 I'm very glad to have this opportunity to speak here. And as you can see later, the title of my talk is very much related to this workshop in the next slide. But before that, let me just mention that I will talk about self-similar, discretely self-similar, and their rotating versions. So these are special solutions of Navier-Store equations. So the connection to this workshop is that this name was learned from this workshop. So this name didn't exist for Navier-Store's community before I attended this workshop in 2009. So that's the connection with this workshop. I was very happy to find the name finally. And that talk is in cosmology, which I will explain later. So these are my two collaborators. Mikhail Kolopkov, he is a young guy in Russia based on the Biasq. He is an expert in new analysis. And Zagre Brescia, he is a postdoc at UBC. So he has several joint work with me. I can just online. So first I will give an introduction. So I will give three constructions for self-similar solutions. So the first one is all of them are based on a prior bounds. But the pros are different. So the first construction is based on herd estimate near individual time for local delivery solutions. So that was first done by GRSFRAC for self-similar solutions and then for myself for DSS solutions. And this construction works only in R3. And the same construction works only in half space. And the prior bounds are based on method of contradiction and reduction to a real field theorem for all the equations. So this is a classical method invented by LeRae. And so it's a joint work of Kolopkov and myself. And this works only for self-similar solutions, not for DSS solutions. And the third construction is based on a new explicit applied estimate. So it's a weak solution approach. And it works in every cases. So self-similar and discrete self-similar and also whole space and half space was done last year. And then we work out the rotating version this year. So my original title did not have this part. But CJ was, in my audience, three months ago in a talk in Harvard. So I decided to give this actual new part to make this more interesting to her. And finally, I will remark on the backward case. So here's the table. So with three methods, they are based on different things. The first method is based on non-field stock equation itself. The second one and third one are based on the LeRae equations. And so this is a, OK, so the applied bounds are put in different ways. So the first one is based on the graduate criteria. And so the bounds, we get is a point-wise bounds. The solutions here are strong. The second way is based on contradiction and Euler equation. It's H1 bound. And the last way is a explicit cut-off. And it's usual energy bound. OK, so introduction. So here's the usual non-field stock equations in R3 times positive time, with the usual data U0, divergent free. So here are the notation. And you drop a linear term, we call it a stock system. So it has two basic properties. The first property is the applied bound. The linear term is anti-symmetric. So you multiply this linear term by 2 U integrate, it disappears. So this is a usual applied bound for non-field stock equations. And here we don't see the linear term. So it's the same as the stock system. And it was based on this applied bound, the LeRae contracted solutions in whole space R3 for any L2 data and hope did it for domain case. And now open questions, since LeRae's work is uniqueness and regularity. And the second property of the equation is a scaling variance. So if I have one solution, and for any lambda greater than 0, I can define U0 superscree lambda to be given by this formula. Then this U superscree lambda is also a solution. And related to this property is the ideal mild solutions. It's treated a linear term as a source term to a stock system and contract solutions by the Picard iteration. And so if U0 is in LQ, then that's a unique local in time solution called mild solutions. And for Q equals 3, this solution would be global if L3 no is small. And so now comes the self-similarity. So it would say it is self-similar if U and U lambda are equal for every lambda. So in this case, the value of U decided by its value at any time moment. And then the discrete version is that U equal to U lambda only for one lambda. This does not need to be for every lambda, but just for one lambda. So in this case, the value of U is decided by its value in this time interval t between 1 and lambda squared. So maybe draw a picture. So for self-similar solutions, the value of U is decided by the value of this time instance. And then the value of U at any later time is decided by a corresponding point at the earlier time. And then if it's discrete self-similar, then the value is decided in this time slice. So self-similar object appears in minimal surfaces and gas dynamics. And discrete self-similar object appears in factors. So a counter set is discrete self-similar with factor 3. And in cosmology, so here's a quotation from a survey paper. It says that there's evidence that discrete self-similarity occurs at the mass threshold for the formation of a black hole. So when I look at this mass threshold, I think about the work of Burrow and Kenick. But anyway, so I don't