article we go, talking about free money is gone, man. The headline out here, the free money experiment is over, carnage from the bond market where the route is worse than anything you'll find in the history books is spreading and the implications are nasty. Now here's what they say is that, excuse me, you got Bank of America, recently got their hands on US bond market data going all the way back to the founding of the nation and it shows they say that never before has there been an extended period of losses like the past three years. They also say how often could bonds have traded during the war of 1812? Are the figures as accurate as they are right now? Worth mentioning, right? The reliability of some of that data, but you don't gotta be a market wizard, man, to go over some of these charts, right? I mean, we've talked about it. It makes what the banks did like Silicon Valley even more astounding in terms of there's your 30 year. Here's where they dumped all their money into this investment, okay? And that's as far as you go back to 1996 on the 30 year, I think it is, yeah, it is. But some of the charts in here, so we'll get over the article, okay? They talk about the debt, of course, and they're talking about the supply the US government is going to markets with, okay? There's just an, it's a vicious cycle as rates rise. As rates rise, deficits grow, as deficits grow, the amount of debt we need to service goes up, of forcing us to issue more treasuries, okay? US government's marketable debt, I mean, 1983, man, 937 billion could have gotten a few, we could have gotten a few people. I mean, Trump can pay that off tomorrow, man, right? It was under a trillion dollars back in 1983. You get Bezos, you get Elon, you get Trump himself. They pay it all off America's debt-free and we owe three people that service. Can't do that anymore when we're at 25 trillion and they weren't worth that kind of money in 1983, okay? But yes, there's quite a change of the trajectory of this and it's real when you look at this market, okay? I mean, do the real math of loaning the US government some money for 30 years, do you have the reliability that you're getting paid reliably for 30 years? Cause I don't. I don't think I would make that calculation that I'm getting paid reliably for 30 years. I don't think you should. I think that would be an irresponsible assessment and I think that you would probably wanna demand a greater interest rate for talking about the reliability of a 30-year payment. Do I think the country's gonna default on their debt? I hope not for sure. But jumping through some of these charts here, okay? This is talking about the annual return of the Bloomberg US aggregate index bonds, bond benchmark index returns. Three years like nothing we've ever seen combined, minus one and a half, minus 13, and we're down again this year at probably like 2% or something like that. Yeah, check it out, right? That the 10-year got as low as 0.3%. Imagine that, man. And those poor people that pulled their money out of equities and dumped it into that 0.3% cause that is correlating to, excuse me, ZN, the 10-year. Like when the market was at lows, right? You would have been buying the 10-year at 140, 24 as rates collapsed, markets collapsed all in one. And meanwhile, rates have continued to rise as the market basically has continued to rise. Cause you back up where the S&P was at the marshmallows, you were down to 2,200. So you had people selling their equities at 2,200, you had 2,500 and you had them putting their money into fixed income at a price of 140. So what did you do? You forego a 100% acceleration in equities to lock in a 30% loss in fixed income. It's a quick lesson of fixed income, folks, if you don't hold to maturity. And a lot of people don't realize that when they're buying, whether it's bond funds, et cetera, that they're not holding to maturity and you get a move like this and boy, are you exposed. So we got three years of returns that have never happened before and yeah, what makes this moment all the more shocking? We were at 2% day after day, year after year in the aftermath of the 2008 financial crisis. So rock bottom rates made sense, right? They don't make sense anymore. Okay, and that's why I keep going back to this 30 year. Here's 2008 on the chart. Okay, here's where we're trading at right now. We're basically where we came into the financial crisis at. Don't get misguided by the price action that's occurred over the last 12 years because that might never come back. It's not coming back anytime soon. That's for sure. Let's hope it's not, because boy, if it does, we're all in trouble in terms of what that means about growth, about potential, about everything because a financial crisis brought that upon. But yeah, you back this up any further. I bring this comparison, okay? If you think that rates can't go any higher, consider this, on a 30 year basis, I graduated college in the year of 2002. Look at where the price action was from 1996 to 2002. We're trading right now at 110. The year I graduated high school, what were we pushing? I graduated in, yeah, the middle of 1998. The 30 year was at 101, right? My junior and senior year, it was as low as 89. We're living in a different world, man. So don't get caught up in the fact that this thing traded to 190. Remember how normal things seemed during the late 90s. Can't believe I'm saying that, because it's 2023. How the heck did I get this old? But for those of you that don't know that, okay? Nobody was talking about that the world was ending because of high rates. This was not the astronomical rates that we saw in the early 80s and inflation of 18% mortgages. That's not what was happening there, okay? It was not that high. This was a more normal environment. And I feel like that might be something that we're coming back to right now.  To get back to there. But you're talking about CDs in the den. Let's take a look at where we are in CDs. Excuse me, because yeah, the ladder is a good deal. We'll continue to talk about it. Where's our five year ladder right now? 5.13%, the five year ladder. You got a one year, two year, three year, four year, and a five year CD. That way you reset that rate every year. You don't expose yourself to duration as much. When you do that, and right now, when you add up the one, two, three, four and five year, you give yourself an average percentage yield and APY of 5.13%. The one year and two year are basically at 5.45 because you have the three month, six month, nine month somewhere near about 5.5%. You go out a year or two years, you're getting about 5.5% because the market knows the Fed's staying there for about a year or two, especially CD rates.  