Wellposedness and scattering for the Zakharov system in four dimensions
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Nakanishi, Kenji
This is joint work with Ioan Bejenaru, Zihua Guo and Sebastian Herr. We consider the Cauchy problem for the Zakharov system in four space dimensions, extending the local wellposedness by Ginibre, Tsutsumi and Velo to wider range of Sobolev exponents, together with scattering for small data. We observe distinct phenomena at two extreme points of exponents. One is at the energy space, where our nonlinear estimates suffer from divergence related to the critical Sobolev embedding. We can however prove the results for small data, without any improvement of the estimates, but from the existence in a smaller Sobolev space and the uniqueness in a larger one, together with the conservation law. At another corner point of exponents, the critical space is also intermediate, but we obtain a strong illposedness result in terms of instantaneous exit or non-existence of weak solutions.
 So I'd like to thank the organizers for inviting me to this nice conference and nice please. So I'm going to talk about well-posedness and scattering question for the Zaharoff system in space four dimensions. This is joint work with Iwan Bejnaru, Zifua Guo and Sebastian Her. So let me start with the equation. This is a kind of classical nonlinear dispersive equation which is a system of the Schrodinger equation and the wave equation. And it's a kind of model equation for the plasma which is called Langmuir turbulence essentially generated by the nonlinear interaction between electric oscillations and the hydrodynamic oscillation. So here U is somehow describing a slow motion of the electric field and N is describing the motion of ion density. So it is a spacetime function, U is complex and N is real. Here alpha is a parameter which will be fixed throughout this talk. It's a kind of positive constant for the ion speed, sound speed. Anyway so this is our equation. The main result is a kind of classical question and simple. We consider this specifically in space dimension four and prove some local and global existence more precisely well-posedness result together with scattering to free state as time goes to infinity. So this dimension four is a kind of weird for this model because this is physics in the real three dimensional space. So usually people consider this only in three dimension or lower. But somehow we find an interesting mathematical question in this special dimension because it brings several sort of criticality for the analysis. So just for simplicity of the notation it is better to transform the wave equation into the first order just by decomposing as usual to the incoming and outgoing waves. So that we can regard it's a kind of shredding type equation. Then we have this type of system of first order equations. And so capital N is just a complex spied wave component. And small n is the real part. Then let me review some basic facts of the Zaharoff system. First of all it is Hamiltonian system and Hamiltonian is given in this form. And mass or the L2 norm for the shredding component is also preserved. Therefore as a solution space or the function space for the initial data the most natural choice is to choose the shredding component from H1 and the wave component from L2. So that we can make sense of both quantities at least in lower dimensions. And then you immediately see already here the four dimension is a kind of critical case because to make sense the energies in this H1 cross L2 space you need to control the nonlinearity and then just by using the Helder and the Soboreff inequality this can be controlled if and only if the dimension is 4 or less. And the dimension 4 is the critical case where this has essentially the same scaling. However, since our equation is not scaling invariant it is not very clear what is the scaling critical exponent. However, if you take the so-called subsonic limit where you regard that the motion is much slower than the sound speed or mathematically it corresponds to taking the limit alpha going to infinity then because let me go back because we have this equation for the wave component if you send formally alpha to infinity then you see that n should be equal to u square. Then plugging this into the first equation you get simply the standard focusing cubic nonlinear Schrodinger equation. And for this equation it is scaling invariant and well known that this is critical in the energy space for four dimensions. So in this case it is clearly a scaling critical as long as we want to have solutions in the energy space. That's the first thing, however, at least from a technical viewpoint the Zaharoff system is even more critical than the Schrodinger equation and it is more related to the use of Strikatz estimate. And here somehow because it is a system of the wave and the Schrodinger equation the exponent which we can use for the Strikatz estimate is somehow limited and we are forced to use the end point Strikatz estimate. It is in 4D L2 in time and L4 in x and this is the exponent if you consider L2 solutions. If you consider H1 solution then you get one regularity in this space so you consider the first derivative in L4 in space. Then again this space is critical and it's even more critical in the sense that the Sobolev embedding fails because this corresponds to the case where it should be embedded into L infinity but this is well known to be false. And this causes a lot of trouble at least if we go in a straightforward way. So let me just recall other difficulties which are actually somehow classical and essentially solved before. However, it is not only for the four dimensional case but in general for the Zaharoff system. First problem is for the local existence of solutions. Even for the local existence it is slightly more difficult than the Schrodinger equation in the sense that or just the wave equation in the sense that the non-linearity has the derivative. And so for example if you consider the simplest estimate that's just you estimate the left hand side by the linear energy inequality then you lose the regularity because of this derivative on the non-linearity. That can be easily seen because for the first order equation if you apply the energy inequality then it means the right hand side or the source term should have the same regularity as the solution. Then for example you assume that your solution is in HS space for S large enough then U square is in HS therefore the derivative of U square should be HS minus 1 and that implies the wave component should have one less regularity. However if you plug it into here then n times U would have HS minus 1 regularity and you cannot go back to HS for the Schrodinger component. But this is a problem in the Zaharoff system or in general