A De Giorgi Argument for L∞ solution to the Boltzmann Equation without Angular Cutoff
Yang, Tong
In this talk, after reviewing the work on global well-posednessof the Boltzmann equation without angular cutoff with algebraic decay tails,we will present a recent work on the global weighted L∞-solutions to the Boltzmann equation without angular cut off in the regime close to equilib-rium. A De Giorgi type argument, well developed for diffusion equations, iscrafted in this kinetic context with the help of the averaging lemma. Mores pecifically, we use a strong averaging lemma to obtain suitable Lp estimates for level-set functions. These estimates are crucial for constructing an ap-propriate energy functional to carry out the De Giorgi argument. Then weextend local solutions to global by using the spectral gap of the linearized Boltzmann operator with the convergence to the equilibrium state obtainedas a byproduct. This result fill in the gap of well-posedness theory for the Boltzmann equation without angular cut off in the L∞framework. The talk is based on the joint works with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun.
 Okay, thank you. Min Hai and Si for the kind invitation. So my great pleasure to give a talk in this conference, even though I cannot go there. So today I will talk about a joint work with Ricardo Alonso and Yoshio Morimoto, Wei Yang Shen. Okay, and then the main film that I would like to present today is to show that in the perturbative framework, okay, so if we assume the initial perturbation, VL, VTL in VLOM is more than the solution, it sits in the same function space. So this approach is new compared to the existing approach for assistance theory in the perturbative framework. Okay, so this is the main result. So all of you are experts, so I don't think I need to talk about the equation. So before I go to the L infinity solution, I'm going to recall the work on the L2 first. So both these two results in fact are for the patients with algebraic decay, decay tail in the velocity variable. So even though the back of the decouple is decay exponentially, but the perturbation has much slower decay. So these are the three collaborators, Alonso in Texas A&N, Katta and Morimoto from Kyoto University and Wei Yang from Simon Fisher University. So this is the equation, we all know the Boltzmann equation, and then the purpose of these results are for the case when this course session B has singularity, angular singularity. So the angular singularity in fact is assumed to be in this form. So it has two parts, kinetic part, v minus v star to power gamma, and we assume that gamma is positive, so we consider only the half potential case. So the singularity in the angle covers the full range. So the power of theta is negative 2 minus 2s. So in the results, we consider the case when s is between 0 and 1. So this is motivated by the inverse power law and so on. So now there are a lot of work here, only this very few, about searching for function space with very minimal regularity in space variable x. So in the boundary domain, there's a series of work initiated by Yan Guo using the L2L infinity approach. So a lot of very good results. And then in the whole space, we started many years ago with Alessandro, Morimoto, Seiji Wukai, and Cao Jiangxu, and trying to show that the perturbation, if the perturbation is in xs, s is greater than 3 over 2 because it's three dimension, L2 in v, and try to find the global assistance. But at that time, we fail and only got the local assistance. So recently, there's a very interesting result by Renjun Don, Shuangqian Liu, and Sakamoto and String. And they use this function space, use the vener algebra in x, so they take Fourier transformation in x and take the summation after taking the supermom in t. So using this norm, they can show that the initial data is bounded in this norm and the solution is also bounded. So well-posedness. So in all these results, including some others that I didn't mention, is based on this inequality. So you need to find a norm such that the space is algebra because the Poznan equation is nonlinear. So you have f times g. So in this norm, you have this kind of inequality. So and then use the priority bound, construct some functionals, and then to close the boundaries of the perturbation in this kind of norms. And then for l infinity, if we take x to be l infinity in x, and this kind of estimate is missing for Poznan equation without angular cutoff. So this is true. This is a walk by Seiji Okahi long time ago. Because if you assume the angular is cutoff, then the linearized operator can be written in two parts, a multiplier operator and a compare operator. Then you can write the equation in a mild form and then you can take l infinity norm and so on. So this can be done. But we all know that if the conditional operator is singularity, then you cannot decompose the game part and the loss part. So this kind of decomposition does not exist. So this is the difficulty. So the Poznan equation without angular cutoff is still valid in l infinity function space. This is the goal. So our approach is slightly different in terms of analysis. The approach is different from the previous walk. So what we did is the following. Instead of applying the algebraic structure to cause the booster argument, the idea is to apply the discharge argument to first walk on the l2 estimate on the level set. And then use a localized strong averaging lemma. Then combine these two estimates, we can construct a functional so that the discharge argument works. And then it will give us the infinity estimate on the perturbation. So the l infinity estimate on the solution is obtained by using this approach. So I will elaborate this a little bit more later. So now the discharge argument for PDE and Kaliate equation, in fact, there are a lot of interesting work. I will not mention all of them here. So here I just mentioned for the diffusion equation, there is a paper by Kefirere, and then for Kaliate equation, for example, Lendel equation, for Kefirere equation, when the Kaliate operator behaves more or less like a lapasse. So there's also a lot of work by Gou's, Inver, Moe, Waxiu, and Kim, Go, Huang, and Moe, Gou, Hart. And then in all these work, they show the order continuity and also the Harnet inequality. So for Boltzmann equation, for space homogeneous, there's also a work by Alonso two years ago. So there's some difference between the Lendel equation, Fokker-Plan equation, and the Boltzmann equation, because the Boltzmann equation basically is a singular integral operator. So even though it has some kind of gain of accuracy, but this can be achieved, okay, can be not directly just like the Lapecheun, but through the strong Ever-Vision number. So before I present the result, I also would like to mention there's a big difference between the perturbation in the Gaussian tail. So this is the standard decomposition, the background West Varian square root of Mu times perturbation, little f, okay. So if you simply write the decomposition as Mu past F, and F only has some kind of algebraic decay, so the estimate, the analysis is very different, okay. So for the later, for this algebraic decay, in fact, with angular cutoff, there's a very important progress, Goddani, Michelin and Moe, okay, two years ago, published, was online for many years, and then about the special gap, okay, in some abstract banana space. So without angular cutoff, there's also two group of people obtain the result, okay, a horn, a toron, a chiselini, for mild singularity, and we also did the case for the strong singularity, okay, cover the whole range when s between 0 and 1, okay. So here's the short presentation about the difference between the Gaussian decay and algebraic decay. So if we assume the solution is, can be written in this form, then the linearized operator is a standard self-adjoint, and then you have a very good strong cohesivity estimate for angular cutoff, okay, because this arbitrary long F, L mu F, in fact, gives you the gain of the grade, the s, s is the parameter in the angle, and also gain of moment, okay, so it's strictly negative. Now if you only have polylonged decay, F is mu past F, and then the linearized operator is no longer self-adjoint, okay, so the cohesivity estimate is much weaker, okay, so all you can show is that this quantity has a very good cohesivity term, gives you the gain of regularity, but no gain of moment, okay, because the gain of moment, in fact, comes from the difference, the differentiation of the Gaussian, okay, but for algebraic decay you don't have it, okay. So error term, okay, L2, and gamma O2 comes from the kinetic part of the course session, and in fact in this analysis, you will find out that this integral plays an