Wellposedness and scattering for the Zakharov system in four dimensions

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Wellposedness and scattering for the Zakharov system in four dimensions
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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.
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possible problem is formatted that if you want to know how scattering theory in solid space and then there is a clear arms for that all decay of ammonia Come as you to 0 and so it is given the full illustrating the question by the power which is called the into critical powerplant plus for all but the so so this is the main problem if you consider three-dimensional or Amendment case however for the it's like getting in on the borderline cases and we're so Wendy's for this is just too and we have erratic wind direction so from this you do not really expect that the nonlinear interactions calls is a big trouble however you can see so this is adding a kind of Greek got case but that this is not really so much critical because by taking account of the nonlinear and knowledge but the only question and follow the nonlinear is not as strong as as quadratic interaction for the trading and therefore this program is not to the summit's read about that OK now and it may
show you our results that it's a very simple statement although the Rangel exponent it's a bit complicated you don't have to consider this but I would say to enable the picture so for some range of exponent and this is for shooting a component consider is for the wave components and we can prove 1st and that's scattering federal whatever deposits and a scattering of timing the polls motivate and so forth any data in desolate space than ages for this trading of born in its signal that wave components and if you assume it is sufficiently small a in cat in a kind of critical space so this regularity one-half under 0 is the weakest possible choices and in this range and then we have a unique solution which could continuously depend on the initial data and understand mostly BAT it composers that it's not a simple to our others so this is a very Carrasco state a member of the Personal Data skeptic and that they say that while half under 0 is always the number in this range so this is forced to resign Apfel I should mention that their energy spaces when cases while uncivilized 0 it is 1 of the experiment .period of this region they that lets for
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reports mysteries out went the way of a component is required to have a higher regularity and this is 1 of the whole extreme .period 12 hour range and at this point we have which is kind of a quite strong and you can regard the plants by the the non-existence of a solution or loss of regularity immediately after NGO so far the precise statement that is that you don't find that some initial data back and even for a lump sum it probably Saturday there are there is no solution in the appropriate for the function space and week we should be there if we had not opened with boldness and we can't even when we got there recollect requirement before the solution it is just the distribution of voters and with much at all regularity and and and than the initial and even with this assumption that we can prove that there is no such solution and the however since this exponent is a relatively high and activity and there is also a unique solution in a different function space and indeed the main problem here is that the labor component is required to have a much regularity as compared with string component and indeed we can construct that this result implies that our solution is in for example in a 2 2 crusts and it's too but the solution is not in the same space of the innocent victims it's 3 as soon as that time is different from the so somehow the solution instantaneously exhibit their intended that so I understand that this is a very strong but it wasn't topics of the I want results and some of it to me
in could a recall some previous results independent a similar result but I also think wimpishness local ripples nest and there was no reason to buy broader and coriander and so they consider three-dimensional are and then they local reports missing energy space and as well as girl existence force Monday in the United States so this is in that I was a guns results for loneliness reading equation using existing space and then it was later extended by edging boots to me and then all of the wider range the modern around range of exponents of analysis for generous space dimensions and it was further improved by the addition of a pair of Clermont back into under visionary in 3 D to the even more regularly but I don't think it's the speaker of the Baltic so here is the range in Geneva which to me and the veil in 4 dimensions that and the point is 1 half of 1 0 and the dates that long and 0 that did this is the energy space but they're they're buying collecting them it is not was not included and the 1 that this is 1 and I don't want the it's state of the art of minds so our results it's
extending the their so that these lines was plotting that Geneva to me and the they're so we are able to include these 2 agencies and in particular while half on 0 as late as the energy space and we can also extend the region to the to the both sides where the high-dose reading of component parts more regularly all the way from 1 and have more residents under the stairs to unfreeze was on the corner .period the extreme point where we have the people OK so this is about political and broken would oppose and the format stand behavior debacle that Saddam results concerning and thus getting silly but starting with the free provide a given timely Infiniti and you try to construct a unique solution asymptotic that so it's the construction of the wave opened the and it was consider the main being 3 so far to also understood me through the centuries out in making the Soviet space and assuming that the fine and guy was small or are there some unknown resonant conditions on the political from provide and so this exceeding being out of is somehow strong reason is reasonable because we're sitting up employment and wave component properties in the same speed suggestion that the that component is but a single at the time Infinity I just kind of non-generic conditions and it was proved that the move would aid the Shimomura an but he has to assume have lost among other conditions on the floor component for you and transform all the wave component that these vanishing of the 0 frequency that also in the sense that so this kind of a conditional form used in the construction of label operator but this also means that to date was not generic and finally do us all to be moved by Geneva on the battle and that there are also able to extend their functional space for their final state it was essentially a kind of solid space-based offered to and in 1 and they considered oversold blue dimension but I don't know of any paper about 14 inches so there was another haven't had the exciting that brought results that