for this kind of equation and in particular for the Zaharoff system this can be easily solved by a few different methods. Taking account that the non-linear part the interaction is not really resonant meaning these two products of two functions do not really interact if you take account of the oscillation and there are easy ways to remove this problem. Another possible problem is for large data. If you want to have scattering theory in the Soboreff space then there is a clear lower bound for the low decay of non-linear term as U goes to zero and so it is given for the Schrodinger equation by the power which is called the L2 critical power 1 plus 4 over D. So this is the main problem if you consider three dimensional or lower dimensional case. So 4D is luckily on the borderline case where so when D is 4D this is just 2 and we have quadratic interaction. So from this you do not really expect that the non-linear interaction causes a big trouble. However, you can see so this is again a kind of critical case but this is not really so much critical because by taking account of the non-linear non-resonance property of the equation our non-linearity is not as strong as quadratic interaction for the Schrodinger. Therefore this problem is not so much relevant. Okay now let me show you our result. It's a very simple statement although the range of exponent is a bit complicated. You don't have to consider this but I will show you later the picture. So for some range of exponent S is for Schrodinger component and sigma is for the wave component. We can prove first the scattering, global well poseness and the scattering at time infinity for small data. So for any data in the sovereign space Hs for the Schrodinger component, Hs for the wave component and if you assume it is sufficiently small in a kind of critical space. So this regularity 1, 1, 0 is the weakest possible choice in this range. Then we have a global unique solution which continuously depends on the initial data and as time goes to infinity it converges, it is asymptotic to a free solution. So this is a very classical statement of small data scattering. So as I said 1, 1, 0 is the lowest number in this range. So this is the first result. So I should mention that energy space is when S is 1 and sigma is 0. It is one of the extremal points of this region. That was for small data. For large data we can construct time local solutions unless the exponent is in the energy space. So except for this special case for the other range of exponents we have local solutions for the same initial data in the same space but not necessarily small. So this is a simple local well-posed result. And we have also a ill-posed result when the wave component is required to have higher regularity. This is another corner extremal point of our range. And at this point we have ill-posedness which is kind of quite strong. You can regard it as either non-existence of a solution or loss of regularity immediately after time 0. So the precise statement is that you can find some initial data and even for a one parameter of initial data there is no solution in the appropriate function space which should be there if we had the local well-posedness. And we can even weaken the requirement for the solution. It is just a distributional solution with much lower regularity than the initial data. Even with this assumption we can prove that there is no such solution. And however, since this exponent is relatively high actually there is a unique solution in a different function space. Indeed the main problem here is that the wave component is required to have too much regularity compared with the Schrodinger component. And indeed we can construct this result in price that our solution is in, for example, in H2 plus H2. But this solution is not in the same space as the initial data H3 as soon as the time is different from 0. So somehow the solution instantaneously exits the initial data space. So in this sense this is a very strong ill-posedness result. So they are our results. So let me quickly recall some previous results and some independent similar results. Considering local well-posedness there was a result by Bruggen and Coriander. So they consider three-dimensional or lower and then they proved local well-posedness in the energy space as well as global existence for small data in the energy space. So this is in the line as Bruggen's results for non-linear shredding equation using excess space. And then it was later extended by Geneveld, Stumi and Vero to wider, to more general range of exponents as well as for general space dimensions. And it was further improved by Veznalherr, Colmar-Tatar in 2D and Veznalherr in 3D to the even lower regularity. I don't make it specific. But OK, so here is the range in Geneveld, Stumi and Vero in four-dimension. The end point is 1, 1, 0 and there is 1 and 0. This is the energy space. But the line connecting them was not included. And this is 1 and 1 and there is a strip of parallel lines. So our result is extending the region. So these blue lines were in the Geneveld, Stumi and Vero. So we are able to include these two edges, and in particular 1, 1, 0 as well as the energy space. And we can also extend the region to the both sides where either shredding component has more regularity or the wave component has more regularity. And this 2 and 3 was another corner point, an extremer point where we have the irreposeness. OK, so this is about local well-poseness. And for large-time behavior, there were several results concerning the scattering theory, but starting with a free profile given at time infinity. And you try to construct a unique solution asymptotic to that. So it's a construction of the wave operator. It was considered mainly in 3D. So first, Ozawa and Tsutsumi proved such result in weighted software space, either assuming that the final data was small or there is some non-resonant condition on the asymptotic profile. So this cb being alpha is somehow strongly resonant because where shredding component and wave component propagates in the same speed. So they assumed that component is vanishing at the time infinity. It was kind of non-generic condition. It was removed later by Shimomura, but he has to assume another condition on the fully transform of the wave component that is vanishing of the zero frequency. So in this sense, this kind of condition often appears in the construction of wave operator. But this also means that data was not generic. And finally, it was removed by Geneva and Vero. And they were also able to extend the function space for the final state. It was essentially a kind of sovereign space based on L2 and L1. And they considered also two dimension, but I don't know any paper about four dimension. So another direction is the blow up results. We don't have such result, but