(03/03/2026) - Seminário de Equações Diferenciais e Parciais - Aula 01

[00:00:00]Hey, so since Filipe asked me to introduce myself, I'll introduce myself to whomever does not know me. I'm Jang.

[00:00:07]Uh I'm a professor here. Have been a professor here for 10 months, more or less, give or take. And so this is uh so Filipe asked me to to speak on a recent very nice paper that me, Simon, and Andrea Shaoto uh put on archive maybe a month and a half ago.

[00:00:29]And so after talking a little bit with Philipe and uh you know after reflecting upon the fact that the paper is tough I decided to give some lectures about it and uh not only one lecture but uh so my initial plan was to speak like to divide it between like a couple of days. I think right now we're looking more towards three days uh maybe four even. Let's try to make it work with three dates. Right. So this week, next week, and then and in two weeks time.

[00:01:03]>> Yeah.

[00:01:03]>> Is it going to be on YouTube?

[00:01:05]>> I think so. Yeah, I think this is going to be on YouTube. This is not an invitation for everybody to just leave right now. So So if you guys could stay, that would be nice. Uh anyway, so yeah, let's see uh what I can what lies I can I can try to convince you that they're true during this this time. So yeah, let's uh let's just remind ourselves of what we're going to be talking about today. So we're going to be talking about gauge transforms for KDV and well posess below h minus one. Right?

[00:02:00]Okay. So in order to talk about that I need to remind you guys what this equation this court of act degrees equation is that I'm interested in and that is this equation right so I'm going to refer to it as KDV and KDV is I take a time derivative on U I sum it with a triple space derivative on U and this is equal to DX of the square of U. Right? So I'm using this normalization for KDV.

[00:02:33]Um and this is the equation of interest to us and some facts about this equation ina in case you do not know much.

[00:02:42]So this is completely integraable and as it's a completely integraable equation then we have infinitely many conserved quantities.

[00:03:07]Some of them are the following right. So the first one that you can see directly from this is mass. So if I take this time quantity here which is the integral on the real line of my of my solution for each fixed slab of time this is constant. The second is momentum or the O2 norm. Right? So if I take the O2 norm of my function as a function of time from this equation you can also see that this is conserved. And the third well out of many many more is what we call the energy.

[00:03:55]And for this equation, I'm going to call energy this thing here. Right? So I'm taking an L2 norm of the gradient and I'm summing with the cube power of my function. So this is also conserved and there is many many more and uh in with increasing degrees of complexity as I go down. Okay.

[00:04:20]And so what do we want to um what we want to study regarding this equation? So what we want to study today is well if I give you the IVP the initial value problem that I have a solution that satisfies this equation with at time zero my solution being a certain fixed function you not.

[00:04:55]So my question if let's say that you not is an hs of the real line. So if you not is in a certain subvove space of certain regularity on the real line for which S can we say that you let's say for instance the most basic question remains in that case for an interval of time.

[00:05:53]Right? So the most basic question is let's let's say that we have this evolution and I'm not going to uh pretend that I know the physics behind this equation.

[00:06:05]Let's just uh look at this equation and let's start with some initial data.

[00:06:11]Let's evolve this initial data according to this law. And what we want to describe is when uh if we start with a certain regularity when can we propagate this regularity forward for some kind of small interval of time. So this is the most basic question and of course if I can do that.

[00:06:33]The next question is when is this property continuous on the initial data that is if I take a look at the curve that my solution is drawing on the space on this space here hs if I perturb my initial conditions. How much am I perturbing this curve? Is this curve depending smoothly on the parameter on the initial data that I'm perturbing? Is it not? And so this is kind of the question that we want to study for this equation. And basically what we're we're going to be calling this we're going to be working with several different concepts of well positedness here. And this is basically a loosely defined uh uh concept that we're going to be using here. So we're going to be calling we're going to be saying that this is locally well posed in hs if well if I can find if for each function I can find some tiny interval of time such that evolving through this initial value problem remains in that space uh in this space hs for that tiny interval of time. And moreover, the curve that I draw in the in my space HS is a continuous curve.

[00:08:03]Right? So I could I could ask that this curve depends smoothly on the parameter ellipse shits and whatnot. But I'm just going to be looking for this very weak kind of uh um definition of wellosiness.

[00:08:16]So let's say that so we're going to be looking for local wellosidedness.

[00:08:29]the map. So, u of t belongs to h s for t is small. So in loose terms, right? And the map that takes you not to this curve u of t 0 t knot continues right and um well let's say that I also that I also that I take all this plus some sort of uniqueness to make sense of well the solution exists it is unique and uh if so then it depends continues on the initial data right so this is the concept that we're going to be working with um for the series of lectures and as I said we're going to be working with the cord of egg de equation and let's just start then our our climb towards understanding this equation right so what is the first thing that we have to note so this equation satisfies certain scale invariance right so if I take a u a solution to my KDV equation then if I define u lambda of xt by lambda 2 u of lambda cubed t lambda x.

[00:10:14]It is also a solution to KDB.

[00:10:25]Right? So if I start my equation, I can hunt for its symmetries.

[00:10:30]If I didn't have this term, of course, this would be silly. any kind of scaling that I did would be uh basically any kind of scaling would be nice as long as I kept this kind of structure here. So I I didn't have to include this term but I have to include it because of the nonlinear term that I have on the right hand side but still we find that there is a non-trivial symmetry there is a non-trivial scaling symmetry of this equation and it's that one and so our question here of finding the best we want the lowest s right so um what we want let's recall is a smallest S real parameter such that we have well supposedness for KDV in my space hs of the real line. Okay, so it's pretty it's it's a pretty simple task. So my question previously was well for which ask can I actually construct this unique solution make it continuously depend on the on the initial data for a certain small amount of time. Now I want well what is the smallest s I want to ask what is the smallest such that it works right and let's make some remark directly related to the scaling invariance it's that if I take a look at this space h minus three halves homogeneous space on the real line this leaves the norm of U lambda and variant right so if I take the norm of u lambda h minus three halves this is equal to the norm of u h minus three halves right so this is a kind of whenever we have this kind of symmetric behavior going on. It's a sign that something is kind of critical that there's some criticality going that some uh behavior should be different below either below or above this threshold. And of course, if we look at the equation again, so we have several derivatives, it is not too hard to believe that if I start with a lot of regularity, this equation should preserve a lot of regularity, right? Um hence uh let's say that I start with a function that is of class I don't know C 100.