important role, okay, in the analysis. You cannot use simply the Hs norm with weight, other than that you have to use this as a good term, okay, so this good term is not equivalent to the Hs norm, okay, because the weight is slightly different, upper bound and lower bound is different, okay, but with this you can indeed cause the error rate, okay, so now for L2, the film can be stated as follows, okay, so if L2, if we assume the, we have a regularity, strong regularity in space variable X, so that means the second order derivative in X is bounded, okay, with some weight, okay, so let's, don't worry about the weight at this moment, and then the film can be stated as follows, okay. So we can say that when the parameter s is between 0 and 1, gamma also between 0 and 1, so if the weight is the decay, L2-bay decay, the parameter is big enough, okay, so if the perturbation is small in this L2, H2 in X space and contains no microscopic component, okay, in average, then the solution to the Boltzmann equation exists and converges to the equilibrium, is potentially in time, okay, so this is the L2 theory, so the question is whether we can remove or replace the H2S, H2X by L in 3D in X, okay, so this is the question we want to answer, okay, so to prove that from, in fact, there are three main components, okay, the first component is to show that the moment is, can be propagated in time, okay, so you always, because the net of strong cohesivity estimates, so you always have this kind of error, so, but it can still propagate in time. Now because the singularity, the semi-group generated by the linearized operator, in fact, gives the gain of the singularity H-s, okay, so the L2-bay on solution to the linearized Boltzmann equation, in fact, is bounded by the H-s norm on the initial data, okay, of course the coefficient depends on square-negative, 1 over square t, okay, so we saw by Kotani-Michelon-Mow for cut-off, holds for long cut-off, this special gap, can buy these two free estimates and you can, we can prove the field, I just mentioned, okay, so the idea is how to, now the question is how to, with less this H2x to L infinity, because H2x is too strong in some sense, okay, so this is the film we want to prove, okay, so the, let me see, no, no, okay, this is the film I would like to present today, yes, so the same assumption on the gamma and S, okay, so if we assume the initial perturbation, okay, F0 minus mu, the perturbation is bounded in L infinity and L2, okay, with some weight, okay, for example K log, this is more, and then for higher moment it is bounded only, okay, and K log and K, I don't think I have time to talk about it, but just some numbers, okay, then you can show that the solution exists, capital F, and the solution in both L infinity long and L2 long is volentially decay to mu, okay, so this is the main film, okay, so how to understand the proof, so let me look at this chart, okay, so the main idea is not to show that the L infinity long has the property, the solution is bounded by initial data and then itself squared, okay, other than that we first assume, maybe locally in a very short time, the supermum in both X and T, the L1 long of perturbation is more, so this is assumption, we need to close it, okay, so under this assumption, then we can perform the L2 estimate, okay, so you can obtain the L2 estimate and also the L2 in Hs in V estimate because of the singularity in angle, okay, and the similar estimate can be performed also on the level set of the solution, okay, now with these two L2 estimates, one can apply the ever-vision number that we'll talk about later to have the some kind of gain in X variable, okay, so you have Hs prime L2 in V estimate on both the solution and the level set, now based on this L2 estimate and also the Hs prime estimate, one can construct a functional on the level set and then apply the George argument to show that in fact the solution is bounded in L infinity long, okay, and this long depends only on the initial energy, okay, so then you can close the loop, okay, in some sense, okay, so this is the idea, okay, so to achieve this, there are three main steps, the first step is the L2 estimate on the level set, okay, so here I just give you the statement, okay, so the statement says the following, okay, so suppose this capital G is mu plus little g, you can build little g is given and f is mu plus f, okay, f is the solution we are looking for, okay, so for this capital G we assume he has this, satisfy these two properties, okay, so these two properties are in fact crucial because with these two one can have the uniform co-eativity estimate, okay, L1 and L6, okay, now under this assumption, so if the weight is sufficiently large, okay, and this or fLk plus is the f if the weight v to the power L minus k, the positive part, okay, I think I'll throw it somewhere, I'll throw it somewhere, sorry, and then, so you can have this, yes, so to have the L2 estimate on fLk plus, okay, this term, the level set component, this is the f greater than k, okay, with the weight of course, then to do this, we multiply both sides by this quantity, then you have to estimate q times this one, okay, but under this assumption you can show that in fact this is has upper bound, this upper bound can be written like this, okay, so here you can see that it involves the L1 bound on the little g, okay, L1 bound on the little g, and then by the assumption, okay, the loop, the chart I just mentioned, this quantity we assume it is small, so this is small, okay, small, small, small, so you have a very good term, okay, so this is the error term, and then somehow with this, you, also the assumption you hope you can spread the, you can cause the L2 estimate on the level set, okay, now to apply the eversion lemma, then we also need the L1 norm on the condition operator, and is this one, okay, so we need to show that for the same setting g and capital F, capital G and capital F, then with all these parameters, okay, this kappa is greater than 2, okay, so we want to show that this integral, okay, has a uniform bound as like this, okay, so in fact this is true, okay, this is true, can be proved, we put extra negative derivative in v on this part because we need to kill the singularity in q in order to have a uniform estimate, okay, now with these two estimates, then we can, with this estimate we can apply the famous eversion lemma, okay, so here's the, just a list of the, so the velocity eversion lemma is introduced, was introduced by Gauss-Pertrand-Santis in 80s, okay, so there's a lot of, what in different function space and improvement, okay, and Gauss-Bell-Yong's-Pertrand-Santis and so on, okay, and also Bouchure-Devilleet and Gauss-Bouchure-Purenti and Pertrand, so here I cannot mention all, okay, so here I would like to say that what we need is the localized, localized in time version of the estimate obtained by Bouchure, okay, in 2002, okay, so the estimate can be stated as follows, okay, so if we consider this kinetic equation, Ft plus V dot gradient is F equals the source term, so if we assume that we know that the solution has some vectority in V variable belongs to LP and then you can always find this AS thread, okay, a small number defined like this, okay, so that the following estimate holds, so this estimate holds, okay, so you have gain of vectority in X and T, okay, and it is bounded by the solution at the two endpoints because it's localized in time, okay, and then also the source term, okay, so this Ft, basically is the Q we are looking for, okay, so now we've, of course, there's also some extra term on the LP, now assume we all, we have all, we assume that all this estimate holds, then we can construct a functional and this functional contains the L2 long on the level set and the HS long in V on the level set comes from the covisivity and also the LP long of the LP long S double prime, no, WP S double prime in X, this term, okay, comes from the everywhere in the number, okay, so now if we work on this functional to see how it evolves in time, then you can show that the functional in fact has satisfied this inequality, okay, so this part, this is the, take the supermode in T, so if we, without taking the supermode, you can show that when T2 is greater than T1 and this functional is bounded by initial data LP long and pass this term, okay, so this term, the important part is that this power beta i is greater than 1, okay, and the ai, of course, is positive, okay, there are several terms comes from Q and the commutator and so on, so there's a summation in form, okay, and this is homogeneous in K, now once we have this estimate, then we can take the supermode in T, between T1 and T2, then we have the estimate, we have the estimate on the function, okay, so here let me