miss them in the talk here in 2009. And this is also basic. It's a Schaefer's example, which I will explain later. And so the forward and backward distinguishes positive time and negative time cases. And when it is stationary, we say stationary, there's no time dependence. So here, we'll take a parameter a to be 1 minus 1 and 0 correspondingly. So it's different from other equations discussed in this workshop. Navier-Stokes equation is time irreversible. So forward and backward are different. And so I want to talk about the decay of a typical solution. So if the self-similar, or if the u0 is self-similar, then it's very simple. u0 will be, if u0 is self-similar, then u0 of x will be equal to its value of uniball divided by its length. So it's mass homogeneous. And so it has to have this decay. And so the natural space for u0 is L3 weak. And the corresponding forward solution will satisfy this point-wise decay. C0 first goes to t plus x. So this t could be part of the negative. But here, I'm talking about the forward solution. And so because of this problem, u is already beating in time and L3 weak in space. So a backward solution, satisfyingly above bound, is called the type 1 singularity, if it is singularity. So the existence is an open question, except actually a similar case, which I did with Bob a few years ago. So I decided to highlight every person who is present in this workshop. So now here's the real equation. And it comes from the similarity transform. So for a equal to plus or minus 1, we define the similarity variables. So u xt now changes to capital U with variable y and s. So here a is plus 1 or minus 1. And y is x over root t. And the scalar time s is a log of t. Then this capital U will satisfy the following equation. And this box part is a new term compared to a log of t equations. So this comes from the time derivative of this guy in this factor and in this factor y. So that's always new. So the behavior of capital U as s goes to positive infinity in caused behavior of little u at infinity. So that means the time asymptotics when a is positive. And it's in caused behavior near a singular time if a is negative. And there's an important difference here. So little u is self-similar only if capital U is independent of s. And little u is discretely self-similar if only if capital U is periodic in s. So a little bit history. So the stationary equation was written down by the Ray in his 1934 paper. And then motivated by this problem, Gigan-Kong introduced similarity variables. And he used that to study nonlinear heated equations. And then it was extended by many people including Merle and Zag and also to energy critical wave equations by Koenig and Merle. And so now it's a bit about the Schaefer's example. So Schaefer in 85 has a paper, which is a factor field defined for negative time over nonlinear stocks equations with a singular force. However, this singular force is speed-reducing. So f dot u is non-positive point-wise. So normally speaking, this f should help to make u smooth. But this f u is actually a singular has a singularity at the origin. And his u and little f are both backward, discretely self-similar. So f has a difference getting low. So this was 85. So that was 30 years ago. So it was there. And so now the question about the existing problem for forward self-similar solutions. So for given self-similar initial data, want to contract the self-similar solution. And the natural space here is a low infinity in time. And then x should be L3 weak or some other critical spaces. And for small data, so when c star is small, and then we have this unique, we can use a unique existing theory for mild solutions. And so this has been done for many answers, including Giga, Miyakawa, Kato, Kanon-Plenchon, Paraza, and Kog-Tatalu in different spaces. So the theory for small data is very satisfactory. So this also includes discretely self-similar solutions and the rotating versions. So now the question is, what want to construct solutions for large data? What puts c star large? So when c star is large, we don't have an existing theory for mild solutions. And the usual low-week solutions does not contain such new data. And so a theory, due to the Mahi Newset, local data solutions is what is useful. But then this is a weak solution theory, so it has no uniqueness result. So for given self-similar data, we cannot guarantee that the solution itself is self-similar. So in 2012, Giga and the Sepharic contracted self-similar solutions for every self-similar initial data, even small or large, which is required to be locally heard continuous. So I highlight Giga because his work on wave equation was a topic of a technique. So what do we care about these guys? So here is one page notes on the reference to the non-uniqueness problem. So we say that the uniqueness problem for the Navier-Store equations for any L02 initial data is an open question. So here is one way to control non-uniqueness. So we consider a self-similar solution w sigma corresponding to initial data sigma times u0. So u0 is fixed. A sigma is available. So for sigma small, w sigma is unique. However, for one increase sigma, what might get by vacation? And by vacation has two different types at least. So if by vacation is a