[00:13:38]It should not be a surprise that if I start to evolve this my function through this uh evolution KDV and it it will remain a function in C 100 for as as as many units of time that I want to continue right and that I'm allowed to continue. So on the one hand I have this scaling critical parameter h minus three halves and on the other hand for a large enough s I should have the losiness that I'm seeking. So what I want to convey to you guys is that if I define SC let's say SC to be minus three halves this is the scaling critical parameter and what I was telling you is that well there should be a threshold for when a certain behavior stops working and some other kind of regime kicks in.

[00:14:44]And let's put a big maybe here.

[00:14:50]We should have wellosedness for S greater than SC. many many questions question marks here right so this is kind of the phenomenon that we expect in several dispersive equations right so in several dispersive equations we take a look at this so it's a standard trick for those who have not seen it yet we try to look so first you I'm giving I'm given an equation let's say if not the KDV I'm given the shortinger equation or something like that and then I try to to hunt for the specific scaling parameters that leave that equation invariant right and I hunt them down. I find all of them and then I'm like look let's try to hunt for the specific so of space that makes that specific uh scaling invariant and then that is our critical space. So then the natural conjecture in any kind of equation is pretty simple. So this is just a scaling argument which should be hey um well above that threshold we have looseness below we don't right. Um and let's see if that is uh indeed uh the case. Right? So uh the spoiler for for some equations it is but for KDV it's not. It's a it's a fun uh it's a fun fact.

[00:16:18]So a bit of of history here.

[00:16:24]So if we take s to be greater than three halves then people have managed in in the 70s to prove that KDV is well posed locally uh well posed and this is due to the work of several people since I'm on record here I better not uh forget any names and let's write some of them so bonus myth Oh, the work of Kato influenced on that.

[00:17:00]So, and Timo Sutsumi and Mukaza.

[00:17:13]And the thing to be looked for here is that there was no dispersive properties used, right? So kind of the idea behind uh those techniques is well maybe we could we should regularize our equation by small parabolic perturbation use parabolic uniqueness and existence theory and then take the scaling parameter to zero and get uh wellosedness. So it's simple, it's for free. It's the way many people in the 70s and 80s did this kind of things before the fer base technique started kicking in which capture oscillation a bit better. But it's uh it as I said it used kind of like a parabolic um perturbation technique and not the dispersive character of this of this equation which is what we're going to be using uh today and in the next couple of weeks. Right?

[00:18:21]So then the next improvement well we can say that let's write it like this. So this is locally well posed for and here the next kind of jump is for s greater than than 3/4 and this is by using some more standard for your method such as maximal function estimates threearts estimates and this is due to the work of again kato to and kenic ponen Vega.

[00:19:13]And then finally we get to our third and let's say uh more recent and more modern in terms of conceptual uh framework improvement which was due to the work of well I think it's s greater than or equal to zero due to the work of Jean Borgan who was using the now so-called Borgan spaces in order to to to work with this equation, right?

[00:19:46]Okay. And then so for the really interesting things, so let's recall that we saw that s critical should be minus three halves, right? So the critical case the critical scaling for that equation. We got all the way from three halves to zero with Borgan.

[00:20:05]So how hard could it be from getting from zero to minus three halves, right?

[00:20:11]Let's see. Hello. So, we got the fourth contribution was S greater than or equal to minus 3/4.

[00:20:21]Oh, that's so cool. 3 halves 3/4 0us 3/4. Oh, nice. Cool. Right guys, we're getting closer. And this is again due to the work of Kenik Ponga with some bilinear estimates.

[00:20:38]And the end point by Cristander and Tao with also contributions by Gu and Kishimoto, right?

[00:21:00]But then there is a catch. People discovered a catch in this thing and the catch is if s is less than minus 3/4 then we don't have uniform continuity of this guy, right?

[00:21:37]So people managed to show that in a certain in a certain sense and most of the techniques here actually all of the techniques here they were based on fixed point methods. Right? So we're using we're come up we're coming up with the norm. We're coming up with the space that has for cert for each slab in in time I have an hs space and it's kind of an integral of of those spaces over time.

[00:22:06]And what happens is uh this thing here this kind of notion of looseness that I defined to you guys if we're whenever we're running a fixed point argument on the relevant spaces it is always going to give us not only a uniform dependency on the initial data but an analytic one right so if we have this kind of endpoint arguments we do we have for this kind of sorry uh fixed point arguments we always get for free from the method we always get analytic dependence on the initial data and what people showed here by looking at the interactions of this bilinear estimates that people used was well this cannot be gotten this cannot be achieved simply because we don't have even uniform continuity of this map right so there is this fundamental change in understanding of the well posess of this equation. Well, the methods that were all well come up with a nicer and nicer space such that we can go down on the on the scale each and every time this breaks down here. Here the game changes. We have to use something different.

[00:23:28]And what people used here for the next kind of quantum leap which is not to minus three halves it's to minus one was this modified energies by Kilip and Vishan which used the completely integral structure. So let's recall what we were saying in the beginning there we have infinitely many kind of uh conserv conserved quantities for that equation.