introduce some notations, okay, so let's choose a K0 to be sufficiently large, depends on initial data, and MK is K0 minus 1 minus 102 to the power K, okay, K is from 0 to infinity, so when K goes to infinity is K0, so if we define the level set FK to be F minus MK with weight positive, and the functional I just define is for MK 0 to T, okay, and then with this estimate on the functional, then we can show that the EK in fact is bounded by the initial data plus EK minus 1 to the power beta or K to the A, okay, but beta is positive, beta is greater than 1, that's the key point, so for the first two terms it's easy because K0 can be chosen large enough so that the level set for the initial data is zero, okay, because the initial data is assumed to be bounded initially, so we end up with this inequality, so EK is bounded by this summation, okay, now for once we have this then it's almost done because it's done because you can construct a E star to be this quantity and then you can show that the E star satisfies the E star K satisfies the equality in the opposite direction, okay, so you can compare by comparison argument then you can see that the EK is in fact bounded by EK star, okay, and what's EK star? EK star by our construction goes to zero when K goes to infinity because EK star this Q0 is greater than 1 because beta i is greater than 1, okay, so this is less than 1, 1 over Q log to the power K, so when K goes to infinity E star K tends to zero, okay, so we have the level set, the functional goes to zero, so the level set the L2 long goes to zero and then this impies the give you the L infinity bound on the solution, okay, now so far this argument up to this point is very locally in time, okay, very locally in time, so basically once you have the local in time estimate by the special gap then the result holds for arbitrary time, okay, so by the special gap and then combine the local L infinity estimate then we have the L infinity bound on the solution go for in time and also have the exponential decay in both L infinity and L2 long, okay, so this is the end of my presentation, so thank you. I have a small question, okay, thank you, hi, hi Tong Yang, hi, I can hear, you can hear me, thank you for the nice talk, I wanted to ask you two small questions, the first one is, so if I understood your talk correctly, you give the first L infinity perturbative solutions in the non-cutoff case, so that's the, you improve the perturbative theory by just using L infinity as the space for the smallness, right, yes, yes, do you think if I'm thinking too, if I'm just comparing with the Navier-Stokes theory, which you know you could imagine that your theorem could be connected to the Navier-Stokes theory by a fluid limit, in Navier-Stokes we can go below L infinity, I'm just wondering if you thought about that, if you think you could go to some LP and if it if it possible to go to some LP, then there should be a critical P maybe, a smaller smallest one so that the perturbative problem can be solved in a well-posed way, so that's the main question and maybe I just said the second one because you maybe you can answer both at the center, I was just wondering whether part of the estimates you use are similar or not to those in the recent papers of Ciri Lambert and Luis Silvestre on non-cutoff-Poltzmann equation, I mean I'm not compared really, I was just curious, so for okay thank you for the question, so for the first question I think this is a very good point, so in fact the L infinity is not necessary, okay, so but I think the argument was also for LP, but for the Boltzmann equation I can so far I cannot find the critical LP, so this is the, so yes, so let me see, so this is at one point I yes I try to understand this LP, we base this by LP but I don't have the precise statement, so this is okay, but in principle your methods can go a little bit below L infinity, in principle, I think so, it seems to me as well, yeah, thank you, thank you, and then for the second question I don't know which paper you refer to, you can look at, well there are two or three papers, it's a series of papers, they are not really focusing on the same goal as you do, so they prove similar things, I participated this program more on the Landau side and a bit with the on the on the moments for Boltzmann, the idea is to prove a conditional regularity, any solution that has, but so it's not exactly the same the same target as you, yes you're right, I know those papers, they are part of the charities rather than the assistance, absolutely, so they don't prove the same theorem as you, but I was just curious because the estimates, the Georgie estimates are similar to the ones you use, it seems to me, I see, but I, yes, so for example for the solution we obtain in this way more, how to show that it is C infinity, I think it's still not covered by the, the C infinity, this is definitely covered in another way in their work, you should have a look, yeah, what they do not cover for sure is your theorem of perturbative solutions, the global existence of perturbative solutions in low regularity space, because they do not focus on the perturbative theory, they do the Georgie-Schauder and infinite bootstrap to get C infinity, right, I know, I know somehow I read their paper and then they mentioned the result covers the solution for example obtained by Thorne-Dewen's frame, yes, but anyway we can discuss this because this is not, okay, thank you, thank you, there are a couple of questions for Ben for us, one on the chatroom, can you read it, can you see it Tom? Oh yes, here you use the D-George argument for L infinity, bang, not the regularity, I know that you work on the Nandau equation and Fokker plan, you have the holder and so, yeah, I'm sorry, I'm sorry, I missed your answer, oh sorry, so yes, we only use the D-George argument for the L infinity, bang, not the regularity for the Nandau or Fokker plan for holder inequality and so on, yeah, but you think it will be useful in this context to try to push the, you think it would be possible to push the the George argument a bit further in order to I think so, yes, yes, I think it's possible to use oscillation numbers and so on, okay, okay, thank you.
 Okay, thank you.  Min Hai and Si for the kind invitation.  So my great pleasure to give a talk in this conference, even though I cannot go there.  So today I will talk about a joint work with Ricardo Alonso and Yoshio Morimoto, Wei Yang  Shen.  Okay, and then the main film that I would like to present today is to show that in the  perturbative framework, okay, so if we assume the initial perturbation, VL, VTL in VLOM  is more than the solution, it sits in the same function space.  So this approach is new compared to the existing approach for assistance theory in the perturbative  framework.  Okay, so this is the main result.  So all of you are experts, so I don't think I need to talk about the equation.  So before I go to the L infinity solution, I'm going to recall the work on the L2 first.  So both these two results in fact are for the patients with algebraic decay, decay tail  in the velocity variable.  So even though the back of the decouple is decay exponentially, but the perturbation  has much slower decay.  So these are the three collaborators, Alonso in Texas A&N, Katta and Morimoto from Kyoto  University and Wei Yang from Simon Fisher University.  So this is the equation, we all know the Boltzmann equation, and then the purpose of  these results are for the case when this course session B has singularity, angular singularity.  So the angular singularity in fact is assumed to be in this form.  So it has two parts, kinetic part, v minus v star to power gamma, and we assume that  gamma is positive, so we consider only the half potential case.  So the singularity in the angle covers the full range.  So the power of theta is negative 2 minus 2s.  So in the results, we consider the case when s is between 0 and 1.  So this is motivated by the inverse power law and so on.  So now there are a lot of work here, only this very few, about searching for function  space with very minimal regularity in space variable x.  So in the boundary domain, there's a series of work initiated by Yan Guo using the L2L  infinity approach.  So a lot of very good results.  And then in the whole space, we started many years ago with Alessandro, Morimoto, Seiji  Wukai, and Cao Jiangxu, and trying to show that the perturbation, if the perturbation  is in xs, s is greater than 3 over 2 because it's three dimension, L2 in v, and try to  find the global assistance.  But at that time, we fail and only got the local assistance.  So recently, there's a very interesting result by Renjun Don, Shuangqian Liu, and Sakamoto  and String.  