sudden not type, then we will get two self-similar solutions, w sigma and w sigma prime with the same initial data. And we could get the whole by vacation. And then the new solutions would be time periodic in the similarity variables in the linear equations. So they correspond to DSS solutions. And so these guys are L03 infinity valued. And just first show that if we have such solutions, we can prove non-uniqueness in the real-hope class by cut-off. So that's why they are interesting. But so far, it's not easy to prove by vacation. So these are introduction. So what do I do? Can I just? No, it refuses to echo. I don't believe it. OK, this is my part one. So I need to speed up. So it's a first construction using heard estimate in linear initial time. So the basis of GR and SPRIC is they have a local heard estimate for local linear solutions. So it's iterated in a framework of local linear solutions. So which you will give estimate here and also here. It's a short time estimate. But then by self-similarity, they have estimate below the parabola. And then for the middle part, they just use regularity CRV for stationary solutions. And then, the only way to get a prior estimate and they apply the rich auto theorem. So this I skip. For DSS solutions, the story is a bit different. Because in the middle, in this part, it's not stationary. I wanted to have regularity. So I had to require short time. So it cannot be arbitrary. So here is my theorem. I say that if lambda minus 1 is sufficient small, which says that this time interval is sufficient small, respect to the size of data. Or if u0 has some symmetry, then we have a regularity theory to guarantee bounds in this middle region. OK. The rest is the same, the rich auto theorem. And here it's important to note that the regularity gives us companies, which is essential for the rich auto theorem. And then the same construction, we want to work on R3 high space. In high space, there is no local solution theory. The difficulty is that it's very hard to infer the formula for the pressure from a solution if the solution is not very well localized. And then what we do is that we look at the regularity equation directly. And then we want to control the solution. And then so here, this equation is coupled with the boundary condition u0x. u0x is the Taiwan map of the usual data u0. So the usual data for original little u becomes the boundary data for the Ray's equation. And then u0x is declared as 1 or x. What we expect is that the difference here has a better decay is in L2. And so the theory says that, OK, so A is a Stokes operator in high space. For any self-similar, for any self-similar usual data u0, which is c1 local, satisfying the necessary condition, then there's a subunit self-similar model solution in this usual class, L3 infinity valued. And then the difference between the solution and the linearity part is decays in L2, like t to 1 quarter. And our proof actually would work for any call. So the proof works for R3 and R3 plus. If you work for any call, you can prove this data. This image says that for A to be the Stokes operator in omega, u0 is the Taiwan map of usual data, u0. 1, u0 to be L6, and it's gradient to be L2. So this is where we use that u data is c1. So L2 is a better decay than L3 weak. And I don't know any semi-group theory which guarantees a better decay for the gradient. So that's why we can only do this in high space, not in any other call. So this beta decay is essential for our prior estimate. We could do this. We could prove something actually much better using Solikov's point-wise estimate for green tensors in high space. So here's sketch. So we look at the equation for the difference. So the equation for difference is this one. And then the idea is the proof has three steps. So step one is to show that solutions to this equation with a parameter lambda multiplied to a right-hand side has a uniform bound, Cr. And we'll prove this by contradiction. I will mention contradiction later. And second step to show that in the previous step, the constant Cr is actually independent of r. And so if we can do these two steps, we can apply the rich-older theory to get solutions in a half ball r and then take limits r to infinity to get a solution in r3 plus. So that is the lowest method called invading domains. So the key here is step one, two, by basic contradiction. So suppose we don't have a uniform bound. We need a solution vk to have a h1 bound jk. And look at the normalized vk. And then we'll converge it weekly to a solution to the Euler equation, v h1 0 over step 2 h1 half space of Euler equation. But then with this important inequality, v dot greater v dot u0 is at least 1. And we need to show such solutions does not exist. And this problem is easy when the boundary omega is connected. And it's a difficult question if a boundary omega has at least two components. And so there is a design work in a DSS setting. So it does not work in a time periodic case for the Ray equations because the lag of time companies. So I mentioned earlier, for the Ray-Shadow theory, it's important to have time companies. So I'll probably hear the limit of vk hat in the time periodic case does not satisfy any equation because the time derivative just