[00:24:23]Since we have those we can use those to our advantage. we can kind of modify them so that we can you know explore them to bring on some rigidity and then maybe just maybe we have some kind of algebraic miracle going on and we can pass a limit and get a solution in H minus one and that's exactly what they do. So this is oversimplifying a nanos of mathematics paper. I'd encourage you guys to check it out. It's a beautiful argument but it is heavily different from these previous techniques. So recalling those are fixed point methods.

[00:24:58]This is kind of a calculus of variations argument. Right? So you have kind of a sequence you try to approach it. you try to guarantee convergence to something by controlling some quantities and showing that they are kind of stable under this limiting process. So it's a different kind of method right and again so just mentioning that weaker notions of well supposedness here were uh we're not even it's not even clear that uniqueness was achieved. Uh but uh anyway just to mention were achieved by Kapla and Tutsumi and bookm and co right and finally here we wanted this flow this kind of thing this kind of curve that we are that we are drawing on the HS space to be continuous and what was shown is that there is no continuous flow below s= to minus one. So again we violate even this weaker notion. So from here to here we had to lower expectations right we had to say hey fix point methods not going to work have to do something else.

[00:26:40]Then they said fine we did something else we got something we have some sort of way to prove localess for this equation but hey below this not even that is going to work. You don't have uh a right off the bat way to make it work.

[00:26:58]not from the from the equation itself you don't right but then uh me Simone and Andrea we are crazy and uh we decided to say hey what if we try to uh redefine uh this notion right so uh how hard could it be again how uh how uh how insanely difficult could it be to you know take this kind of beautiful result and try to bring it to a slightly lower scale And so yeah, let me just write it down as an explicit question.

[00:27:37]Can we go below in some sense?

[00:27:51]And another question which is directly tied to this case. So in this case as I told you guys this is related to the fact that the KDV equation is completely integraable. So it's something that only works for the specific case where you have these conserved quantities and you have this algebraic miracles.

[00:28:13]So the second question is what happens to perturbed versions of KDV there we have to resort back to the endpoint meth to to the uh fixed point methods right there's nothing in the literature that allows us so far to to to go all the way below to this this modified energy just won't cut.

[00:28:50]Uh and yeah, and these are the questions that somehow we're going to be trying to answer in these lectures.

[00:29:00]And in order to do that, I need some sort of a background uh notation. So the first thing that I'm going to define are the so-called fyer leagg spaces and this is basically only the temporary distributions on the real line such that whenever I take c to the s u hat and I measure the lp norm this is finite and here of course this is the Japanese bracket it defined like this, right?

[00:29:44]Okay. So, this is just um just recalling this is the same thing as a soil of space.

[00:29:53]If we take P equals 2, right? So if P equals 2, so if P equals 2, then this fouryear leag spaces they are equal to the usual soil spaces defined in the fourear sense, right?

[00:30:16]Okay.

[00:30:20]So let's look at the scalar invariance again right so if you use if you take that u lambda that we just said yeah >> taking the norm of the transform >> I'm taking the LP norm this this is an LP norm right so of the for transform yeah because when I take pals has to to coincide with >> Yeah. Yeah. I thought you were taking the open >> I have >> you apply derivative and then you take it back and then you take >> uh yeah so in order for me to do a sov norm I have to just take the for transform multiply by something then take uh then take an L2 norm I don't have to go back >> all right >> uh yeah thanks uh so I think it's it's a good moment to ask for other questions anyone else.

[00:31:23]If not, we can go on for a bit. You can interrupt me at at any point in time.

[00:31:31]>> For people is the same, but for not= >> uh I'm going to use this norm.

[00:31:38]>> I'm going to use this norm always >> without going back.

[00:31:40]>> Without going back. Yeah. Yeah. Exactly.

[00:31:43]Yeah. Yeah. If I if I were going back.

[00:31:46]Exactly. I think that's that ties that back to the newest question.

[00:31:50]If I'm going back then I'm actually defining a subable of space right by Calderon Ziggman theory. No the importance here is that no we're not going back. We're working on fundamentally fiery quantities. So for using the for a transform is important to us, right? Okay. I think that's that's kind of your question, right?

[00:32:07]Yeah. Okay. Thanks.

[00:32:10]Um anyone else? Let's go on then. Okay.

[00:32:17]Okay. So if we take the u lambda that I have here and I look, hey, when is a norm like this for fixed B when is a norm like this invariant with respect to that?

[00:32:39]Well, what I have is that if I define SCP as min -1 -1 / P is the natural scaling scaling critical threshold.

[00:33:06]Right?

[00:33:08]So just a sanity check we take P equals 2 we get minus three halves right and this is the so this basically says hey below that you can't get anything so this is kind of the best possible best case scenario for us uh and let's define an analogy to how we had h minus one for KDV P. So the analog for us is going to be the min - one/2 minus 1 / P scale right. So again for P= 2 this is just minus1. This becomes H minus one itself.

[00:33:51]So this is kind of the scaling critical uh indices uh for our equation. This is the true scaling critical threshold and this is the one induced by the Kip Vishan kind of argument. Right? So for for LP and our question is well when can we work with elosidness in these spaces can we just instead of using the usual subolive spaces can we use these spaces instead right so this is our way to bypass the h minus one and to try to go below so the first uh the first step in our program to go below hus one will be to replace the so beloved spaces by these kind of spaces these ferang spaces here Right.

[00:34:35]Okay. So, uh, right. So, let's maybe mention that this is not out of the blue. People have worked with this before.

[00:34:48]So, Axel Grunhawk actually did something. So he he proved that there is well supposedness in those spaces.