I just mentioned. In 2D, there is a result by Grange-Dass and Merl. They constructed the blow up solution in finite time using the critical, sorry, non-linear shredding equation as an approximate. And for 2D and 3D, Merl also proved for negative energy, the solution blows up either in finite time or infinite time. But I guess the three dimensional case, the existence of it is still open. And for scattering, starting from the initial data, it's a relatively recent progress, as far as I know. So in 3D with the go, we proved the scattering result as in our result here. In 3D, but assuming radial symmetry in the energy space. And almost simultaneously, around the same time, honey, pustateri, and shutter proved without assuming symmetry, but imposing decay at space infinity. And then with the go, we somehow improved our first result by removing the radial symmetry, but assuming that angular derivative is also in the energy space. So if the data is sufficiently decaying at infinity, of course, the angular derivative is fine. And in particular, these two results are contained in this case. In the 4D case, I don't know other result, but Kato and Tsugawa obtained independently, essentially the same result. However, they had to, they consider only on this diagonal case, which is this green line starting from the lowest regularity point. And the proof is different. OK, so this is a quick review about the previous result. So let me talk about some ideas in the proof. With our proof, it's pretty simple and essentially it's classical and non-argument. So first we use, in order to treat this problem of derivative loss, we use the argument known from Shata. It is just by using integration by parts to remove some nonlinear interaction, which is oscillatory in time. So it can formulate as follows. You first consider the fully transform of each space, each solution, and filter the free evolution, or you filter out the free oscillation from the solutions. Then the remainder is supposed to oscillate slowly in time. And you can inject just this expression into the Duamel formula in the full side. Then to extract the time oscillation compared with the free evolution, and you immediately see that it is given by this formula. So of course, for generaxi and eta, you may have a bit complicated set of zero of this function. But here we just need to consider the case when the wave component has higher frequency, because wave component is expected to have lower regularity than the Schrodinger equation by the previous argument. So if you assume that the wave component has much higher regularity, then you can essentially ignore eta, and then this is just c square minus c square plus alpha c. So this can vanish only if c is the same size as alpha. This is the same set as in the result of other to me. If you avoid this special set, then this oscillation in time is always large. And if you go higher frequency, it grows like c square. And this means if you integrate in time, then you gain one over c square. It's like one minus inverse of minus Laplacian. So you gain a lot of regularity by this oscillation in time. And also by this special integration, you either differentiate this part by time, or you get some boundary term of the integral. That means you gain also decay in time, because you either remove the integral or by differentiation, you use the equation, and then you get the cubic non-linearity. So just by using this simple argument, we can transform our equation, which is a bit ugly, but much better than before. And so it is essentially of this form. We lift some part of the interaction, which is when the wave component has lower frequency. But the other part is coming from this integration by parts is cubic. And there is some quadratic part, but for which we don't have time integral. And then for this equation, it's essentially like nonlinear Schrodinger equation in the sense that you don't have to take care of the regularity, semi-linear Schrodinger equation. And so for example, if you assume that the wave component is in L2, and the Schrodinger component is in HS, then you can estimate it by very simple, helder type inequality. For example, for this part, HS and L4 over 3 is just a helder inequality. And since N is low frequency, the derivative hits only on the Schrodinger component. For the components which we applied in partial integration, the derivative is hitting the wave component. However, here we have minus, plus, inverse, thanks to this integration by parts. So it is even better that the derivative hits the wave component, and essentially the same helder type inequality works, and this allows us to go from L4 to L4 first. These exponents appear in the end point strickers estimate. More precisely, we need to use Coffman Bayer-Byrini, sorry, Byrini fully multiplier estimates, but it's essentially like helder inequality. And this is particularly simple because we are putting the wave component in L2, and here we gain two derivatives. That means this part is essentially like H2, and H2 is embedded into L infinity, almost in 4D. That's why we get a very simple numerology here. However, you should note that this is not embedded into L infinity. It is again a critical property of our argument, and in particular this estimate fails if you choose the Schrodinger component in H1. But as long as S is below 1, these estimates are good enough. Of course, we need to take care about the wave component, wave equation, but anyway, very simple helder type estimate together with end point strickers estimate, which is if you have the Schrodinger equation in the dual space, L2 in time, L4 first in space, then you get your solution in L2 in time, L4 in space. Just by this estimate, we can get a closed estimate, and by iteration, we get solution. But as I said, this estimate just fails at the energy space. Typically, these two estimates are simply false if we choose S equals 1. And this is because we cannot put the lower frequency part into L infinity. So it is not for this part, but more precisely because of this. As I said, it is related to the end point strickers estimate and together with failure of so-called inequality. So you might wonder, maybe you should change the estimate for the wave component, but there is absolutely no other choice for the wave component if it has zero regularity. Because for example, if you use the strickers estimate for the wave component, then you are going to lose some regularity because the regularity for the Schrodinger strickers is always better than the wave strickers estimate. So if it's just on the borderline, if you shift slightly to the wave, then you immediately lose the closed estimate. That's why we have to stick to the end point strickers estimate. That's the best regularity for the Schrodinger strickers estimate. However, if