And they use this function space, use the vener algebra in x, so they take Fourier transformation  in x and take the summation after taking the supermom in t.  So using this norm, they can show that the initial data is bounded in this norm and  the solution is also bounded.  So well-posedness.  So in all these results, including some others that I didn't mention, is based on this inequality.  So you need to find a norm such that the space is algebra because the Poznan equation is  nonlinear.  So you have f times g.  So in this norm, you have this kind of inequality.  So and then use the priority bound, construct some functionals, and then to close the boundaries  of the perturbation in this kind of norms.  And then for l infinity, if we take x to be l infinity in x, and this kind of estimate  is missing for Poznan equation without angular cutoff.  So this is true.  This is a walk by Seiji Okahi long time ago.  Because if you assume the angular is cutoff, then the linearized operator can be written  in two parts, a multiplier operator and a compare operator.  Then you can write the equation in a mild form and then you can take l infinity norm  and so on.  So this can be done.  But we all know that if the conditional operator is singularity, then you cannot decompose  the game part and the loss part.  So this kind of decomposition does not exist.  So this is the difficulty.  So the Poznan equation without angular cutoff is still valid in l infinity function space.  This is the goal.  So our approach is slightly different in terms of analysis.  The approach is different from the previous walk.  So what we did is the following.  Instead of applying the algebraic structure to cause the booster argument, the idea is  to apply the discharge argument to first walk on the l2 estimate on the level set.  And then use a localized strong averaging lemma.  Then combine these two estimates, we can construct a functional so that the discharge  argument works.  And then it will give us the infinity estimate on the perturbation.  So the l infinity estimate on the solution is obtained by using this approach.  So I will elaborate this a little bit more later.  So now the discharge argument for PDE and Kaliate equation, in fact, there are a lot  of interesting work.  I will not mention all of them here.  So here I just mentioned for the diffusion equation, there is a paper by Kefirere, and  then for Kaliate equation, for example, Lendel equation, for Kefirere equation, when the  Kaliate operator behaves more or less like a lapasse.  So there's also a lot of work by Gou's, Inver, Moe, Waxiu, and Kim, Go, Huang, and Moe,  Gou, Hart.  And then in all these work, they show the order continuity and also the Harnet inequality.  So for Boltzmann equation, for space homogeneous, there's also a work by Alonso two years ago.  So there's some difference between the Lendel equation, Fokker-Plan equation, and the Boltzmann  equation, because the Boltzmann equation basically is a singular integral operator.  So even though it has some kind of gain of accuracy, but this can be achieved, okay,  can be not directly just like the Lapecheun, but through the strong Ever-Vision number.  So before I present the result, I also would like to mention there's a big difference between  the perturbation in the Gaussian tail.  So this is the standard decomposition, the background West Varian square root of Mu times  perturbation, little f, okay.  So if you simply write the decomposition as Mu past F, and F only has some kind of algebraic  decay, so the estimate, the analysis is very different, okay.  So for the later, for this algebraic decay, in fact, with angular cutoff, there's a very  important progress, Goddani, Michelin and Moe, okay, two years ago, published, was online  for many years, and then about the special gap, okay, in some abstract banana space.  So without angular cutoff, there's also two group of people obtain the result, okay, a  horn, a toron, a chiselini, for mild singularity, and we also did the case for the strong singularity,  okay, cover the whole range when s between 0 and 1, okay.  So here's the short presentation about the difference between the Gaussian decay and  algebraic decay.  So if we assume the solution is, can be written in this form, then the linearized operator  is a standard self-adjoint, and then you have a very good strong cohesivity estimate for  angular cutoff, okay, because this arbitrary long F, L mu F, in fact, gives you the gain  of the grade, the s, s is the parameter in the angle, and also gain of moment, okay,  so it's strictly negative.  Now if you only have polylonged decay, F is mu past F, and then the linearized operator  is no longer self-adjoint, okay, so the cohesivity estimate is much weaker, okay, so all you  can show is that this quantity has a very good cohesivity term, gives you the gain of  regularity, but no gain of moment, okay, because the gain of moment, in fact, comes from the  difference, the differentiation of the Gaussian, okay, but for algebraic decay you don't have  it, okay.  So error term, okay, L2, and gamma O2 comes from the kinetic part of the course session,  and in fact in this analysis, you will find out that this integral plays an important  role, okay, in the analysis.  You cannot use simply the Hs norm with weight, other than that you have to use this as a  good term, okay, so this good term is not equivalent to the Hs norm, okay, because the weight is  slightly different, upper bound and lower bound is different, okay, but with this you  can indeed cause the error rate, okay, so now for L2, the film can be stated as follows,  okay, so if L2, if we assume the, we have a regularity, strong regularity in space variable  X, so that means the second order derivative in X is bounded, okay, with some weight, okay,  so let's, don't worry about the weight at this moment, and then the film can be stated  as follows, okay.  So we can say that when the parameter s is between 0 and 1, gamma also between 0 and  1, so if the weight is the decay, L2-bay decay, the parameter is big enough, okay, so if the  perturbation is small in this L2, H2 in X space and contains no microscopic component,  okay, in average, then the solution to the Boltzmann equation exists and converges to  the equilibrium, is potentially in time, okay, so this is the L2 theory, so the question  is whether we can remove or replace the H2S, H2X by L in 3D in X, okay, so this is the  question we want to answer, okay, so to prove that from, in fact, there are three main components,  okay, the first component is to show that the moment is, can be propagated in time,  okay, so you always, because the net of strong cohesivity estimates, so you always have this  kind of error, so, but it can still propagate in time.  Now because the singularity, the semi-group generated by the linearized operator, in fact,  gives the gain of the singularity H-s, okay, so the L2-bay on solution to the linearized  Boltzmann equation, in fact, is bounded by the H-s norm on the initial data, okay, of  course the coefficient depends on square-negative, 1 over square t, okay, so we saw by Kotani-Michelon-Mow  for cut-off, holds for long cut-off, this special gap, can buy these two free estimates  and you can, we can prove the field, I just mentioned, okay, so the idea is how to, now  the question is how to, with less this H2x to L infinity, because H2x is too strong in  some sense, okay, so this is the film we want to prove, okay, so the, let me see, no, no,  okay, this is the film I would like to present today, yes, so the same assumption on the  gamma and S, okay, so if we assume the initial perturbation, okay, F0 minus mu, the perturbation  is bounded in L infinity and L2, okay, with some weight, okay, for example K log, this  is more, and then for higher moment it is bounded only, okay, and K log and K, I don't  think I have time to talk about it, but just some numbers, okay, then you can show that  the solution exists, capital F, and the solution in both L infinity long and L2 long is volentially  decay to mu, okay, so this is the main film, okay, so how to understand the proof, so let  me look at this chart, okay, so the main idea is not to show that the L infinity long has  the property, the solution is bounded by initial data and then itself squared, okay, other  than that we first assume, maybe locally in a very short time, the supermum in both X  and T, the L1 long of perturbation is more, so this is assumption, we need to close it,  okay, so under this assumption, then we can perform the L2 estimate, okay, so you can  obtain the L2 estimate and also the L2 in Hs in V estimate because of the singularity  in angle, okay, and the similar estimate can be performed also on the level set of the  solution, okay, now with these two L2 