[00:35:06]Whenever I take P to be any parameter between two and plus infinity and whenever I take my index S to be greater than the maximum of -2 - 1 / 2 P and -4 - 11 over 8 P right so again if I take P = 2 here I just have minus 3/4 right so in this left hand side so this is bounded from below by this kind of KPV result so the best case scenario what Grunhawk gets is the KPV result and that is not by chance what Grunhawk was doing here is just using by linear estimates again so if he's using by linear estimates which is exactly what the same as KPV are using in the in the so space he's at most going to get the same thing Okay.

[00:36:06]And then the problem here that doesn't allow So somebody ask a question. Not really. Okay. Uh the problem is with uh Grunhawk's uh uh with Grunhawk's idea and with also the KPV uh proof is that we cannot really bound the high high low interactions right so sometimes we have in this kind of convolution that we get and I'm going to explain what what this is by the way uh is that sometimes when we are using kind of a bilinear form a method to get our fixed point to work maybe we will get we get a term s 1 and a term si 2 such that their sum is very very small and we cannot use stationary phase in order to get extra decay. This is basically what prevented KPV from going below minus 3/4 and this is basically the main principle behind this lack of uniform continuity here. Building the counter example is just exploiting this high high low problem here. Right?

[00:37:30]So whatever kind of thing that we wish to do in order to conceptually go below h minus one has to take this into account and has to get rid of this bad interaction in a certain way. Right? So, our idea maybe I'll leave this here.

[00:38:06]So our idea it's going to be to remove to manually remove the bad interaction by using a gauge transform.

[00:38:36]Right. And now I spent the first part of this talk talking about what it what it means or hinting at what it means to go below h minus one. Now let's talk about what is the gauge transform that we're going to use. Right?

[00:38:52]And the gauge transform is that we're going to use is going to be defined as follows. So if I define G and I want it to be a map from F SLP to FSLP I'm going to define it by its formal inverse.

[00:39:22]And why is it formal is that well sometimes it might not be invertible but hey we're going to work with the in with the inverse in any case and we're I'm going to give us a couple of words on that later. So let's imagine that there is a map. So imagine that life is beautiful and that we can uh fulfill our dreams and that there is a map G from these four bank spaces to themselves such that whenever I apply their inverse to F this just spits back F plus the inverse space for a transform of minus 1/3 integral over this notation here is something that we're going to see a lot in this lecture which is just we're integrating over one of these two variables but I'm going to use this notation to mean that we're integrating over either Sai 1 or SI 2 so this is just a formal notation if you will this just means this just means that we're working with this convolution plane I multiply it by a certain indicator of a certain set and this set a is exactly what's going to encompass the low the high high low interactions I take fhat of i1 fhat of i2 divided by i12 so this is my gauge transform right so I'm taking a convolution of these two properties of these two functions Here I'm multiplying them by a certain set which is which is going to tell me that hey in that set the problematic frequencies interact I take it for a transform back and I sum with my function right this should somehow I mean it's complicated to see like this when it's written like that but this should somehow kind of cancel those those those interactions and I'm going to motivate to you guys how that cancels those interactions and how that actually makes our lives much much easier in the long run, right?

[00:41:38]>> Yeah.

[00:41:41]>> change scaling, right? You divide by two size integrated with respect to one.

[00:41:46]>> Uh that is uh correct. Yes, it is going to change scaling. As a matter of fact, this is connected and it's a good point to to uh to open up a kind of a parenthesis here is to say that this is connected to something called the MUA transform. So the MUA transform is something that uh connects the KDV equation with the modified KDV equation. Right?

[00:42:15]um that transform in particular has a nicer kind of structure than this which is basically if I raise this multiplier that is exactly the mura transform so it can be written if I take for a transform on on both sides it can be there is there you don't even need to take for a transform on both sides I can I can actually write this transform this mura transform uh in terms of the function itself if I'm not mistaken this is just like you take a solution you sum it with like its derivative squared or something like that. And this gives you uh a uh so this mirror transform gives you a way to go between modified KDV and KDV and this of course changes scaling and but this of course allows us to employ methods that are useful for one equation to the other. This does the same but the fact is that the mutter transform is very equation dependent right so it's 100% dependent on which KDV equation on the on the specific form of the KDV equation that we're working with that we have the specific nonlinearity and that we can go uh to the to to this other modified KDV equation and there we also have a specific structure and so on this here this is more flexible just because we're putting this indicator of the bad set here. We're basically cancelling the bad interactions and that actually allows us to work with way more uh general models than this. I'm going to make a remark on that later, but basically you can put like some sort of pseudo differential operator here in front and things work in the same way. Thanks.

[00:43:57]Okay.

[00:43:58]So then let's imagine that we have a transform like that.

[00:44:04]Sorry.

[00:44:07]>> Oh, this is going to be a set that I'm going to define later.

[00:44:10]>> Uh, probably we're going to see only the definition of that next lecture. But it this is basically so the idea A is the set of good frequencies.

[00:44:32]So in the sense that well whenever we try to work with the usual bilinear estimates or multilinear estimates we try to use some sort of oscilly integral principle right and this is going to be the set where the oscilly integrals are nice and well behaved and we can get some decay and uh the complement is going to be the set of bad frequencies where in general everything just kind of crashes down I don't have this kind of oscilly integral behavior anymore and this is actually directly related to uh the the composition that I want to do uh in a bit. I just wanted to state uh the theorem uh that we will prove in these lectures and then maybe I will uh talk a bit more on this decomposition.

[00:45:23]Okay. Okay. So the theorem that I want to prove is first I'm going to take this gauge transform and I'm going to apply it on you.

[00:45:39]Then I'm going to observe that this new variable zed, this satisfies a new KDV kind of equation which which has a specific form like this, right?

[00:46:06]And what is this form? It looks complicated.

[00:46:09]And it isn't really that complicated after I tell you what we're doing specifically. So what we're doing is we're replacing KDV which is that equation with its very nice form by this one. But this one I have these guys and these operators they are kind of J linear in a sense J linear operators.

