Thank you for inviting me. It's been really nice to be in Paris this month.

Really nice talks to watch as well, and just really great, except the weather, but everything else really great. No, I mean, to be honest, the weather, in a way, it's a 100-year experience, supposedly, so that's nice, too, I guess.

OK, so in this talk, I'll talk about the three. So this is the equation that I will be discussing in this talk,

and it's the cubic wave equation in three dimensions. And so this equation is called an H to the 1 half

critical problem, and a lot of people already know this, of course. And in fact, Rohan talked a bit about criticality

on, I think, Tuesday. But in any event, just to get into, just to say what it is, it means that if you have a solution to this problem, or a solution to the equation, then in fact, you have an entire family of solutions, because you can insert a lambda.

And in fact, then if you check, then you'll see that the norm that's preserved under that transformation is the H 1 half norm of your initial data,

and H dot minus 1 half norm of your initial velocity, and thus it's called a H 1 half critical problem. And this is by no means an unimportant thing. Again, as if I remind you of what Rohan was talking about,

he sort of sketched out a way that, for example, this argument that Christ, and Collyander, and Tao used to prove that if you have data that is in fact ill-posed for data less regular than H to 1 half. So H to the 1. So this criticality really is important.

Now, of course, he mentioned some stuff about how in the negative orders, then you sort of Alice in Wonderland type thing or something. But for s bigger than 0, we know that in fact that's exactly

the right space to think about, because we have this local well-posedness result. And this is by Lindblad and Sog, I think.

It's by Lindblad and Sog, who are also from Johns Hopkins. And we have local well-posedness for U0 and U1 for U0 in H dot 1 half.

Oh, sorry. So we have local well-posedness for U0 in H dot 1 half and U1 in H dot minus 1 half. And just to remind what that is, there exists a T of U0 U1 greater than 0,

such that a unique solution, this is sinus T comma T. U

is in LT loc, LX loc, or LX, just LX. And the solution depends continuously on initial.

OK, so this is important, because you have this T of U0 U1, because it can't just depend on the size. Because if it did, then you could always do some rescaling

to show that in fact you had a global solution. And so then anyway, this is the sort of the, I don't want to talk too much about the functional analysis and all that of this, but in any event, this is sort of the standard definition of local well-posedness.

Now, the goal that I was hoping to prove, and it's not done yet, is to prove that there's

global well-posedness and scattering for this problem. That's the goal. So global is hopefully pretty clear from local and then make it all R, right? But then scattering is that there exist U0 plus or minus in H dot 1 half and U1

in H dot minus 1 half, such that U of T minus S of T U0

plus comma U1 plus H dot 1 half cross H dot minus 1 half goes to 0 as T goes to plus infinity.

And then the same minus infinity S of T is the solution operator to just the linear solution.

So we have our solution. So we have a solution operator. It starts to look like a linear equation and the non-linearity is just the perturbation of the linear equation.

So this is the goal of the defocusing case. So focusing, we know that that doesn't happen. We know that because we saw a talk about that. We saw a talk about some type 1 blowup solutions and blowups on a pyramid and other things like that.

And but that was for the focusing problem. For the defocusing, there's no such counterexample. So this is what I think a lot of people believe the defocusing problem scatters.

And in fact, that's equivalent to saying that U is in L T x4 R x3 plus 1.

So this is equivalent to scattering. And yeah, it's a nice exercise to try to show that. But it doesn't take very long.

I'm not going to do it. So then we have then two types of blowup, two types of ways that our function, our solution, can blow up. And those are, appropriately enough,

called type 1 and type 2 blowup. So the first type is that you have the H1 half

norm going off to infinity. T plus is the, you've got some maximal interval of existence. And T plus is the endpoint of that. And we're saying that the H1 half norm goes to infinity

in one of those two time directions. So why is that automatically, I rule out scattering, right?

Because you just look at your wave operator, and you see that, oh yeah, so sorry, I didn't write it. I'm not going to bring it back down. But I should have wrote a plus or minus in front of the U1 as well as the U0. OK, but I'm not going to bring it all the way back down just to do that.

OK, that's what I should have done. But you see that, I mean, your wave operator is a unitary operator on subless bases. So if U of T goes, if this norm goes off to infinity, then automatically the game is over, right? You can't scatter. But even still, it might still fail to scatter,

even if this type 1 blowup doesn't happen. And that's called the type 2 blowup, where you have less than infinity of the soup.

But nevertheless, you still have the L4 norm equals infinity. Yeah, so these are sort of our two types of blowup solutions.

Now, at this point then, I want to put this then in the context of a couple of other problems that have been solved by a number of people. And the reason why I want to do that

is to sort of talk about what we have and what we don't have here. So basically, we have these that the results, for the most part, at least for the defocusing case,

ignoring the leaving, the focusing, there's of course the channels of energy, which is, I think, a little bit different than the, or I should say it's a little bit, it seems to me to be a new idea. So that's different from how I've done some other ideas.