estimates, one can apply the ever-vision number that  we'll talk about later to have the some kind of gain in X variable, okay, so you have Hs  prime L2 in V estimate on both the solution and the level set, now based on this L2 estimate  and also the Hs prime estimate, one can construct a functional on the level set and then apply  the George argument to show that in fact the solution is bounded in L infinity long, okay,  and this long depends only on the initial energy, okay, so then you can close the loop,  okay, in some sense, okay, so this is the idea, okay, so to achieve this, there are three main  steps, the first step is the L2 estimate on the level set, okay, so here I just give you  the statement, okay, so the statement says the following, okay, so suppose this capital  G is mu plus little g, you can build little g is given and f is mu plus f, okay, f is  the solution we are looking for, okay, so for this capital G we assume he has this,  satisfy these two properties, okay, so these two properties are in fact crucial because  with these two one can have the uniform co-eativity estimate, okay, L1 and L6, okay, now under  this assumption, so if the weight is sufficiently large, okay, and this or fLk plus is the f  if the weight v to the power L minus k, the positive part, okay, I think I'll throw it  somewhere, I'll throw it somewhere, sorry, and then, so you can have this, yes, so to  have the L2 estimate on fLk plus, okay, this term, the level set component, this is the  f greater than k, okay, with the weight of course, then to do this, we multiply both  sides by this quantity, then you have to estimate q times this one, okay, but under this assumption  you can show that in fact this is has upper bound, this upper bound can be written like  this, okay, so here you can see that it involves the L1 bound on the little g, okay, L1 bound  on the little g, and then by the assumption, okay, the loop, the chart I just mentioned,  this quantity we assume it is small, so this is small, okay, small, small, small, so you  have a very good term, okay, so this is the error term, and then somehow with this, you,  also the assumption you hope you can spread the, you can cause the L2 estimate on the  level set, okay, now to apply the eversion lemma, then we also need the L1 norm on the  condition operator, and is this one, okay, so we need to show that for the same setting  g and capital F, capital G and capital F, then with all these parameters, okay, this kappa  is greater than 2, okay, so we want to show that this integral, okay, has a uniform bound  as like this, okay, so in fact this is true, okay, this is true, can be proved, we put extra  negative derivative in v on this part because we need to kill the singularity in q in order  to have a uniform estimate, okay, now with these two estimates, then we can, with this  estimate we can apply the famous eversion lemma, okay, so here's the, just a list of  the, so the velocity eversion lemma is introduced, was introduced by Gauss-Pertrand-Santis in  80s, okay, so there's a lot of, what in different function space and improvement, okay, and  Gauss-Bell-Yong's-Pertrand-Santis and so on, okay, and also Bouchure-Devilleet and Gauss-Bouchure-Purenti  and Pertrand, so here I cannot mention all, okay, so here I would like to say that what  we need is the localized, localized in time version of the estimate obtained by Bouchure,  okay, in 2002, okay, so the estimate can be stated as follows, okay, so if we consider  this kinetic equation, Ft plus V dot gradient is F equals the source term, so if we assume  that we know that the solution has some vectority in V variable belongs to LP and then you can  always find this AS thread, okay, a small number defined like this, okay, so that the  following estimate holds, so this estimate holds, okay, so you have gain of vectority  in X and T, okay, and it is bounded by the solution at the two endpoints because it's  localized in time, okay, and then also the source term, okay, so this Ft, basically is  the Q we are looking for, okay, so now we've, of course, there's also some extra term on  the LP, now assume we all, we have all, we assume that all this estimate holds, then  we can construct a functional and this functional contains the L2 long on the level set and  the HS long in V on the level set comes from the covisivity and also the LP long of the  LP long S double prime, no, WP S double prime in X, this term, okay, comes from the everywhere  in the number, okay, so now if we work on this functional to see how it evolves in time,  then you can show that the functional in fact has satisfied this inequality, okay, so this  part, this is the, take the supermode in T, so if we, without taking the supermode, you  can show that when T2 is greater than T1 and this functional is bounded by initial data  LP long and pass this term, okay, so this term, the important part is that this power  beta i is greater than 1, okay, and the ai, of course, is positive, okay, there are several  terms comes from Q and the commutator and so on, so there's a summation in form, okay,  and this is homogeneous in K, now once we have this estimate, then we can take the supermode  in T, between T1 and T2, then we have the estimate, we have the estimate on the function,  okay, so here let me introduce some notations, okay, so let's choose a K0 to be sufficiently  large, depends on initial data, and MK is K0 minus 1 minus 102 to the power K, okay,  K is from 0 to infinity, so when K goes to infinity is K0, so if we define the level  set FK to be F minus MK with weight positive, and the functional I just define is for MK  0 to T, okay, and then with this estimate on the functional, then we can show that the  EK in fact is bounded by the initial data plus EK minus 1 to the power beta or K to the  A, okay, but beta is positive, beta is greater than 1, that's the key point, so for the first  two terms it's easy because K0 can be chosen large enough so that the level set for the  initial data is zero, okay, because the initial data is assumed to be bounded initially, so  we end up with this inequality, so EK is bounded by this summation, okay, now for once we have  this then it's almost done because it's done because you can construct a E star to be this  quantity and then you can show that the E star satisfies the E star K satisfies the equality  in the opposite direction, okay, so you can compare by comparison argument then you can  see that the EK is in fact bounded by EK star, okay, and what's EK star? EK star by our construction  goes to zero when K goes to infinity because EK star this Q0 is greater than 1 because  beta i is greater than 1, okay, so this is less than 1, 1 over Q log to the power K, so  when K goes to infinity E star K tends to zero, okay, so we have the level set, the functional  goes to zero, so the level set the L2 long goes to zero and then this impies the give  you the L infinity bound on the solution, okay, now so far this argument up to this point is  very locally in time, okay, very locally in time, so basically once you have the local  in time estimate by the special gap then the result holds for arbitrary time, okay, so by the  special gap and then combine the local L infinity estimate then we have the L infinity bound on the  solution go for in time and also have the exponential decay in both L infinity and L2 long, okay,  so this is the end of my presentation, so thank you. I have a small question, okay, thank you,  hi, hi Tong Yang, hi, I can hear, you can hear me, thank you for the nice talk, I wanted to  ask you two small questions, the first one is, so if I understood your talk correctly, you give  the first L infinity perturbative solutions in the non-cutoff case, so that's the, you improve  the perturbative theory by just using L infinity as the space for the smallness, right, yes, yes,  do you think if I'm thinking too, if I'm just comparing with the Navier-Stokes theory,  which you know you could imagine that your theorem could be connected to the Navier-Stokes theory by  a fluid limit, in Navier-Stokes we can go below L infinity, I'm just wondering if you thought about  that, if you think you could go to some LP and if it if it possible to go to some LP, then there  should be a critical P maybe, a smaller smallest one so that the perturbative problem can be solved  in a well-posed way, so that's the main question and maybe I just said the second one because  you maybe you can answer both at the center, I was just wondering whether part of the estimates you  use are similar or not to those in the recent papers of Ciri Lambert and Luis Silvestre on  non-cutoff-Poltzmann equation, I mean I'm not compared really, I was just curious,  so for okay thank you for the question, so for the first question