[00:46:36]They're non-local and nonlinear operators, but they're jinear. And the the the thing about them is that well, although they're non-local and nonlinear, they're very well behaved on the fer side. So if I want to do estimates on them, it is much much easier than doing estimates on for instance the term that I come that I come up with if I take for transforms on KDV in particular since these terms they're going to come up with a lot of cancellation since we're killing off the bad terms here the these terms they're going to induce us a lot of good sets and a lot of good sets for us is very good because we're inducing a lot of decay. So these are going to defi define like a converging series and basically these are going to allow us to define a solution even though v might be a very low u might be a very low regularity. So if I start something barely uh regular enough apply this gauge transform get this thing I'm going to get some kind of result that allows me to say hey there is some sort of solution in a certain sense and the theorem is exactly that right so the theorem which uh this is on archive already by the way so this is by Andrea Shapoto Simon Koha and myself and this this from a month and a half ago.

[00:48:00]Uh this is the following. So I'm going to state it in the P equals infinity case. But I will talk a bit more on what happens in the in also the other regimes.

[00:48:12]Um if I take my parameter s to be greater than minus2/3 and I take some sort of time parameter t greater than zero then there exists a delta depending on my time such that whenever I take some initial data on my fouryear bag space with the norm of this guy small enough.

[00:48:52]So if I start with initial data that is small enough in terms of my for bag space and in terms of this time parameter then there exists a unique strong solution to this equation which I'm going to call the G KDV the gauge KDV equation.

[00:49:22]on the interval 0 t.

[00:49:26]And this satisfies moreover that e if I take the area of evolution and apply to zed of t. So this function here it is a continuous it defines a continuous curve in my f lsp space and moreover we recover analytic dependency again right so moreover all these things that I said that below that minus 3/4 scale could not be achieved. We managed to achieve with this change of variables.

[00:50:25]Z depends analytically on ZN KN.

[00:50:34]Right? So this is a legit uh wellosedness result. Can I say that G is kind of distance?

[00:50:43]>> Um >> because you're killing you're killing some frequencies.

[00:50:48]>> That is kind of a way to see it. Yes.

[00:50:51]Yeah.

[00:50:52]>> Yeah. I think that's a good interpretation. Yeah. Yeah. Exactly. So that is the idea. Yeah.

[00:51:00]Um Yeah. Thanks.

[00:51:03]Okay. Yeah. So this is the so our statement >> why you writing there this exponential of third derivative >> uh I have to kind of kill the evolution by the group. So I'm just I'm just taking like the evolution by the linear group here at times t and if I kill the evolution of zed by that other evolution this is a continuous uh this is something continuous right but I have to use it I have to kill it because uh well the evolution the linear evolution here so if I if I had zero here instead then the linear evolution would be e to the minus t dx cub something right times the initial data so this effective effectively what this is doing is just killing off uh the the dependency on the on the linear term. Right? So this this is basically kind of resetting my my evolution uh at each given moment in time. Uh but this this basically says that hey if I readjust my equation by the corresponding suitable time parameter I'm I can manage to land on a continuous curve.

[00:52:09]And this is was something that was not uh that uh uh Kilip and Vishan said they they proved that if I don't do a gauge transform if I just work with regular KDV this cannot be done you have the curve is discontinuous right so we're saying that hey we can't correct things and the way to correct things this is the interpretation of this result is let's say that we use this kind of gauge transform that kind of corrects the course of how things are going This basically just switches things around.

[00:52:41]It jumbles things around so that from a from a certain solution u to KDV assuming it exists a priority we get this zed and the zed satisfies this. Of course it might be that u is just a distribution or something but it says we don't really care whether u is a distribution or not. Maybe we cannot really invert this map. What we can say is like well there there is always a solution if you start with small enough initial data this everything is nice the dependency is analytic sometimes I may not be able to go back but it doesn't matter it gives us a way to define specifically what a weak solution is in in low regularity >> what means not not able going back like not able to take some inverse transform >> no this uh this is a good question so this map is sometimes that it's not really invertible. So this is the first lie that you guys kind of swallowed here that this map here is in general not invertible. It's just invertible whenever these things all match, right?

[00:53:46]Whenever we can make sense of everything. So this is the way that we have to make sense of the solution.

[00:53:51]Right? So uh well as I said I cannot make miracles happen. I'm not Jesus Christ. So I cannot uh you know make some kind of curve becomes become continuous. Whenever Vishan and Kilip and Vishan said well no curve is continuous. So tough luck. I'm out of luck there. I cannot uh make uh I cannot come up with continuous curves out of nowhere. What I can say is like well if I switch my notion of well posess a bit then I can come up with a continuous curve. As a matter of fact it's a very regular curve and it depends analytically on the initial data. So if I change my variables in a nice way, I can get a very nice a very nice solution. This this is basically what this this result is saying. It's basically saying well uh you cannot contradict mathematics but sometimes if you change variables you can get nice stuff.

[00:54:41]>> It's not invertible means probably that sort of projection and on this projection everything is nice. Curve becomes continuous.

[00:54:50]>> Yeah. Exactly. Yeah. It's it's kind of it's it's not really a projection >> but it kind of if you restrict to the space where things are invertible then everything matches right so that that's what I'm saying but then there is like two branches there is the space of things where where where things are not invertible when you cannot really talk about the inverse gauge transform and there is the space of gauge transforms of things where we can talk about something this is the latter which I'm looking at so I'm kind of looking at something that is not invertible but I'm looking hey what about this side this side still is related in some sort of distributional sense to some initial data maybe this is going to you know make a curve maybe trying to invert this thing is make is making my curve go wild it is going to be a continuous curve in the space of temporary distributions but not in the fer spaces but I'm saying hey if I correct course with the gauge transform it becomes continuous so what I'm saying is that if I so this g to the minus one it's in generally going to be invertible in like let's say tempered distributions right and if I invert it if I construct my zed like this I invert it as a temper distribution I take a look at the V that I obtain it might be something wild it's going to be a temper distribution or something but it might be something very poorly behaved but what I'm saying is like well for me it doesn't really matter what V is what matter is what the gauge transform of V is and the gauge transform I'm saying it's very nicely it's very nicely behaved So this is the way that we're this result should be interpreted. We're not only constructing solutions below the regularity threshold. We are making them uh we're making them regular if we kill the bad frequencies. So we're basically saying the only obstruction is really this high high low interaction and if we kill that off we get something.