So because it's primarily analyzing linear behavior, right? It's analyzing linear behavior at the exterior. So, but in general, in any event it's,

so let's say elements in common between the NLS and nonlinear wave equations, let's say. In the elements of proof, for example,

you have the energy critical wave. So this is UTT minus delta U in 40, right?

In 40, then if you do your rescaling, you're going to have the norm is h1 cross L2.

Critical, and this is in 40, this is 2D.

And then the h dot 1 half critical NLS, which is in 3D.

So for these types of problems, the elements of proof have generally centered around conservation laws.

And then from that, then you have a sort of a stress energy tensor along with that. So for example, the energy critical wave equation.

So when you solve, for example, the energy critical wave equation, right, I mean, like let's say you have radial data. So you have your Morowitz estimate is bounded by the,

and then you have an energy. So in the de-focusing direction, you have, you know,

you combine your conserved quantity and your Morowitz estimate, and then you just use the, for the radial case, then all you would do is just use the radial cell

blood embedding, right? And then you'd integrate it and you'd get an L5 bound. And then, but then for the non-radial energy critical wave equation in the h1 half critical NLS, so these are works by Kenning and Merrill, you have that,

you know, do you have the profile decomposition, right? So you find a minimal blow up element and then you show, well, this is a minimal blow up element that can't be concentrated because again, you have your Morowitz estimate, whether that's for the wave equation or the Linn-Strauss

Morowitz estimate. So yeah, and then, and then of course, then there's the work of the energy critical NLS and mass critical NLS as well. And you know, a lot of different people have worked on this in the audience and so on and so forth.

But I think that the interesting thing to me about these problems is in a way they're very much type two blow up results. If you really boil, excuse me, if you really boil down to it,

oh, I have to move, okay. It's your cloth, it's scraping against the microphone. It's not feedback, it's not feedback. Now you can, it's okay. Move it up your collar. No, no, no, no, no, no, no, no. Not feedback. Okay, well, maybe we'll get some feedback on that

and see if it, if it. It's the wave equation in any case. Yes, that's right. We're studying the wave equation, doing an experiment right now. But in any event, this has been very interesting to me because we have these results of, for example,

we have these, or let's just say these three results for the energy critical wave equation, energy critical NLS, mass critical NLS, we have a conserved quantity, right? We have a conserved quantity, whether that be the mass, in the case of the mass critical NLS, that's just U squared, or the energy for the energy critical NLS.

And so automatically we know type one blow up doesn't happen, at least in the rate, at least in the defocusing case, right? We didn't really do anything. I mean, whoever came up with these conservation laws, of course, did something, but there's no, it doesn't seem to me like to be very,

that's sort of given to us, but then we have to prove type two blow up doesn't happen for the mass critical or for the energy critical or what have you. Whereas in the case of, but then, and then recently people have been working on now results for

where you don't have these conserved quantities, right? Which is, of course, the result of Kenig and Merrill as well. They just assumed that the H1 half norm was uniformly bounded. But the interesting thing now is that, in a way,

the only thing that those ever prove is no type two blow up. And some people, and I've gone back and forth in my mind about this. Now you have to listen to me argue with myself because I'm still not completely decided in my mind about this. But showing no type two blow up, it's really the same issue, right?

Whether it just so happens that these problems have a conserved quantity and that, for example, the H1 half critical NLS doesn't have a conserved H1 half norm. But, I mean, like when I did the mass critical NLS, I didn't do any work to show that the mass was conserved, right?

I mean, that was easy. So, I mean, I figured out like first day I was doing that problem, right? You just integrate it, right? You integrate it by parts. Oh, by the way, so just as a reference to Vlad, we're talking about, in all these cases, we're talking about, we can approximate with smooth data, right?

So we never have to worry about like what Vlad talked about with, you know, do you have a conserved quantity? Because we're dealing with strict arts estimates and all these results are perturbative, we don't ever have to think about a situation where we might not be able to integrate by parts or something. We can always integrate by parts as much as we want. Yeah, but then the other thing that's interesting then

about problems where you don't have a conserved quantity is that you, in a way, you have these conserved quantities actually give you two things if you really think about it, not just one. They give you a conserved norm,

but they also give you the mechanism by which they're conserved. So for example, this H 1 half critical problem of Kennegan-Merrill, they have a momentum that's conserved. Well, that doesn't control the H 1 half norm, but they have conserved momentum, which is used for the Lind-Strauss-Morwitz estimate, right?