blows up. So it's related to this following open problem. So for any domain 2D, 2D is easier than 3D. But this 2D is open. So for any open domain with the most multi-connected boundary, and then with periodic boundary data satisfying the necessary condition, do we have a periodic solution? And so this is the open question. Part three is my third construction. So it's important to know that this is the weak solution theory. So my data would be quite general. My data is only required to be L3 weak, not heard continuous, or C1. So we start with the DSS initial data U0, which is divergent-free. But we don't require symmetry over small lambda minus 1. So we don't have regularity. So what can I apply the Ray-Schilder theorem? So what we do is we want to study the Ray's system and look for time periodic solutions. So I denote U1 to be the corresponding vector field after similarity transform of this linear solution. So this t, the periodic t, is equal to 2 log lambda. So it's a log of lambda squared. So here's a new observation. So here's an equation for the difference v, u minus u1. So this is similar to what I wrote in part two, which is the time derivative here. This is only thing, which is new. So you multiply this equation by v and integrate, from time 0 to time t. So in this way, this term will drop out. So formally, this left-hand side equal to this right-hand side. So this is quadratic. And this part is cubic in v and u1. So here, this term in this majority color, v dot greater than u1, dot v, is usually the trouble term for non-field story equations. Because this term can be very, very large and cannot be controlled by this term. And there are examples showing that this term is bigger than this term. And what our new observation is that, unlike usual non-field story equations, we have this term here. This is only available for the linear equation, not for non-field story equations. Well, this new term could help us. And also, this u1 has a case like 1 over x. So these two combined enable us to control this term. So this is much better the situation for non-field stocks. So we'll replace u1 by a cut-off w. kasiya basically keeps the outside part for large radius u1. And then this correction term to make w diverges free. And this correction term is global. It's not localized. It's convenient this way. So w is divergent free. And its difference with u1 is coming from this part. It has a decadent y to a negative 2. So that's acceptable. So for error 3, OK, I can say that u0 is in error 3 in this annulus if u0 is in error 3 weak. That's a theorem for DHS factor fields. So if u0 is in this space, then we can prove that the LQ term of w for any q bigger than 3 is actually decaying when r goes to infinity. So when we take r large enough, w is small in LQ. So for a difference term, this v dot gradient v dot w turn is controlled by this. And then this term is not small. And then we get a prime bound. So this is a new observation for the Ray equations. So again, this term is important. Without this term, here one can only take error 3 weak, not anything else. So here's our theorem last year. So for any divergence free in your data, which is DSS and in error 3, v lambda minus b1. So as I said, this is equivalent to that. It is in error 3 weak in the whole space. Then there's a local solution, which is also DSS with the same lambda. And the difference between u and the linear path in error 2 is decayed like t to 1 quarter. So what is the interesting of this solution is that, yes? That sort of position that replaces like the initial condition, right? You are saying like. That's the initial condition. You know when you write this, and you are also saying that u of u at equal 0 is equal to u 0, right? Let's give it by this, right? Yeah, so is it enough to say u at u is equal to u 0 at 9 0? It depends on which sense. You have to talk about your definition of initial data. So usually, this is a global norm. But for local solutions, for example, is that the converges error 2 in any compact set. So it's a weaker. It's OK to be a weaker. And for the theory of error 3 weak, it's usually weak convergence is sufficient. But here, this is stronger. This is error 2. So you have a strong information. Right. But it's so similar, or DSS. So any other question? I have recovered my time, so I should slow down. So that's an interesting scenario. So only in this part 3, this serve construction, the solutions could be singular. So the previous two constructions, the solutions are all have a point-wise bound. But this solution could have singularity. And suppose it is singular at a particular point, x0, t0. And then, by DSS property, it's also singular at all this risk-gilled spacetime point. So it's from time 0 to up to time infinity. So we say that uxt does not become regular for t sufficiently large. And its after-statement is called eventual regularity. So this says that it's possible for a solution to lose eventual regularity. So in contrast, there's a simple theory saying the following. So any local linear solution, any, is any. We don't have any structure, but we just need data to be in L3, R3, not L3 weak. Then this solution must have eventual regularity. So it's this borderline difference which allows the failure of