I think this is a very good point,  so in fact the L infinity is not necessary, okay, so but I think the argument was also for LP,  but for the Boltzmann equation I can so far I cannot find the critical LP,  so this is the, so yes, so let me see, so this is at one point I  yes I try to understand this LP, we base this by LP but I don't have the  precise statement, so this is okay, but in principle your methods can go a little bit below L infinity,  in principle, I think so, it seems to me as well, yeah, thank you, thank you, and then for the second  question I don't know which paper you refer to, you can look at, well there are two or three papers,  it's a series of papers, they are not really focusing on the same goal as you do, so they  prove similar things, I participated this program more on the Landau side and a bit with the on  the on the moments for Boltzmann, the idea is to prove a conditional regularity, any solution that  has, but so it's not exactly the same the same target as you, yes you're right, I know those  papers, they are part of the charities rather than the assistance, absolutely, so they don't  prove the same theorem as you, but I was just curious because the estimates, the Georgie estimates  are similar to the ones you use, it seems to me, I see, but I, yes, so for example  for the solution we obtain in this way more, how to show that it is C infinity, I think it's still  not covered by the, the C infinity, this is definitely covered in another way in their work,  you should have a look, yeah, what they do not cover for sure is your theorem of perturbative  solutions, the global existence of perturbative solutions in low regularity space, because  they do not focus on the perturbative theory, they do the Georgie-Schauder and infinite bootstrap  to get C infinity, right, I know, I know somehow I read their paper and then they mentioned the  result covers the solution for example obtained by Thorne-Dewen's frame, yes, but  anyway we can discuss this because this is not, okay, thank you, thank you, there are a couple  of questions for Ben for us, one on the chatroom, can you read it, can you see it Tom? Oh yes, here  you use the D-George argument for L infinity, bang, not the regularity, I know that you work on the  Nandau equation and Fokker plan, you have the holder and so,  yeah, I'm sorry, I'm sorry, I missed your answer, oh sorry, so yes, we only use the  D-George argument for the L infinity, bang, not the regularity for the Nandau or Fokker plan for  holder inequality and so on, yeah, but you think it will be useful in this context to try to push  the, you think it would be possible to push the the George argument a bit further in order to  I think so, yes, yes, I think it's possible to use oscillation numbers and so on, okay, okay, thank you.
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Okay, thank you. Min Hai and Si for the kind invitation. So my great pleasure to give a talk in this conference, even though I cannot go there. So today I will talk about a joint work with Ricardo Alonso and Yoshio Morimoto, Wei Yang Shen. Okay, and then the main film that I would like to present today is to show that in the perturbative framework, okay, so if we assume the initial perturbation, VL, VTL in VLOM is more than the solution, it sits in the same function space. So this approach is new compared to the existing approach for assistance theory in the perturbative framework. Okay, so this is the main result. So all of you are experts, so I don't think I need to talk about the equation. So before I go to the L infinity solution, I'm going to recall the work on the L2 first. So both these two results in fact are for the patients with algebraic decay, decay tail in the velocity variable. So even though the back of the decouple is decay exponentially, but the perturbation has much slower decay. So these are the three collaborators, Alonso in Texas A&N, Katta and Morimoto from Kyoto University and Wei Yang from Simon Fisher University. So this is the equation, we all know the Boltzmann equation, and then the purpose of these results are for the case when this course session B has singularity, angular singularity. So the angular singularity in fact is assumed to be in this form. So it has two parts, kinetic part, v minus v star to power gamma, and we assume that gamma is positive, so we consider only the half potential case. So the singularity in the angle covers the full range. So the power of theta is negative 2 minus 2s. So in the results, we consider the case when s is between 0 and 1. So this is motivated by the inverse power law and so on. So now there are a lot of work here, only this very few, about searching for function space with very minimal regularity in space variable x. So in the boundary domain, there's a series of work initiated by Yan Guo using the L2L infinity approach. So a lot of very good results. And then in the whole space, we started many years ago with Alessandro, Morimoto, Seiji Wukai, and Cao Jiangxu, and trying to show that the perturbation, if the perturbation is in xs, s is greater than 3 over 2 because it's three dimension, L2 in v, and try to find the global assistance. But at that time, we fail and only got the local assistance. So recently, there's a very interesting result by Renjun Don, Shuangqian Liu, and Sakamoto and String. And they use this function space, use the vener algebra in x, so they take Fourier transformation in x and take the summation after taking the supermom in t. So using this norm, they can show that the initial data is bounded in this norm and the solution is also bounded. So well-posedness. So in all these results, including some others that I didn't mention, is based on this inequality. So you need to find a norm such that the space is algebra because the Poznan equation is nonlinear. So you have f times g. So in this norm, you have this kind of inequality. So and then use the priority bound, construct some functionals, and then to close the boundaries of the perturbation in this kind of norms. And then for l infinity, if we take x to be l infinity in x, and this kind of estimate is missing for Poznan equation without angular cutoff. So this is true. This is a walk by Seiji Okahi long time ago. Because if you assume the angular is cutoff, then the linearized operator can be written in two parts, a multiplier operator and a compare operator. Then you can write the equation in a mild form and then you can take l infinity norm and so on. So this can be done. But we all know that if the conditional operator is singularity, then you cannot decompose the game part and the loss part. So this kind of decomposition does not exist. So this is the difficulty. So the Poznan equation without angular cutoff is still valid in l infinity function space. This is the goal. So our approach is slightly different in terms of analysis. The approach is different from the previous walk. So what we did is the following. Instead of applying the algebraic structure to cause the booster argument, the idea is to apply the discharge argument to first walk on the l2 estimate on the level set. And then use a localized strong averaging lemma. Then combine these two estimates, we can construct a functional so that the discharge argument works. And then it will give us the infinity estimate on the perturbation. So the l infinity estimate on the solution is obtained by using this approach. So I will elaborate this a little bit more later. So now the discharge argument for PDE and Kaliate equation, in fact, there are a lot of interesting work. I will not mention all of them here. So here I just mentioned for the diffusion equation, there is a paper by Kefirere, and then for Kaliate equation, for example, Lendel equation, for Kefirere equation, when the Kaliate operator behaves more or less like a lapasse. So there's also a lot of work by Gou's, Inver, Moe, Waxiu, and Kim, Go, Huang, and Moe, Gou, Hart. And then in all these work, they show the order continuity and also the Harnet inequality. So for Boltzmann equation, for space homogeneous, there's also a work by Alonso two years ago. So there's some difference between the Lendel equation, Fokker-Plan equation, and the Boltzmann equation, because the Boltzmann equation basically is a singular integral operator. So even though it has some kind of gain of accuracy, but this can be achieved, okay, can be not directly just like the Lapecheun, but through the strong Ever-Vision number. So before I present the result, I also would like to mention there's a big difference between the perturbation in the Gaussian tail. So this is the standard decomposition, the background West Varian square root of Mu times perturbation, little f, okay. So if you simply write the decomposition as Mu past F, and