[00:56:44]Maybe one more thing this um inequality here in the statement of theory was should be the norm should be smaller than delta right >> what does it exactly mean it's supposed to mean that uh okay so you have uh your initial condition >> do go transform >> then you do spectrum of this thing and the spectrum should not contain something which is too too bad something which is too too rapidly oscillating how how would uh so I think it's like let's let's maybe go to the s so this is to for p equals infinity right so let's go maybe for p equals 2 right where our where our results actually match the subolive spaces this is actually saying that uh look I'm not talking about the spectrum per se I'm talking about the regularity of the solution I'm saying that if I have a certain regularity of solution and if I'm assuming that my norm in terms of this regularity norm is small enough then I can get something uh here of course I'm I'm going to get something like this and this is going to I mean LP norms of weighted things of for a transform these should in a sense um let's imagine that you're that you're working purely formally this by the housef young inequality it's kind of dual so working with the for transforms in LP it's kind of dual to working on space in LP prime So this is basically working with WSB prime. So this is basically the idea behind this. So this is a much harder space to work with. The norms are hideous. It's terrible. But instead of working with these spaces which are really really bad, we're going to work with these ones where the structure is nicer. So basically it's saying well if we have certain regularity in terms of these other subolive spaces and if you have the norms in terms of those other subolive spaces become small enough then we can get something. So that is basically how everything should be interpreted here. Well you you should it should be noted that I'm always working with P between two and infinity.

[00:58:54]So this guy always embeds into into a space like this. So if I'm working in terms of regularity then then you can you can think of the results in terms of like WSB prime. So I'm looking instead of I think this is ties back to the question by denilo I think it's it's this this is not really measuring a weighted 4year norm. This is measuring regularity just in a disguised form.

[00:59:18]So the the result should be interpreted as let's suppose that a certain regularity norm of my gauged uh initial data is small then my gauged equation is satisfied and I I still be I still belong to my certain for space.

[00:59:36]Good. Thanks Pablo. Uh so we are can I go a bit over time? Do you guys mind if I if I speak for 15 more minutes today and then uh if you if you do mind well I uh you know as somebody noted already you all noted already this is going to be on YouTube so no hard feelings if you have another if you have something else to do right now uh let's maybe uh talk a bit more about this today so just some remarks that the first remark arc that I have here is that this holds for P greater than or equal to 2.

[01:00:28]But then here I have to ask my parameter S to be greater than - 2/3 - 1 / 6p. So whenever I plug P equals infinity I get minus 2/3. again the second remark is that this is monus is necessary is necessary and this is in a particular sense which is well if I had stated this result right here not for gdv but for KDV itself the smallness would be absolutely not necessary why because I can do scaling and by scaling with that u lambda scaling that I told you if I choose lambda appropriately I'm going to shrink the norm and if I'm in the super critical regime then I'm shrinking the norm then I get into a kind of a threshold like that and I always get a uniqueness uh a uniqueness a localess result here these operators are jinear they're known local and uh and nonlinear. So I don't have a scaling uh I don't have a scaling theory to go by.

[01:01:46]I don't have scaling to save me this time. So I really do need to get this kind of smallness condition here. Right?

[01:01:54]So this is kind of in a non-local nonlinear setting. This is kind of the the the counterpart to the usual local opposess estimates that we have. Right?

[01:02:03]So the the usualness generally I have like okay yeah my time of existence depends on a certain norm of my solution here that's the way to make sense of that in a kind of a non-local nonlinear problem.

[01:02:17]Uh and the third I think I have three or four remarks. So the third remark is that same holds the same result same theorem holds for well let's say the analog of so the gauged analog of this equation. So if I if I write it like this, I have KDV.

[01:02:52]All I'm saying is that I can put a zero order operator here. So this is a zero order pseudo differential operator. Right? So if I take a pseudo any pseudo order differential operator any multiplier of order zero homogeneous of order zero on the on this on this on this symbol right here and I apply it to the nonlinear term. So this might become already very non-local the nonlinear term I also get I can do the same thing and the same result holds. So this is by no means dependent on the complete integrability of KDV. So our result is basically uh it's much more amunable to perturbations in particular.

[01:03:38]We don't need uh the complete integrability of KDV.

[01:03:44]Uh oh, I actually have five remarks. So the fourth remark is that this map is a local bjection on F lsp. So that inverted G inversion on the gauge transform this is legal if and only if S is greater than the scaling critical H minus one of P regime.

[01:04:25]So this this makes sense right. So this is what kilip and vision were saying in the kilip and vision regime minus h minus one you can try to make sense of this gauge transform this still makes sense you can still talk about well posedness below that you cannot and that is exactly what we're seeing here here we don't have uh this kind of uh uh uh of of of corres direct correspondence is just like a more conceptual correspondence And in a sense so this implies that we can so our result improves grindros to this regime. So if I take s to be greater than the maximum of -1 / 2 - 1 / p - 2/3 - 1 / 6p.

[01:05:30]This number right here is always smaller than this number right here. So I'm getting a you know a higher a larger amount a lower kind of sub poor bag space than green rocks result did.

[01:05:45]And the final remark uh this is the final remark.