we have more regularity for the Schrodinger component, we can do that. So we can somehow sacrifice part of regularity by Schrodinger strickers. And instead, we use the strickers estimate for the wave equation. That is why we can somehow go slightly to the exponents where Schrodinger component is better. Although in this case, we have to restrict the range of strickers norm in our estimate. I don't get into much detail, but in particular, we cannot prove that the solution satisfies the same strickers estimate as the free equation, and I suspect that they indeed do not satisfy all of them. And so this somehow suggests why in the previous results, this case where the Schrodinger component has too much regularity was not covered. Because if you use, for example, the Brugans-XSP norm, then it implies that the solution has the full range of the strickers estimate automatically, as long as you choose the usual exponent. So therefore, if you want to close all of the estimates just by using Brugans-SP, then and if the solution does not satisfy such strickers estimate, then it is, of course, a difficulty. And other results concerning the local variable poseness always using this kind of space. So this may be a kind of difficulty, but we don't know if the solution really fails to have the full range of strickers estimate. Okay. So what about energy space? So in this case, all the previous arguments fail because of the critical Sobolev embedding. So essentially, we have no nonlinear estimate. However, this very critical case is not like the critical case of the scalar equation in the sense that we have actually solutions either both in a smaller function space and also in larger function space. So this is where the energy space where we have the problem. But for example, if you raise the regularity for the wave component, it's smaller than of course we have our results. Also if you lower the regularity for the Schrodinger component, then this is a larger space. But we have also the local variable poseness always as well as the scattering just by the previous simple argument. So this critical space is somehow in between. And this structure allows us to use the following argument. It's also a kind of classical argument by using weak compactness together with the conservation law. It's a very simple argument. So we can not estimate in this space, but we can start with a smaller space and try to approximate the solution by a sequence of solutions from there and then by using energy and mass conservation law, assuming that the solution is small, so this is only for small solutions. The energy and mass conservation law gives us a purely bound global in HM1 and L2. Therefore, we can take the weak limit using the equation globally in time. And in general, the weak limit may have a smaller energy, but anyway, it has this inequality. And by using this larger space, we can say that this loss of energy cannot happen because we know that the limit solution is unique in this larger space. Therefore, if the energy decreases, then you can solve the equation again backward to time zero and then you get a contradiction, right? Because the energy always decreases and then you cannot come back to time zero. Therefore, this loss of energy cannot happen, and that means that convergence has to be strong. Therefore, and the same argument works for the mass. As we get strong solution and also strong continuity in the energy space, and the same argument works for large time to get the scattering in the strong sense. So just by a cheap trick, we can close this part, although we have essentially no nonlinear estimate. However, it should note that this structure alone does not tell you that the well-posed space should hold in the intermediate space. Indeed, another corner point two and three has the same structure, but where we have a strong ill-posedness, one reason is that we don't have the conservation law at this point. And the proof here is also pretty simple. After applying this normal form argument, we can extract somehow this transformation of the solution is actually the worst part in this special case. And then the problem is reduced to get to find initial data for which this bilinear form is worst. And it is related again because this is like you have a product and make one derivative up. It is related to this basic estimate. Also it is related to the critical sub-ref embedding. H2 is not embedded into L infinity, therefore it's not an algebra. The product of them is not embedded into the same space. It's essentially the same for this bilinear transformation. And therefore we can essentially choose counter example to this inequality with a small modification to avoid some cancellation between similar terms. But essentially something like this, very simple, gives us a counter example. So I stop here. Thank you. Any other question or comments? In the I-Times paper on energy critical 3D shredding, they obtain the 4, infinity stric arts inequality by 3 analysis, not by 3, if you were at 4, 3 and then sub-ref would fail. If you try to prove it in an inequality by first doing stric arts and then doing sub-ref, you fail for example 4 infinity shredding at h1 and 3D. But if you prove it directly, it's true. And this is observed. The result of stric arts followed by sub-ref can be solved. But then you can't do sub-ref. But I'm claiming that sometimes stric arts sub-ref is true around the legs. Actually, it's not always together with the stric arts. So we have this B-town, by linear transform, for which we have to use directly the estimate for this kind of by linear expression without any reevolution. So yeah, it may help, but it's not clear if we can. So I have a question, a remark. In the energy space, can you do the scheme to increase your small data? Yes, that's what we want to do. And yeah, we think probably the natural conjecture is the first grand state. But the difference from the critical shredding equation is that we don't even have any nonlinear estimate in the energy space. And I guess the first thing what we should do is to get a nice estimate for the shredding equation, but including the wave component as a potential. And maybe we can get a nice estimate for such a shredding equation with a potential, assuming that the wave component has less L2 norm than the grand state. That's what we are trying now. Okay, so we use this piece. Okay. Thank you. And one more question. Okay, but this I know the answer is no, but you haven't tried to construct the solution of the grand state. Yes, because we don't even have the local wave poisonous for large data. First, we need to have... You can... Yeah, we may have some strange statement that we don't have... Either we don't have the solution or blows up, but... You can remove the bubble and let the wave of small data. But in order to go close to the bubble, we need the existence of the solution. No, but you can work directly on the linearized equation. Okay. Don't we need any perturbation argument? Yes, of course. We don't have any estimate here. Okay. Thank you. Let's finish the speaker again.