eventual regularity. So at the beginning of my project, I was actually trying to prove the opposite. I want to show that any local linear solutions with new data in L3 weak has eventual regularity. And then I realized that it's not possible. Schedule proof. So this, so I need local linear solutions. So I need pressure. So this is usually the Ray multiplication. But I do this in the Ray equation side. I do multiplication here. And then I had to solve a periodic, time periodic solution from 0 to type of t. And then this is in a gargoyle approximation. And so the Ray equation has some damping. So the size of v at the type of t is less than some sigma, less than 1, theta less than 1 per constant. And then one can use for all of this point to construct a periodic solution. And then take a limit first in k. And then we still have epsilon, back to physical variables to recover the pressure p epsilon. And then take the limit. So here we could call slim, who is somewhere here. But there are many people who did these kind of things. So that was part c. So I talk about all three constructions. So the part four is a new part, notative versions. So here I need to fix my notation. My notations will be notation about the x-axis. So you just discuss. And so these notations, they commute with each other. So their derivative has a simple form. It's jr or equal to rj, j, just this matrix. So these kind of solutions was first proposed by Grisha Perlman for backwards solutions a decade ago as a candidate for singularity. He asked, does this kind of solution exist in the critical space? But they both couldn't solve it. So here I managed to set Yifam. So I was working with him on the new Schrodinger equations at the time. So I have set it all three organizers of this workshop in this transparency. Is there a fourth one? No. OK, very good. So here's the definition. We say u is a notative self-similar. If for some alpha real number, uxt is equal to this formula for every space point for any time and for any scaling factor lambda. So here the new part is this notation part, outside and inside. So you can see that first they have opposite sign here. And then they are both. So it's a number of alpha times the log of this scaling factor lambda. So we actually mentioned why log. So the choice that the angle theta of lambda is for alpha log lambda is natural because we want this to be true for every lambda. And so because of that, we need the sum of theta lambda plus mu has to be equal to theta of the product. So because then this gives us this formula here. And then notative self-similar solutions is always discrete self-similar. Just choose any lambda, such that alpha times log lambda is 2 pi times the integer. And if alpha is 0, it is reduced to self-similar. And then so this is a formula here. Notative self-similar. So this formula works for forward case and also for positive case, for backward case. So now for forward case, we say lambda to be t to negative 1 half. Then this u is given by this formula here now. So what it says is that, so this is time 1. So it's the same as this picture. The value of u is determined by its value at the end of fixed time, in particular at time 1. And now rotate it is quickly self-similar factor fields. So its definition is actually easier. So we need two numbers, lambda and theta. We require UST to be equal to this formula here. So we are theta inside and are negative theta outside. And this is true for ABXT, but only for 1 lambda. So there are some particular cases. If theta is 2 pi times the integer, or then r theta is just identity. So it's lambda dSs. If n theta is 2 pi times the integer, then u is dSs with vector lambda to power n, which is iterative formula n times. And then rotation part will return to identity. However, if theta over 2 pi is irrational, then u is not dSs. In this case, I think it's quasi periodic in the Ray equation side. And so for any time positive, we can choose tau between 1 and lambda squared, which is lambda to 2k times t. And then UST can be expressed by the value of u at time tau. So u is also decided by its values at the slice from 1 to lambda squared. So now, similarly, transform. So first, u0 is the rdss with vector lambda. And the phase theta, we have to choose the alpha. And this alpha is not unique. We can choose any alpha k for any integer k, just to be this guy. And then we define the new variables. So this part is the auto one. But now we do a rotation here, r, alpha s, outside of v. And then in the inner variable, we'll do an inference rotation. This is an inference rotation of y. And then the equation for original Laffy-Stoy equations, u is equivalent to this new equation. So this part is there for the Ray equation. And this box part comes from the rotation. So y is from outside. y is from inside. So at the beginning of my project, I chose z to be y. And thankfully, I thought that that's a rotation. That's good enough. But then the Launier terms does not change well. So they have to be coupled. Otherwise, Launier term is not preserved. And divergent free-taste is also not preserved. And this is actually quite natural. So I forgot to put in the slides the study for rotating fluid with a rotating obstacle. It's a very well-studied object. So