And that's why I talked about this energy critical radial problem first. That's really easy because in this problem, you have both. In the radial data, it's very clear that you have both. You have conserved H 1 half, H 1 norm, and you have the mechanism, you have a positive definite stress energy tensor, which gives you a Morwitz estimate.

You just put them together and use the fact that the radial symmetry, you use it and it's, I mean, this is the proof. This is the whole proof of the scattering for the radial wave equation energy critical. And so then the question is, can we extend these results to other situations

where we don't have a conserved quantity anymore? And that's a, to me, that's an interesting question because, and so the type two blow-up results are in a way removing the harder of the two obstacles, but it's still an addition, an obstacle removed

because you've removed away the, you don't have a mechanism by which your conserved quantity is conserved, and thus you have to do some more work, I think, to build up to it. You know, even the mass critical NLS,

you have the conservation of the L2 norm, which is extremely useful for the interaction Morwitz estimate. It's extremely useful. Anyway, I hope I haven't said anything controversial, but if not, I will move on.

I suppose if I have, I should just move on too. From now on, I'm talking about results in the radial case. Let me set up the spot again.

In the radial case, and there's a result of myself

and Lowry that proved no type two blow-up. And this is for radial data. So we know that there's type one blow-up for the focusing problem.

We don't know about the defocusing. We know about the defocusing. We know there's type one blow-up in the focusing case, but we show that in either case, there's no type two blow-up. And the second result is we have the cubic scatters.

So if I can make a brief mention, again, I'm very happy about Caleb going before me because now I can just mention. So we have data in h to the one half plus epsilon,

but now look, this x to the two epsilon, that's equivalent to saying it's in, that's like saying it's in h to the dot one half minus epsilon, right? Because you have one half plus epsilon minus two epsilon, right? So we're just slightly around the critical regularity, both lower and higher, right?

So that's good. I mean, we want to, we don't want to be, because of our scaling, we want to have a norm that somehow interpolates the h to one half norm, or interpolates to the h to one half norm, but then we also,

so we have just an epsilon, right? So if epsilon could go to zero, that'd be great, right? Because that would mean that that would be the whole result, right? But of course, as probably most people know, just removing an epsilon, not always so simple, right?

It's often the hard part. I guess now it's the hard part. But in any event, you know, so yeah, so let's see. So let me discuss how these results are proved.

So the first result with Lowry is proved by concentration compactness. So we have a quantity, and this is just a L2 inner product,

u sub t equals the energy, right? This is the result of energy. So you've got this, this gives you a nice plus conservation of energy

plus additional regularity shows that your minimal blow up solution satisfies

e u of t equals zero. So maybe this is even the wrong order. So first you show that, in fact, your minimal blow up solution not only is in H1 half, it's in H1. And then from there, then you know, okay, that energy is always conserved. And then you've got this nice Morowitz inequality,

which says that you can, and then you take a time out, because it's a derivative. So you take a time average and estimate the endpoints and then say, oh, well, then that shows actually the energy has to be zero. And then so in the devocusing case already, you know, then that means u is zero. In the focusing case, then you have a result of Stovall and Kilip and Vichon

that say, well, if its energy is zero, it has to blow up in both time directions. But you've carefully chosen the minimal, you've taken the minimal blow up solution that doesn't blow up in both time directions. So as I like to, as an analogy, I like to take an analogy to the Sir Walter Scott, who said,

ah, what a tangled web we weave when first we practice to deceive. And now it's falsely attributed to Shakespeare a lot, but I looked it up once and it's actually Sir Walter Scott. So what it means is that, so basically the idea of these concentration compactors, you'll say, okay, let's assume that it doesn't blow,

that it does blow up, there's a blow up solution. Well, then you've got this, then that actually means you've got a whole family of blow up solutions because of your scaling and things like that. And they're compact, so you can take limits and get another solution. And eventually this blow up solution is caught in its web of lies that it's created. And then you realize that it's not true.

And so then it's done. And usually the more people know about concentrated compactness, the funnier they think it is, or at least that's what I, maybe it's the heat, I don't know. But I try to tell myself that it's a good analogy. But this is the standard recipe, right?

But what I want to point out is that for this type of thing, we have to get additional regularity, right? Because we don't have any Moore-Witz estimate that scales at the H to one half level in the wave equation.

So we have to do this work. But then the other proof is a bit of a different argument because this actually uses the I method quite heavily.

And along with the hyperbolic coordinates. So this is a work that at least goes, doing it in a hyperbolic coordinates at least goes back to Tataru,

who proved some nice estimates for data that started in a compactly supported region. So let's take our data that we have that's in this norm over here. Well, we know that it is radial. So we know that, so what Tataru did was he said,

well, okay, so here's our light cone. And let's say we start with data in here, in this part region here, and then they evolve forward in time and we know it's going to stay inside this light cone. And thus it's inside this, if he draws a hyperbolic,

it's going to lie, and Tataru's wasn't a radially symmetric result, but it's going to lie inside this hyperboloid then. And so if you calculate then the wave equation in hyperbolic coordinates, this is going to get you some nice estimates on the wave equation. But for this result over here, we don't know that it's compactly supported.