 So I'd like to thank the organizers for inviting me to this nice conference and nice please.  So I'm going to talk about well-posedness and scattering question for the Zaharoff system  in space four dimensions.  This is joint work with Iwan Bejnaru, Zifua Guo and Sebastian Her.  So let me start with the equation. This is a kind of classical nonlinear dispersive equation  which is a system of the Schrodinger equation and the wave equation.  And it's a kind of model equation for the plasma which is called Langmuir turbulence  essentially generated by the nonlinear interaction between electric oscillations and the hydrodynamic  oscillation.  So here U is somehow describing a slow motion of the electric field and N is describing  the motion of ion density.  So it is a spacetime function, U is complex and N is real.  Here alpha is a parameter which will be fixed throughout this talk.  It's a kind of positive constant for the ion speed, sound speed.  Anyway so this is our equation.  The main result is a kind of classical question and simple.  We consider this specifically in space dimension four and prove some local and global existence  more precisely well-posedness result together with scattering to free state as time goes  to infinity.  So this dimension four is a kind of weird for this model because this is physics in the  real three dimensional space.  So usually people consider this only in three dimension or lower.  But somehow we find an interesting mathematical question in this special dimension because  it brings several sort of criticality for the analysis.  So just for simplicity of the notation it is better to transform the wave equation into  the first order just by decomposing as usual to the incoming and outgoing waves.  So that we can regard it's a kind of shredding type equation.  Then we have this type of system of first order equations.  And so capital N is just a complex spied wave component.  And small n is the real part.  Then let me review some basic facts of the Zaharoff system.  First of all it is Hamiltonian system and Hamiltonian is given in this form.  And mass or the L2 norm for the shredding component is also preserved.  Therefore as a solution space or the function space for the initial data the most natural  choice is to choose the shredding component from H1 and the wave component from L2.  So that we can make sense of both quantities at least in lower dimensions.  And then you immediately see already here the four dimension is a kind of critical case  because to make sense the energies in this H1 cross L2 space you need to control the  nonlinearity and then just by using the Helder and the Soboreff inequality this can be controlled  if and only if the dimension is 4 or less.  And the dimension 4 is the critical case where this has essentially the same scaling.  However, since our equation is not scaling invariant it is not very clear what is the  scaling critical exponent.  However, if you take the so-called subsonic limit where you regard that the motion is  much slower than the sound speed or mathematically it corresponds to taking the limit alpha going  to infinity then because let me go back because we have this equation for the wave component  if you send formally alpha to infinity then you see that n should be equal to u square.  Then plugging this into the first equation you get simply the standard focusing cubic  nonlinear Schrodinger equation.  And for this equation it is scaling invariant and well known that this is critical in the  energy space for four dimensions.  So in this case it is clearly a scaling critical as long as we want to have solutions in the  energy space.  That's the first thing, however, at least from a technical viewpoint the Zaharoff system  is even more critical than the Schrodinger equation and it is more related to the use  of Strikatz estimate.  And here somehow because it is a system of the wave and the Schrodinger equation the  exponent which we can use for the Strikatz estimate is somehow limited and we are forced  to use the end point Strikatz estimate.  It is in 4D L2 in time and L4 in x and this is the exponent if you consider L2 solutions.  If you consider H1 solution then you get one regularity in this space so you consider the  first derivative in L4 in space.  Then again this space is critical and it's even more critical in the sense that the Sobolev  embedding fails because this corresponds to the case where it should be embedded into  L infinity but this is well known to be false.  And this causes a lot of trouble at least if we go in a straightforward way.  So let me just recall other difficulties which are actually somehow classical and essentially  solved before.  However, it is not only for the four dimensional case but in general for the Zaharoff system.  First problem is for the local existence of solutions.  Even for the local existence it is slightly more difficult than the Schrodinger equation  in the sense that or just the wave equation in the sense that the non-linearity has the  derivative.  And so for example if you consider the simplest estimate that's just you estimate the left  hand side by the linear energy inequality then you lose the regularity because of this  derivative on the non-linearity.  That can be easily seen because for the first order equation if you apply the energy inequality  then it means the right hand side or the source term should have the same regularity as the  solution.  Then for example you assume that your solution is in HS space for S large enough then U square  is in HS therefore the derivative of U square should be HS minus 1 and that implies the  wave component should have one less regularity.  However if you plug it into here then n times U would have HS minus 1 regularity and you  cannot go back to HS for the Schrodinger component.  But this is a problem in the Zaharoff system or in general for this kind of equation and  in particular for the Zaharoff system this can be easily solved by a few different methods.  Taking account that the non-linear part the interaction is not really resonant meaning  these two products of two functions do not really interact if you take account of the  oscillation and there are easy ways to remove this problem.  Another possible problem is for large data.  If you want to have scattering theory in the Soboreff space then there is a clear lower  bound for the low decay of non-linear term as U goes to zero and so it is given for the  Schrodinger equation by the power which is called the L2 critical power 1 plus 4 over  D.  