[01:06:04]So this is just to drive the point home is that this number that appears here and it's minus 2/3 in the case of P= infinity.

[01:06:17]This is below this guy right here - 1 / P for P greater than or equal to 5.

[01:06:34]So what we're doing is well for P large enough we're actually beating the scaling uh we're beating we're we're kind of like cheating the system right so we're beating the the the kilip vision result uh uh for such p and of course it begs the natural question can you bring this all the way to p equals to 2 we don't know so this is a conjecture for you guys so maybe a question.

[01:07:04]Can this be improved to?

[01:07:13]Of course, we would have to change the result here. But can you use the same methods in a way to kind of beat the kilip vision result all the way up until their result by some sort of gauge transform?

[01:07:25]uh with our current methods it seems not to be the case but uh we don't know right so just before I stop I'd like to talk at least about the strategy of proof that we're going to be looking at in the next lectures so basically so let's say that we write the duh formula for our solution of KDV. Right? So the duamemed formula and I never know whether this has one or two amps but who cares. So this at time t should be something like e to the minus t dx cub x kn plus the integral from 0 to t e to the minus t - t prime dx cubed and here I have to insert the nonlinear term right so here I have dx of u ^2 d t prime Right. Okay. So this is the duh formulation.

[01:08:40]And what am I going to do? Well, I seek to kind of u counteract the linear effect. So we just apply this group on both sides, right? So we killed this guy and you know I killed that guy too. So I have this guy here like that. Pavo is finding it funny for some reason. I don't know.

[01:09:08]>> Okay. Sure.

[01:09:11]Okay. So then I have this new equation which let's say that this is a V of T.

[01:09:22]Let's take for a transforms on both sides. Right? So if you take for transforms on both sides, let's maybe differentiate in T first, right? So if I differentiate in t then this guy does not depend on t anymore. It vanishes.

[01:09:40]This is an integral on t. But then I have only this term. So I will have e to the t prime dx cubed of dx of u ^2 right something like that and then I take if I take for a transform. So if I take for a transform here I take it for a transform here and this gives me exactly the following expression. So it's I * integral and here I have dx of u ^2 which is u * dxu and when I take for a transform this becomes a convolution. Hence I get the integral over a convolution plane showing up. So I'm what I'm doing is you remember that thing that appeared in the gauge transform it appears for a reason right it's appearing because whenever I'm looking at the 4year uh identity that this kind of reormalized version of the dame formula satisfies uh I get a convolution plane on that convolution plane I get e to the minus i t and then here what I have is basically basically I cubed - 1 cub minus i 2 cubed.

[01:11:01]So this is what happens when I take the for transform. If you if you take the for transform of this area group and then do the convolutions properly.

[01:11:10]This is multiplied by times vhat of ti 1 v hat of t si2 right so whenever so this is d should be d x so whenever I take the for transform I should get an equation for the v variable And then what is what we're going to do next time is well we're going to work with this equation here.

[01:11:46]It's already on the fer space. It's already very nice. And that's generally how you know all of these techniques go.

[01:11:53]They work with this equation here. And then somehow they manage to make this function here which is the resonance function of this interaction.

[01:12:05]They make they make use explicit use of the the parts where we have stationary phase and the parts where we don't have stationary phase generally well we read and weep I don't know in our case we're not going to do that in our case what we're going to do is like well let's suppose that we have a certain set A where we can use stationary phase then we index this by a we freeze it and we take an integral on the complement on the complement I recall that this is actually an integral of v hat and v hat here and I have some expression for the derivative. So I can integrate by parts and when I integrate by parts in time maybe I can get a derivative here or a derivative here and I can reiterate my formula inside itself. So this is all about so this is what what the idea behind this this infinite normal form expansion is. I can get one equation like this for my v hat.

[01:13:02]And then by trying to you know make this multiplier appear I can kind of integrate by parts in time. And integrating by parts in time I can make the derivative fall in either of these two functions and then reiterate the equation reinserting it it into itself and making kind of like an an infinite uh series expansion for my solution. Of course, at each at each step, I'm going to increase the increase the complexity of these uh series expansions. And that is the reason why I have this J linear operator showing up here. I'm I'm increasing the uh the I'm increasing the the complexity by a lot. And as I said, at each moment that I'm doing this, I need to be able to go to the next one. And whenever I'm going to the next one, I have to restrict on the bad set and then do an integration by parts again. So I'm doing so this is the basis of the iteration procedure that I'm going to show to you guys next week. Um yeah but that is uh maybe I won't spoil the fun for next week then.

[01:14:04]So let's maybe finish here. Thanks.

[01:14:14]And those nonlinear terms they come from this iteration procedure like when you're doing that they appear one by one >> they will basically uh yeah so we're working with KDV and then here we're going to get an an infinite expansion for V. We get many boundary terms they are cumbersome they're bad because you know we cannot estimate them on Borgan space as well. So what we do is the gauge transform is going to rescue us.

[01:14:40]So the gauge transform is going to kill all the boundary terms. So what the gauge transform is going to do is going to be we're going to get basically it's going to kill many of these nonlinear terms that appear here and also all of the boundary terms. So the gauge transform is going to save us massively whenever we're doing this kind of expansion.

[01:14:58]Uh and uh but we're going to see how this is the case next week.

[01:15:08]>> Yeah. Can you tell what what's the reason to call it gauge transform?

[01:15:12]>> Uh, so this is you're going to have to ask Simo. He's the one who came up with that name. Maybe Filipe knows. I don't know. I have no idea.

[01:15:21]>> Has to refer to >> So yeah. So that is a great question. I I'm not the one who came up with that name. My suggestion was well why don't we call it a modified mura transform and uh Simone was like well maybe it's a so he had a kind of reason to for it I uh maybe if he's going to watch this video he's going to be able to say you're you just said a bunch of stupid things this is the reason and I'm coming next week to explain to you guys if that's the case but >> the thing is people use that like the Benjamin This is the G transformation. There is no symmetry there.