F only has some kind of algebraic decay, so the estimate, the analysis is very different, okay. So for the later, for this algebraic decay, in fact, with angular cutoff, there's a very important progress, Goddani, Michelin and Moe, okay, two years ago, published, was online for many years, and then about the special gap, okay, in some abstract banana space. So without angular cutoff, there's also two group of people obtain the result, okay, a horn, a toron, a chiselini, for mild singularity, and we also did the case for the strong singularity, okay, cover the whole range when s between 0 and 1, okay. So here's the short presentation about the difference between the Gaussian decay and algebraic decay. So if we assume the solution is, can be written in this form, then the linearized operator is a standard self-adjoint, and then you have a very good strong cohesivity estimate for angular cutoff, okay, because this arbitrary long F, L mu F, in fact, gives you the gain of the grade, the s, s is the parameter in the angle, and also gain of moment, okay, so it's strictly negative. Now if you only have polylonged decay, F is mu past F, and then the linearized operator is no longer self-adjoint, okay, so the cohesivity estimate is much weaker, okay, so all you can show is that this quantity has a very good cohesivity term, gives you the gain of regularity, but no gain of moment, okay, because the gain of moment, in fact, comes from the difference, the differentiation of the Gaussian, okay, but for algebraic decay you don't have it, okay. So error term, okay, L2, and gamma O2 comes from the kinetic part of the course session, and in fact in this analysis, you will find out that this integral plays an important role, okay, in the analysis. You cannot use simply the Hs norm with weight, other than that you have to use this as a good term, okay, so this good term is not equivalent to the Hs norm, okay, because the weight is slightly different, upper bound and lower bound is different, okay, but with this you can indeed cause the error rate, okay, so now for L2, the film can be stated as follows, okay, so if L2, if we assume the, we have a regularity, strong regularity in space variable X, so that means the second order derivative in X is bounded, okay, with some weight, okay, so let's, don't worry about the weight at this moment, and then the film can be stated as follows, okay. So we can say that when the parameter s is between 0 and 1, gamma also between 0 and 1, so if the weight is the decay, L2-bay decay, the parameter is big enough, okay, so if the perturbation is small in this L2, H2 in X space and contains no microscopic component, okay, in average, then the solution to the Boltzmann equation exists and converges to the equilibrium, is potentially in time, okay, so this is the L2 theory, so the question is whether we can remove or replace the H2S, H2X by L in 3D in X, okay, so this is the question we want to answer, okay, so to prove that from, in fact, there are three main components, okay, the first component is to show that the moment is, can be propagated in time, okay, so you always, because the net of strong cohesivity estimates, so you always have this kind of error, so, but it can still propagate in time. Now because the singularity, the semi-group generated by the linearized operator, in fact, gives the gain of the singularity H-s, okay, so the L2-bay on solution to the linearized Boltzmann equation, in fact, is bounded by the H-s norm on the initial data, okay, of course the coefficient depends on square-negative, 1 over square t, okay, so we saw by Kotani-Michelon-Mow for cut-off, holds for long cut-off, this special gap, can buy these two free estimates and you can, we can prove the field, I just mentioned, okay, so the idea is how to, now the question is how to, with less this H2x to L infinity, because H2x is too strong in some sense, okay, so this is the film we want to prove, okay, so the, let me see, no, no, okay, this is the film I would like to present today, yes, so the same assumption on the gamma and S, okay, so if we assume the initial perturbation, okay, F0 minus mu, the perturbation is bounded in L infinity and L2, okay, with some weight, okay, for example K log, this is more, and then for higher moment it is bounded only, okay, and K log and K, I don't think I have time to talk about it, but just some numbers, okay, then you can show that the solution exists, capital F, and the solution in both L infinity long and L2 long is volentially decay to mu, okay, so this is the main film, okay, so how to understand the proof, so let me look at this chart, okay, so the main idea is not to show that the L infinity long has the property, the solution is bounded by initial data and then itself squared, okay, other than that we first assume, maybe locally in a very short time, the supermum in both X and T, the L1 long of perturbation is more, so this is assumption, we need to close it, okay, so under this assumption, then we can perform the L2 estimate, okay, so you can obtain the L2 estimate and also the L2 in Hs in V estimate because of the singularity in angle, okay, and the similar estimate can be performed also on the level set of the solution, okay, now with these two L2 estimates, one can apply the ever-vision number that we'll talk about later to have the some kind of gain in X variable, okay, so you have Hs prime L2 in V estimate on both the solution and the level set, now based on this L2 estimate and also the Hs prime estimate, one can construct a functional on the level set and then apply the George argument to show that in fact the solution is bounded in L infinity long, okay, and this long depends only on the initial energy, okay, so then you can close the loop, okay, in some sense, okay, so this is the idea, okay, so to achieve this, there are three main steps, the first step is the L2 estimate on the level set, okay, so here I just give you the statement, okay, so the statement says the following, okay, so suppose this capital G is mu plus little g, you can build little g is given and f is mu plus f, okay, f is the solution we are looking for, okay, so for this capital G we assume he has this, satisfy these two properties, okay, so these two properties are in fact crucial because with these two one can have the uniform co-eativity estimate, okay, L1 and L6, okay, now under this assumption, so if the weight is sufficiently large, okay, and this or fLk plus is the f if the weight v to the power L minus k, the positive part, okay, I think I'll throw it somewhere, I'll throw it somewhere, sorry, and then, so you can have this, yes, so to have the L2 estimate on fLk plus, okay, this term, the level set component, this is the f greater than k, okay, with the weight of course, then to do this, we multiply both sides by this quantity, then you have to estimate q times this one, okay, but under this assumption you can show that in fact this is has upper bound, this upper bound can be written like this, okay, so here you can see that it involves the L1 bound on the little g, okay, L1 bound on the little g, and then by the assumption, okay, the loop, the chart I just mentioned, this quantity we assume it is small, so this is small, okay, small, small, small, so you have a very good term, okay, so this is the error term, and then somehow with this, you, also the assumption you hope you can spread the, you can cause the L2 estimate on the level set, okay, now to apply the eversion lemma, then we also need the L1 norm on the condition operator, and is this one, okay, so we need to show that for the same setting g and capital F, capital G and capital F, then with all these parameters, okay, this kappa is greater than 2, okay, so we want to show that this integral, okay, has a uniform bound as like this, okay, so in fact this is true, okay, this is true, can be proved, we put extra negative derivative in v on this part because we need to kill the singularity in q in order to have a uniform estimate, okay, now with these two estimates, then we can, with this estimate we can apply the famous eversion lemma, okay, so here's the, just a list of the, so the velocity eversion lemma is introduced, was introduced by Gauss-Pertrand-Santis in 80s, okay, so there's a lot of, what in different function space and improvement, okay, and Gauss-Bell-Yong's-Pertrand-Santis and so on, okay, and also Bouchure-Devilleet and Gauss-Bouchure-Purenti and Pertrand, so here I cannot mention all, okay, so here I would like to say that what we need is the localized, localized in time version of the estimate obtained by Bouchure, okay, in 2002, okay, so the estimate can be stated as follows, okay, so if we consider this kinetic equation, Ft plus V dot gradient is F equals the source term, so if