So this is the main problem if you consider three dimensional or lower dimensional case.  So 4D is luckily on the borderline case where so when D is 4D this is just 2 and we have  quadratic interaction.  So from this you do not really expect that the non-linear interaction causes a big trouble.  However, you can see so this is again a kind of critical case but this is not really so  much critical because by taking account of the non-linear non-resonance property of the  equation our non-linearity is not as strong as quadratic interaction for the Schrodinger.  Therefore this problem is not so much relevant.  Okay now let me show you our result.  It's a very simple statement although the range of exponent is a bit complicated.  You don't have to consider this but I will show you later the picture.  So for some range of exponent S is for Schrodinger component and sigma is for the wave component.  We can prove first the scattering, global well poseness and the scattering at time infinity  for small data.  So for any data in the sovereign space Hs for the Schrodinger component, Hs for the wave  component and if you assume it is sufficiently small in a kind of critical space.  So this regularity 1, 1, 0 is the weakest possible choice in this range.  Then we have a global unique solution which continuously depends on the initial data and  as time goes to infinity it converges, it is asymptotic to a free solution.  So this is a very classical statement of small data scattering.  So as I said 1, 1, 0 is the lowest number in this range.  So this is the first result.  So I should mention that energy space is when S is 1 and sigma is 0.  It is one of the extremal points of this region.  That was for small data.  For large data we can construct time local solutions unless the exponent is in the energy  space.  So except for this special case for the other range of exponents we have local solutions  for the same initial data in the same space but not necessarily small.  So this is a simple local well-posed result.  And we have also a ill-posed result when the wave component is required to have higher  regularity.  This is another corner extremal point of our range.  And at this point we have ill-posedness which is kind of quite strong.  You can regard it as either non-existence of a solution or loss of regularity immediately  after time 0.  So the precise statement is that you can find some initial data and even for a one parameter  of initial data there is no solution in the appropriate function space which should be  there if we had the local well-posedness.  And we can even weaken the requirement for the solution.  It is just a distributional solution with much lower regularity than the initial data.  Even with this assumption we can prove that there is no such solution.  And however, since this exponent is relatively high actually there is a unique solution in  a different function space.  Indeed the main problem here is that the wave component is required to have too much regularity  compared with the Schrodinger component.  And indeed we can construct this result in price that our solution is in, for example,  in H2 plus H2.  But this solution is not in the same space as the initial data H3 as soon as the time  is different from 0.  So somehow the solution instantaneously exits the initial data space.  So in this sense this is a very strong ill-posedness result.  So they are our results.  So let me quickly recall some previous results and some independent similar results.  Considering local well-posedness there was a result by Bruggen and Coriander.  So they consider three-dimensional or lower and then they proved local well-posedness  in the energy space as well as global existence for small data in the energy space.  So this is in the line as Bruggen's results for non-linear shredding equation using excess  space.  And then it was later extended by Geneveld, Stumi and Vero to wider, to more general  range of exponents as well as for general space dimensions.  And it was further improved by Veznalherr, Colmar-Tatar in 2D and Veznalherr in 3D to  the even lower regularity.  I don't make it specific.  But OK, so here is the range in Geneveld, Stumi and Vero in four-dimension.  The end point is 1, 1, 0 and there is 1 and 0.  This is the energy space.  But the line connecting them was not included.  And this is 1 and 1 and there is a strip of parallel lines.  So our result is extending the region.  So these blue lines were in the Geneveld, Stumi and Vero.  So we are able to include these two edges, and in particular 1, 1, 0 as well as the energy  space.  And we can also extend the region to the both sides where either shredding component has  more regularity or the wave component has more regularity.  And this 2 and 3 was another corner point, an extremer point where we have the irreposeness.  OK, so this is about local well-poseness.  And for large-time behavior, there were several results concerning the scattering theory,  but starting with a free profile given at time infinity.  And you try to construct a unique solution asymptotic to that.  So it's a construction of the wave operator.  It was considered mainly in 3D.  So first, Ozawa and Tsutsumi proved such result in weighted software space, either assuming  that the final data was small or there is some non-resonant condition on the asymptotic  profile.  So this cb being alpha is somehow strongly resonant because where shredding component  and wave component propagates in the same speed.  So they assumed that component is vanishing at the time infinity.  It was kind of non-generic condition.  It was removed later by Shimomura, but he has to assume another condition on the fully  transform of the wave component that is vanishing of the zero frequency.  So in this sense, this kind of condition often appears in the construction of wave  operator.  But this also means that data was not generic.  And finally, it was removed by Geneva and Vero.  And they were also able to extend the function space for the final state.  It was essentially a kind of sovereign space based on L2 and L1.  And they considered also two dimension, but I don't know any paper about four dimension.  So another direction is the blow up results.  We don't have such result, but I just mentioned.  In 2D, there is a result by Grange-Dass and Merl.  They constructed the blow up solution in finite time using the critical, sorry, non-linear  shredding equation as an approximate.  And for 2D and 3D, Merl also proved for negative energy, the solution blows up either in finite  time or infinite time.  But I guess the three dimensional case, the existence of it is still open.  And for scattering, starting from the initial data, it's a relatively recent progress,  as far as I know.  So in 3D with the go, we proved the scattering result as in our result here.  