[01:16:06]>> Right.

[01:16:07]>> Okay. So there is another equation that you have also a G transformation that has nothing to do with with that high high low.

[01:16:16]>> Mhm.

[01:16:17]>> And this one.

[01:16:19]>> Yeah. Just by induction, right?

[01:16:22]People call it a gauge transform in other contexts. I think that uh that should be the main reason why people are calling it here. Yeah.

[01:16:32]many other examples there is an exponential of some >> is possible to write this as an exponential form in some >> I don't think so I don't think so no so this is uh so the best form that you can do is uh kind of write it in terms just like one does the mura transform which is like v plus I don't know derivative of v ^2 or something like that the mur a transform. Uh yeah, so this is the simplest case of this gauge transform, but uh there is no exponential form. So this is I think the simplest way to look at it is really the one that's written there on the blackboard. It's a kind of a way to kill and it's the most intuitive way to do it too. I think it's like we're killing the bad frequencies and that way just basically makes evident that well the bad bad frequency should be indexed by a set a here and we're doing it something like that.

[01:17:31]>> That's that's a common >> Mhm.

[01:17:36]>> you have to kill the >> Yeah. You have to you have to kill the bad part. Yeah. Yeah. Yeah. Exactly. Uh >> yeah. Exactly. So Mura does exactly that as well. And uh yeah so you kill the bad part in a way and so there is a way to define so there's a different kind of mua transform that you can do which is way more palatable than that one so it has a nice form that doesn't be that doesn't have anything to do with f transforms but then you get a system you don't get KDV so basically you pass from KDV to a system and it's not clear whether one is easier than the other so the the whole key here is that we're we're passing from KDV to this guy. And this guy is easier than KDV.

[01:18:28]By the way, this formula you've written, I expect anything but this one.

[01:18:32]Numerical analysis, it goes by the name exponential time differencing.

[01:18:37]>> Uhhuh.

[01:18:37]>> And also massively helps when you do numerical schemes for equations of this sort. You basically split linear and linear parts and you do uh all the work for linear part like more or less analytically which like >> uh >> makes it allows a lot to to handle what is what whenever it becomes stiff.

[01:18:57]>> Yeah. Yeah. No, this is a I mean uh I can I can I speak for myself, right? But this is our bread and butter, right? You cannot solve a PD at least if you if you're aiming for for your analytic methods. uh that's what you go to.

[01:19:15]I mean basically what happens here like the the sketch of a proof you've given it's like more or less sketch of respect to exponential time difference method analysis what >> this idea appears in a lot of places it's a normal form it's a normal form expansion right I I I expect you guys to to be able to you know to be familiar with this idea somehow which is like well you have this equation you iterate it within itself and then you get an infinite expansion. This is a kind of a you know sometimes what people do is like well maybe you don't get an infinite expansion you do like you do three integrations by parts and then you analyze all the terms and things magically work. In our case we need to in order to get to this threshold here we need to do infinitely many integration by integrations by parts.

[01:20:00]Um yeah, but I I expected some of you guys to know >> doesn't go as far as this iteration stops at exactly the first step you written >> and that's why like I I get worried immediately when you that thing that's why here I got my explanation.

[01:20:16]>> Uhhuh. Yeah. Yeah. Exactly. Yeah. Yeah.

[01:20:20]So you see also how how these fer leag spaces will come into play. Right. So here I'm I'm basically using the fer analysis of it all to be able to uh to write it in a very nice equation with not an operator but a multiplier here and then uh integrate by parts.

[01:20:39]>> So it's a question about the question.

[01:20:42]>> Sure.

[01:20:44]>> All right. So, so you you asked us if you can you know maybe improve this to be bigger even bigger or >> Yeah. Right. So if you take in the current formula >> Oh yeah. Yeah.

[01:21:03]>> Yeah. Yeah. So this is >> this is the threshold that you have in front.

[01:21:07]>> Yeah.

[01:21:08]>> To to improve this.

[01:21:10]>> I don't know. It's an idea to to use the key >> that that is you know >> there you go. Brilliant. You have a problem. So go ahead and work on you know I also think so. Then I said to Simone, he basically said, "Well, I cannot repeat his words. They not not at this time at least.

[01:21:30]I'm being recorded. You know, I cannot repeat what he said to me."

[01:21:36]So yeah, but this is a great idea. I think that if you if you kind of, you know, if you're able to merge the techniques that I'm going to show with the complete interability in some sort of way, then you should be able to get to P equals to two. But I don't know how. So these this these seem unbearable, right? So these these seem like this is one world and the the other thing is another different world. Maybe maybe what you need is really like I don't know you get to a certain sort of level then you truncate and you kind of use this the symmetries this complete interability up to a certain point and then you start expanding maybe there is something like that >> yeah I was thinking more about >> I don't know the complete interability techn you kind of it's similar to the parabolic >> right yeah >> so so you basically define a sequence of other equations Yeah, >> maybe you can take one of each of each of those equations on those sequence and try to use the same technique, >> use the patient information, then take a >> brilliant. You see, Philippe, you already got a problem for your student.

[01:22:40]I mean, no, I'm I'm I'm being serious.

[01:22:43]This is a great idea. You should try.

[01:22:45]I'm I'm I'm actually, you know, you know, encouraging everybody, especially Danilo who's asking this question. You should I we haven't tried. So, this is a great question.

[01:22:55]This is generally a great question.

[01:23:02]Okay, if don't have any more questions then um well we are back next week, next Tuesday, right? But then it's going to be at 3:30. 3:30, right?

[01:23:13]>> Yeah. Thank you guys.