That's a remarkable risk-free rate of return right now with everything going on in this market. But guess what? Even a Middle East war cannot hold this market down. Are we gonna be positive by the time I get off the air in about seven minutes? Stay tuned, folks. We'll come back for one more segment. S&Ps only down single digits off by nine. We'll take a look at some of the other equities when we get back. Stay tuned. The Gold Report. As a precious metal, gold is still king. It continues to hold the most effective safe haven and hedging properties across the global major trading hubs of the London OTC market, the US futures market and the Shanghai Gold Exchange. The Gold Report. Tom O'Brien publishes his weekly Gold Report every Monday morning for subscribers, consisting of coverage of the XAU, HUI, GDX, the Dollar, Bonds, the South African Rand, as well as 25 different mining equities with specific buy-sell recommendations. The Gold Report. New subscribers get a 30-day money-back guarantee so you have nothing to risk. Subscribe to Tom O'Brien's Gold Report newsletter now at TFNN.com. You might think that if you want to be successful at trading in the stock market, you're going to need a crystal ball. After all, it's impossible to predict the future, right? Like any endeavor in life, before you decide it's impossible, get some advice from the experts. You might find that it's not so impossible after all. For daily market overviews that give you direction on the key indices, selective stocks and commodities, subscribe to the opening call newsletter at TFNN.com. The opening call newsletter is written by Basil Chapman, creator of the trading methodology known as the Chapman Wave. The Chapman Wave up-down sequence gives you an edge in identifying price turns, finding the peaks and valleys in stock prices.  First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. TFNN.com. Educating investors. Everything in the universe is governed by the Fibonacci sequence. This mathematical principle is responsible for everything from the most aesthetically pleasing artwork to patterns in the stock market. To stay on top of stock patterns you can take advantage of, sign up for the Fibonacci 24-7 newsletter at TFNN.com. When you subscribe, you'll get a weekly report from veteran day trader Larry Pesavento on stocks you need to pay attention to and you can trust Larry's analysis. After all, he's got 45 years experience as a day trader. Larry will also provide daily charts, videos and data on the key markets that he's tracking. Expect notifications from Larry on market movement you need to act on at any time. First-time subscribers also get a 30-day money-back guarantee. If you're not satisfied, let us know and you'll get a full refund within 30 days of signing up. Subscribe to the Fibonacci 24-7 newsletter today. TFNN.com, educating investors. Don't forget, you can listen to TFNN live on your mobile device 24 hours per day. Go to TFNN.com and hit watch Tiger TV. That's TFNN.com and hit watch Tiger TV. We're gonna charge a Disney up here up about 1% this morning. So you got activist investor Nelson Peltz and his firm Tri-Infund Management. Reigniting a potential proxy war with Disney less than a year after dropping the initial battle, they got 30 million shares, $2.5 billion is their position. I think if you jump over here, I mean, what's Disney at now, 80 billion? No, 153, they've had quite an acceleration. Okay, so you're still talking about almost a 2% position of ownership in the company. We had a caller, Charlie from Frammingham last week talking about Disney on a longer-term basis. You know what I'm saying? Listen, you're getting this at prices, I think he called it some point middle of last week, right, in the 70s.  Because this market, loading up the boat on anything in this market. You wanna be diversified, folks, even adding in a little risk-free rate of return of 5.13% on a five-year ladder. But boy, you're at some attractive prices when you look at this thing going back five years. You're at COVID lows when nobody was going to movie theaters and no one was going to theme parks and we didn't even know if business was gonna exist in the current form of how we had it. You come back, you test those lows, what do we do it? We test it on dramatically lighter volume. I mean, you're compared yourself to the COVID lows. That's a weekly chart we're looking at, okay? I mean, look at these numbers. COVID, we did 169 million. The last two weeks, we did 64 and 64. 64 and 64, yeah, 64 and 64, we're popping a bit. But he's back in there, again, pelts. So they got $2.5 billion. The firm plans to push from multiple seats on the board this time, including one for pelts. This is kind of what they were doing earlier, right? And then he dropped that after they made some changes. The nomination window for the new board members opens December 5th, so he's got a couple months and runs until January 4th. So keep Disney on your radar. I was also going over last week when Charlie called in Framingham, the potential movies that they got coming out. And as I was mentioning then folks, you got three Star Wars movies over like 18 months, about two, two and a half years away. But if you look at longer term, right, maybe you scale into this equity, kind of the way you scale into a CD, right? That's what a ladder is. You scale into it, one, two, three, four, five. You scale into Disney, take a position now. Take one in three months, take another in three, take another in three, you scale in over the period of nine months. Nonetheless, Disney up on a negative day. Market relentless to the upside. We are negative at only six points. Stay tuned folks, a lot of market action to come. Basil Chat is up next with the Tiger Technicians


This study aimed to identify new microRNAs, myrna's, in soybean that are regulated by water deficit and rust stress. Eight small RNA libraries were constructed from drought-sensitive and tolerant seedlings, rust-susceptible and resistant soybeans with or without stresses. Solexa sequencing was used to detect 256 myrna's, including 24 families of novel myrna's, six families of conserved myrna's, and 22 families previously reported in soybean. RTQPCR was performed to validate the expression profiles of 11 myrna's across all libraries. The results showed that most myrna's were up-regulated during water deficit stress in sensitive plants, while most were down-regulated in tolerant genotypes. In contrast, most myrna's were down-regulated during rust attack in the susceptible genotype, but not in the resistant one. The study provides molecular evidence for the involvement of myrna's in water deficit and rust stress responses in soybean. This article was authored by Nacimento Leandro Si, Abdelno Riccardo Vi, Mocellino Guimores Francis Marci, and others. We are article.tv, links in the description below.