In 3D, but assuming radial symmetry in the energy space.  And almost simultaneously, around the same time, honey, pustateri, and shutter proved  without assuming symmetry, but imposing decay at space infinity.  And then with the go, we somehow improved our first result by removing the radial symmetry,  but assuming that angular derivative is also in the energy space.  So if the data is sufficiently decaying at infinity, of course, the angular derivative  is fine.  And in particular, these two results are contained in this case.  In the 4D case, I don't know other result, but Kato and Tsugawa obtained independently,  essentially the same result.  However, they had to, they consider only on this diagonal case, which is this green line  starting from the lowest regularity point.  And the proof is different.  OK, so this is a quick review about the previous result.  So let me talk about some ideas in the proof.  With our proof, it's pretty simple and essentially it's classical and non-argument.  So first we use, in order to treat this problem of derivative loss, we use the argument known  from Shata.  It is just by using integration by parts to remove some nonlinear interaction, which  is oscillatory in time.  So it can formulate as follows.  You first consider the fully transform of each space, each solution, and filter the  free evolution, or you filter out the free oscillation from the solutions.  Then the remainder is supposed to oscillate slowly in time.  And you can inject just this expression into the Duamel formula in the full side.  Then to extract the time oscillation compared with the free evolution, and you immediately  see that it is given by this formula.  So of course, for generaxi and eta, you may have a bit complicated set of zero of this  function.  But here we just need to consider the case when the wave component has higher frequency,  because wave component is expected to have lower regularity than the Schrodinger equation  by the previous argument.  So if you assume that the wave component has much higher regularity, then you can essentially  ignore eta, and then this is just c square minus c square plus alpha c.  So this can vanish only if c is the same size as alpha.  This is the same set as in the result of other to me.  If you avoid this special set, then this oscillation in time is always large.  And if you go higher frequency, it grows like c square.  And this means if you integrate in time, then you gain one over c square.  It's like one minus inverse of minus Laplacian.  So you gain a lot of regularity by this oscillation in time.  And also by this special integration, you either differentiate this part by time, or  you get some boundary term of the integral.  That means you gain also decay in time, because you either remove the integral or by differentiation,  you use the equation, and then you get the cubic non-linearity.  So just by using this simple argument, we can transform our equation, which is a bit  ugly, but much better than before.  And so it is essentially of this form.  We lift some part of the interaction, which is when the wave component has lower frequency.  But the other part is coming from this integration by parts is cubic.  And there is some quadratic part, but for which we don't have time integral.  And then for this equation, it's essentially like nonlinear Schrodinger equation in the  sense that you don't have to take care of the regularity, semi-linear Schrodinger equation.  And so for example, if you assume that the wave component is in L2, and the Schrodinger  component is in HS, then you can estimate it by very simple, helder type inequality.  For example, for this part, HS and L4 over 3 is just a helder inequality.  And since N is low frequency, the derivative hits only on the Schrodinger component.  For the components which we applied in partial integration, the derivative is hitting the  wave component.  However, here we have minus, plus, inverse, thanks to this integration by parts.  So it is even better that the derivative hits the wave component, and essentially the same  helder type inequality works, and this allows us to go from L4 to L4 first.  These exponents appear in the end point strickers estimate.  More precisely, we need to use Coffman Bayer-Byrini, sorry, Byrini fully multiplier estimates,  but it's essentially like helder inequality.  And this is particularly simple because we are putting the wave component in L2, and  here we gain two derivatives.  That means this part is essentially like H2, and H2 is embedded into L infinity, almost  in 4D.  That's why we get a very simple numerology here.  However, you should note that this is not embedded into L infinity.  It is again a critical property of our argument, and in particular this estimate fails if you  choose the Schrodinger component in H1.  But as long as S is below 1, these estimates are good enough.  Of course, we need to take care about the wave component, wave equation, but anyway,  very simple helder type estimate together with end point strickers estimate, which is  if you have the Schrodinger equation in the dual space, L2 in time, L4 first in space,  then you get your solution in L2 in time, L4 in space.  Just by this estimate, we can get a closed estimate, and by iteration, we get solution.  But as I said, this estimate just fails at the energy space.  Typically, these two estimates are simply false if we choose S equals 1.  And this is because we cannot put the lower frequency part into L infinity.  So it is not for this part, but more precisely because of this.  As I said, it is related to the end point strickers estimate and together with failure  of so-called inequality.  So you might wonder, maybe you should change the estimate for the wave component, but there  is absolutely no other choice for the wave component if it has zero regularity.  Because for example, if you use the strickers estimate for the wave component, then you  are going to lose some regularity because the regularity for the Schrodinger strickers  is always better than the wave strickers estimate.  So if it's just on the borderline, if you shift slightly to the wave, then you immediately  lose the closed estimate.  That's why we have to stick to the end point strickers estimate.  That's the best regularity for the Schrodinger strickers estimate.  However, if we have more regularity for the Schrodinger component, we can do that.  So we can somehow sacrifice part of regularity by Schrodinger strickers.  And instead, we use the strickers estimate for the wave equation.  That is why we can somehow go slightly to the exponents where Schrodinger component  is better.  Although in this case, we have to restrict the range of strickers norm in our estimate.  