rotating obstacle in fluid. So if you do Google search, you will see several papers on this subject. And there, I think they also have these two terms with different forms. But of course, they don't have these two terms coming from self-similarity. So the new terms, OK, they are new. But they are divergent free by direct computation. And the new terms are suckled to v in no two sense. And so it will have the same applied bounds as the Ray questions. And then RSSU correspond to stationary v. And RDSSU correspond to periodic v. And then so the proof itself is not very hard for this theory. So it's similar. So for any L3 week, usual data U0 in the whole space or in half space, if it uses RSS, then there's a RSS solution. This is a typo. This is RSS. U0 is RDSS. Then there's solution, which is also RDSS. So the only thing I require for data is L3 week. So that's everything on my work. And so I just give two pages like a new mark on the back work case. So what I know is that in the back work case, self-similar solutions do not exist. If U is L5 in time, time here is a negative L3 week. So this was done by my advisor and his advisor and my academic uncle in the L3 case. And then I did a case for L3 week. And then DSS solutions do not exist. If UT is L5 and L3 in space, so this is a separate work of Escalariza, Serakis, and Seferik. But then the existence of DSS solutions in this class is unknown. So here I also mentioned that Schaefer's example of single factor fields DSS. So these are basically what are known. And then the second page, the open problems, problem two. So problem one was the existence of periodic solutions for multi-connected boundary. So problem two is this. So existence of discrete self-similar solutions satisfying this critical bound. And the special case of problem two is the problem of Briecher-Perilman. He said that the existence of rotated self-similar solutions satisfying the same bound. I put question mark here because I didn't talk to him. So what I learned are a few sentences from Serakis. So this is my guess. Here are a few remarks. So are SS solutions you can deal with a stationary equation? So that is good. However, for this stationary solution, there's no maximum principle unlike the case studied by my family. So this is very hard. Finally, problem three is I think it's possible to improve Schroeder's example from DSS to self-similar. But I don't know how to do it yet. OK, and that's the end. Thank you. Right, do we have questions? I just wanted to go back to part one of your talk. You mentioned that you needed these estimates on U log in L6 and gradient U log in L2, yes. And you claim that you know it from the half space because you know the half space, but you don't know on the cone. So what are some difficulties when you're trying to show it on, say, certain types of cones? I say quarter space or? Yeah, I thought about a quarter space. However, Solonium of estimates for green tensors is already very difficult in half space. It's a, his estimate is not complete. His estimate is only for divergence free initial data, which has zero normal component. But then, so that does not apply to non-field storage equations because the source term V dot gradient V will be on the right hand side, and it's not divergent free. So even for half space, the available estimates are very limited. And I don't know any estimate of this sort in quarter space. I know nothing about the non-field storage equations, so that might be a stupid question. But at the beginning of the talk, you have mentioned that your BSS solution is related to the formation of black holes and some applications in cosmology, right? Yes. My question is, I noticed that we are looking at incompressible fluid with viscosity. And in an incompressible case, I think the speed of the sonic speed is infinite. Is this a conflict with the principle of general or special relativity? I think self-similar solutions are. Is there any some properties of finite speed propagation to be in the region of general or special relativity? So your question about general relativity. So general relativity is, in some sense, tiny reversible. But we have self-similar objects in many tiny reversible systems. So that's my question, but how is the relation of this? I say it's not really related to. So I mean the dissipation, the presence of dissipation is not a deciding factor for self-similar objects. Actually, OK, it may have less flexibility. But actually, without dissipation, there is a bigger chance of self-similar object. That's my opinion. And is there some possibility to regard, for example, the compressible fluid? Compressible fluid, yes. It's because compressible fluid in the region, you have finite speed of sun. I see gas dynamics is compressible, right? Is it to relate your rotating self-similar solution with this problem, with the obstacle? The problem you have on the board, the obstacle problem of the obstacle? This one is different, because for rotating obstacles, the actual is rotating. So they choose a coordinate fixed to the obstacle so that the domain becomes the exterior domain and the fixed. So it wasn't necessary for them. OK, so there is no direct relation, right? No, but the derivation of equation are the same. Thank you.