we assume that we know that the solution has some vectority in V variable belongs to LP and then you can always find this AS thread, okay, a small number defined like this, okay, so that the following estimate holds, so this estimate holds, okay, so you have gain of vectority in X and T, okay, and it is bounded by the solution at the two endpoints because it's localized in time, okay, and then also the source term, okay, so this Ft, basically is the Q we are looking for, okay, so now we've, of course, there's also some extra term on the LP, now assume we all, we have all, we assume that all this estimate holds, then we can construct a functional and this functional contains the L2 long on the level set and the HS long in V on the level set comes from the covisivity and also the LP long of the LP long S double prime, no, WP S double prime in X, this term, okay, comes from the everywhere in the number, okay, so now if we work on this functional to see how it evolves in time, then you can show that the functional in fact has satisfied this inequality, okay, so this part, this is the, take the supermode in T, so if we, without taking the supermode, you can show that when T2 is greater than T1 and this functional is bounded by initial data LP long and pass this term, okay, so this term, the important part is that this power beta i is greater than 1, okay, and the ai, of course, is positive, okay, there are several terms comes from Q and the commutator and so on, so there's a summation in form, okay, and this is homogeneous in K, now once we have this estimate, then we can take the supermode in T, between T1 and T2, then we have the estimate, we have the estimate on the function, okay, so here let me introduce some notations, okay, so let's choose a K0 to be sufficiently large, depends on initial data, and MK is K0 minus 1 minus 102 to the power K, okay, K is from 0 to infinity, so when K goes to infinity is K0, so if we define the level set FK to be F minus MK with weight positive, and the functional I just define is for MK 0 to T, okay, and then with this estimate on the functional, then we can show that the EK in fact is bounded by the initial data plus EK minus 1 to the power beta or K to the A, okay, but beta is positive, beta is greater than 1, that's the key point, so for the first two terms it's easy because K0 can be chosen large enough so that the level set for the initial data is zero, okay, because the initial data is assumed to be bounded initially, so we end up with this inequality, so EK is bounded by this summation, okay, now for once we have this then it's almost done because it's done because you can construct a E star to be this quantity and then you can show that the E star satisfies the E star K satisfies the equality in the opposite direction, okay, so you can compare by comparison argument then you can see that the EK is in fact bounded by EK star, okay, and what's EK star? EK star by our construction goes to zero when K goes to infinity because EK star this Q0 is greater than 1 because beta i is greater than 1, okay, so this is less than 1, 1 over Q log to the power K, so when K goes to infinity E star K tends to zero, okay, so we have the level set, the functional goes to zero, so the level set the L2 long goes to zero and then this impies the give you the L infinity bound on the solution, okay, now so far this argument up to this point is very locally in time, okay, very locally in time, so basically once you have the local in time estimate by the special gap then the result holds for arbitrary time, okay, so by the special gap and then combine the local L infinity estimate then we have the L infinity bound on the solution go for in time and also have the exponential decay in both L infinity and L2 long, okay, so this is the end of my presentation, so thank you. I have a small question, okay, thank you, hi, hi Tong Yang, hi, I can hear, you can hear me, thank you for the nice talk, I wanted to ask you two small questions, the first one is, so if I understood your talk correctly, you give the first L infinity perturbative solutions in the non-cutoff case, so that's the, you improve the perturbative theory by just using L infinity as the space for the smallness, right, yes, yes, do you think if I'm thinking too, if I'm just comparing with the Navier-Stokes theory, which you know you could imagine that your theorem could be connected to the Navier-Stokes theory by a fluid limit, in Navier-Stokes we can go below L infinity, I'm just wondering if you thought about that, if you think you could go to some LP and if it if it possible to go to some LP, then there should be a critical P maybe, a smaller smallest one so that the perturbative problem can be solved in a well-posed way, so that's the main question and maybe I just said the second one because you maybe you can answer both at the center, I was just wondering whether part of the estimates you use are similar or not to those in the recent papers of Ciri Lambert and Luis Silvestre on non-cutoff-Poltzmann equation, I mean I'm not compared really, I was just curious, so for okay thank you for the question, so for the first question I think this is a very good point, so in fact the L infinity is not necessary, okay, so but I think the argument was also for LP, but for the Boltzmann equation I can so far I cannot find the critical LP, so this is the, so yes, so let me see, so this is at one point I yes I try to understand this LP, we base this by LP but I don't have the precise statement, so this is okay, but in principle your methods can go a little bit below L infinity, in principle, I think so, it seems to me as well, yeah, thank you, thank you, and then for the second question I don't know which paper you refer to, you can look at, well there are two or three papers, it's a series of papers, they are not really focusing on the same goal as you do, so they prove similar things, I participated this program more on the Landau side and a bit with the on the on the moments for Boltzmann, the idea is to prove a conditional regularity, any solution that has, but so it's not exactly the same the same target as you, yes you're right, I know those papers, they are part of the charities rather than the assistance, absolutely, so they don't prove the same theorem as you, but I was just curious because the estimates, the Georgie estimates are similar to the ones you use, it seems to me, I see, but I, yes, so for example for the solution we obtain in this way more, how to show that it is C infinity, I think it's still not covered by the, the C infinity, this is definitely covered in another way in their work, you should have a look, yeah, what they do not cover for sure is your theorem of perturbative solutions, the global existence of perturbative solutions in low regularity space, because they do not focus on the perturbative theory, they do the Georgie-Schauder and infinite bootstrap to get C infinity, right, I know, I know somehow I read their paper and then they mentioned the result covers the solution for example obtained by Thorne-Dewen's frame, yes, but anyway we can discuss this because this is not, okay, thank you, thank you, there are a couple of questions for Ben for us, one on the chatroom, can you read it, can you see it Tom? Oh yes, here you use the D-George argument for L infinity, bang, not the regularity, I know that you work on the Nandau equation and Fokker plan, you have the holder and so, yeah, I'm sorry, I'm sorry, I missed your answer, oh sorry, so yes, we only use the D-George argument for the L infinity, bang, not the regularity for the Nandau or Fokker plan for holder inequality and so on, yeah, but you think it will be useful in this context to try to push the, you think it would be possible to push the the George argument a bit further in order to I think so, yes, yes, I think it's possible to use oscillation numbers and so on, okay, okay, thank you.
there. So today I will talk about a joint work with Ricardo Alonso and Yoshio Morimoto, Wei Yang like to present today is to show that in the we all know the Boltzmann equation, and then the mention, is based on this inequality. So you need close the boundaries of the perturbation in this without angular cutoff is still valid in l localized strong averaging lemma. Then combine functional so that the discharge argument works. the solution is obtained by using this approach. So for the later, for this algebraic decay, in converges to the equilibrium, is potentially in generated by the linearized operator, in fact, estimate by the special gap then the result holds comparing with the Navier-Stokes theory, which estimates are similar to the ones you use, it solutions, the global existence of perturbative regularity, I know that you work on the Nandau