But what we do know is we know about the finite propagation speed. And so we know that outside of a ball of radius r, our data is going to be small, just from reading off of this. We know it's small h to one half norm outside that ball.

So in particular, out here, the L4 norm of our solution is bounded. So then we're just left with, back in the situation

that Tataru, for example, had to deal with, where we're just inside this light cone now. And then at that point, then there was a good observation of Staffolani and Rupeng Shen, who realized that, in fact, if you were to shift to hyperbolic coordinates,

so you have a V of tau comma s equals e to the tau sinh s over s times u of e to the tau sinh s e to the tau cosh s.

Then if you shift to this, and then you want to get your data on tau equals zero, right, because then V sub tau, V is going to solve minus s over v cubed.

It's going to solve a cubic-like equation.

But then out here, again, and I drew the picture wrong, because these branches should go towards slope one. They should have slope one as you go along, but I didn't draw it right because I don't draw very well. But nevertheless, it's going to be pretty well approximated by the wave equation,

by the linear wave equation evolution, just because of the fact that you're starting with small data, and so your remainder is going to be a small iteration of that. Then there's a result by Staffolani and then a further result by Rupeng Shen,

who observed that. In fact, look at this. So you have that s of v of tau comma s equals one half u naught e to the tau

minus s times e to the tau minus s plus one half u naught e to the tau plus s times e to the tau plus s plus one half e to the tau minus s times e to the tau plus s

u1 of s prime s prime ds prime. And again, we're talking about s big, so we're talking about out in this region,

so we're approximating with our linear evolution. Of course, in here, we're going to have a little bit more difficulty, but we can handle that. Then in fact, this wave equation has an energy e v of tau,

so d to the fourth times s squared ds.

So I might have forgotten an s squared ds somewhere, but that's the polar coordinate integral. We have this energy, but now if you were to calculate the H1 norm of this energy,

so you calculate the H1 norm of this energy, so you would take the derivatives of s times v of s and integrate it from zero to infinity, and then you'd have another term left over from commuting your derivative in space with the s, but same thing.

I mean, it all works out. In fact, if you just directly calculate the derivative of your s v s, it's bounded by integral of u naught and then do a change of variables. It's bounded by u naught squared r cubed, or u naught prime, square r squared times r cubed dr,

and then plus integral of r cubed u1 of r squared dr. In fact, this is precisely your,

if you write it out in just axis, this is a norm that scales like H to the one half, right? So the energy is controlled by a norm that scales like H, the energy of v is, at least out at the wings, is controlled by a norm that scales like H to one half.

And then if you tried to, and then if you played with the ball, you know, the energy, you could rescale, of course. And so if you have u, you have a finite energy, what Shen proved is if you have a finite energy u that also satisfies this, then you have a global result that scatters because he has this energy

and then he also has a Morowitz estimate. But it's at the right scaling, right? It's at the scaling of H to the, it's the Morowitz estimate that's bounded by H to the one, but that scales like H, where the energy scales like H to one half.

And you have integral of v to the fourth times s squared ds, or s squared times sinh or cosh of s over sinh s ds d tau is bounded by your energy.

So, okay, so that's how it scales. But now, but you can do the exact same thing and show that this norm gives you good control over the H to the one half plus epsilon norm on tau equals zero as well. Well, now at that point, what do you do? You've got the H to one half norm controlled, H to one half plus epsilon norm controlled.

You want to use this Morowitz estimate, which clearly controls the L, after you do a, oh, sorry, sorry, s over, after you do a change of variables, this controls the L4 norm of u inside this cone.

So then there's one last thing you have to do. You have to show that your energy, your H to the one half plus epsilon norm is somehow propagated through. And you have to, well, you have to cut off in frequencies and show that, and get a Morowitz estimate and show that that energy of your cutoff,

both one, gives you a good Morowitz estimate, you know, controls the errors pretty well. And two, also the energy is bounded. Well, how do you do that? Well, that's where the long-time Strickart's estimates come to save the day. And they get you good control on the errors, and then you get a nice bootstrap going,

and you can control this for a Fourier truncated piece. And then the long-time Strickart's estimates just say, well, then the linear behavior, the behavior is basically linear at higher frequencies. So you put it together, and you're done. So anyway, that's probably about it.

Do you have any idea of the dependence on epsilon,

about your constant? I would guess it's probably like,

at least one over epsilon to a power of some sort. But it could be even worse, yeah. Yeah, I'd have to get back to you on that.