I don't get into much detail, but in particular, we cannot prove that the solution satisfies  the same strickers estimate as the free equation, and I suspect that they indeed do not satisfy  all of them.  And so this somehow suggests why in the previous results, this case where the Schrodinger component  has too much regularity was not covered.  Because if you use, for example, the Brugans-XSP norm, then it implies that the solution has  the full range of the strickers estimate automatically, as long as you choose the usual exponent.  So therefore, if you want to close all of the estimates just by using Brugans-SP, then  and if the solution does not satisfy such strickers estimate, then it is, of course, a  difficulty.  And other results concerning the local variable poseness always using this kind of space.  So this may be a kind of difficulty, but we don't know if the solution really fails to  have the full range of strickers estimate.  Okay.  So what about energy space?  So in this case, all the previous arguments fail because of the critical Sobolev embedding.  So essentially, we have no nonlinear estimate.  However, this very critical case is not like the critical case of the scalar equation in  the sense that we have actually solutions either both in a smaller function space and  also in larger function space.  So this is where the energy space where we have the problem.  But for example, if you raise the regularity for the wave component, it's smaller than  of course we have our results.  Also if you lower the regularity for the Schrodinger component, then this is a larger space.  But we have also the local variable poseness always as well as the scattering just by the  previous simple argument.  So this critical space is somehow in between.  And this structure allows us to use the following argument.  It's also a kind of classical argument by using weak compactness together with the conservation  law.  It's a very simple argument.  So we can not estimate in this space, but we can start with a smaller space and try to  approximate the solution by a sequence of solutions from there and then by using energy  and mass conservation law, assuming that the solution is small, so this is only for small  solutions.  The energy and mass conservation law gives us a purely bound global in HM1 and L2.  Therefore, we can take the weak limit using the equation globally in time.  And in general, the weak limit may have a smaller energy, but anyway, it has this inequality.  And by using this larger space, we can say that this loss of energy cannot happen because  we know that the limit solution is unique in this larger space.  Therefore, if the energy decreases, then you can solve the equation again backward to time  zero and then you get a contradiction, right?  Because the energy always decreases and then you cannot come back to time zero.  Therefore, this loss of energy cannot happen, and that means that convergence has to be  strong.  Therefore, and the same argument works for the mass.  As we get strong solution and also strong continuity in the energy space, and the same  argument works for large time to get the scattering in the strong sense.  So just by a cheap trick, we can close this part, although we have essentially no nonlinear  estimate.  However, it should note that this structure alone does not tell you that the well-posed  space should hold in the intermediate space.  Indeed, another corner point two and three has the same structure, but where we have  a strong ill-posedness, one reason is that we don't have the conservation law at this  point.  And the proof here is also pretty simple.  After applying this normal form argument, we can extract somehow this transformation  of the solution is actually the worst part in this special case.  And then the problem is reduced to get to find initial data for which this bilinear form  is worst.  And it is related again because this is like you have a product and make one derivative  up.  It is related to this basic estimate.  Also it is related to the critical sub-ref embedding.  H2 is not embedded into L infinity, therefore it's not an algebra.  The product of them is not embedded into the same space.  It's essentially the same for this bilinear transformation.  And therefore we can essentially choose counter example to this inequality with a small modification  to avoid some cancellation between similar terms.  But essentially something like this, very simple, gives us a counter example.  So I stop here.  Thank you.  Any other question or comments?  In the I-Times paper on energy critical 3D shredding, they obtain the 4, infinity stric  arts inequality by 3 analysis, not by 3, if you were at 4, 3 and then sub-ref would fail.  If you try to prove it in an inequality by first doing stric arts and then doing sub-ref,  you fail for example 4 infinity shredding at h1 and 3D.  But if you prove it directly, it's true.  And this is observed.  The result of stric arts followed by sub-ref can be solved.  But then you can't do sub-ref.  But I'm claiming that sometimes stric arts sub-ref is true around the legs.  Actually, it's not always together with the stric arts.  So we have this B-town, by linear transform, for which we have to use directly the estimate  for this kind of by linear expression without any reevolution.  So yeah, it may help, but it's not clear if we can.  So I have a question, a remark.  In the energy space, can you do the scheme to increase your small data?  Yes, that's what we want to do.  And yeah, we think probably the natural conjecture is the first grand state.  But the difference from the critical shredding equation is that we don't even have any nonlinear estimate in the energy space.  And I guess the first thing what we should do is to get a nice estimate for the shredding equation,  but including the wave component as a potential.  And maybe we can get a nice estimate for such a shredding equation with a potential,  assuming that the wave component has less L2 norm than the grand state.  That's what we are trying now.  Okay, so we use this piece.  Okay.  Thank you.  And one more question.  Okay, but this I know the answer is no,  but you haven't tried to construct the solution of the grand state.  Yes, because we don't even have the local wave poisonous for large data.  First, we need to have...  You can...  Yeah, we may have some strange statement that we don't have...  Either we don't have the solution or blows up, but...  You can remove the bubble and let the wave of small data.  But in order to go close to the bubble, we need the existence of the solution.  No, but you can work directly on the linearized equation.  Okay.  Don't we need any perturbation argument?  Yes, of course.  We don't have any estimate here.  Okay.  Thank you.  Let's finish the speaker again.
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