we don't have sent of legislation in good the today's Arizona buy and understand the memo and they constructed a brought servicing finite time using the Greek god of solid illustrating loneliness reading the questions and the approximate and fall to the on 3 D animated also proved that the full negative energy that solution rose up either in prime time .period found in 18 I thought that I guess that the three-dimensional case the existence of 1 in and the scattering their starting from the initial data it's a relatively recent progress as far as I know so that in the car with the ball we proved that the scattering results as knowledge that the here and in the data about the questioning courageous in Italy and in the Kennedy Space and almost simultaneously at around the same time 130 and down proved without assuming the symmetry but involving could decay of space the and then with the glory and 1 week somehow improve our posture is owned by removing through the region symmetry but assuming that the 180 but there is also in the subway attack in the United States some if that they it sufficiently digging up the infinity of course the angular derivative is fine and in particular the school results are contained in this case faults in the fall the case and I don't know what the reason but a couple of months without a lot of obtained independently essentially the same results as high but they have to think that they consider only in on this case which is this green buying that starting from the lowest regularity .period 10 feet to the fire OK so this is what we could do about the of previously
that so let me talk about some ideas in the group and indeed the Alcolu free a pretty simple and that is it's Prescott the known arguments and so fast we use In order to create this problem over there about the also that we use the argument at all from the but it is just that by using the integration by parts to remove some ammonia production which is also that to impact uh so it can as follows you 1st consider the Fourier Transform over each space each solution and feared that the free evolution you feel that ultimately consternation from the solutions then the remainder is supposed postal be posted subtlety in time and the you can inject just that these expression the argument formula in the food inside then it the extract that that time oscillation compared with the free evolution and you immediately see that it is given by this formula that solve of Internet for generic C under a pile you may have a bit complicated Sipple was it all of this function but here we just need to consider the case when the waiver compliment France higher frequency because we've component is expected to have a lower regularity than the shredding the question by the previous government so if we assume that the wave component has much higher regularity you can essentially Illinois it and then this is just the CSE square minus the square plus an epoxy so discount bodies but only if see is the same size as part of this is the same since it has been the result of all outstanding that that is the avoided the special state then this oscillation in time is always and available higher frequency it rolls and I could see square and this means that if you integrating time then you gain 1 overlooks the square it's about 1 of the miners in Brussels minus the process so you gain a lot of regularly denied this oscillation in time and also by the sponsoring integration you either differentiate this time despite the pipeline or you get somebody come on the Internet and that means you gain but also the game that because you either removed the interim report by the politicians use the equation and then you get the QB 1 union so just by using this simple argument we can transform
our equation which is a bit ugly but they're much better than before and some solidity the efficiency of this fall we will miffed some part of the interaction Wheaties when the way component has it all frequency but the other part is coming from this integration by parts is purely and there is some what I think Bob but for which we don't have time in interim and therefore that this equation is essentially that colonias trading rights which are in the sense that you don't have to take care the rewrite as immediate string and so the example if you assume that the waiver component pleasing and grew and the shooting of components is contagious there you go estimated by very simple and the type inequality .period sample of all the for despite and adjacent under forward 3 King added that there ought therefore 3 is just that and inequality and assisted in Maysville frequency that they about the heat on the on the spreading of components for the components through which we are applied possible integration and regulate the and the directive is hitting the wave component of here we have a minus Annapolis in vast thanks to this integration by parts so it is even better than that of the United States the wave component and insisted that same hinted that that we equality wants and this allows us from do you go from there for all to benefit from these exponents of the in the in the point streak consistent I'm always says that we need to use government media biting assault binding a 14 month prior estimates but it's essentially that could hinder equality under this is a particularly simple because we are putting their complementing and and the Here we gain to derivatives that means that the spot is essentially H 2 and H streets embedded into infinity almost in full the and that's why we get the very simple and Middle however the use of the Malta that this is not embedded in Dayton VAT deciding that the critical about the argument and in particular the system 8 favorites if you choose the the string of complementing each 1 but as long as the basis of you know 1 of these system are good enough of course we need to take care of the wave component may be questioned about anyway and simple of ahead of time but if they make good enough to withstand the point is to cut the estimate which is if you have that equation in the United States including time in 4 1st in space then you get your solution including time and for working space and just by this estimated we can get across the estimated and by Teresa McLeod solutions and that much as I say that this case makes just fades there at the end of his face and that the Bacardi these 2 estimates are simply for if he chooses its he was was and this is because we cannot look that frequency part into so solidly small but it is not for this part about the more precisely these because of this as I said the didn't to this and the Wantastiquet testimony and together with the failure of so what if knew what are so so you you might wonder if maybe you shouldn't and change the estimate for the wave component but there is absolutely no other choice for the waiver component in the fleet has said 0 0 regularity and because for example if you use the streak that for the wave upon born and then you are going to lose their stamina guarantees because the regularity for the shooting streak that is always better than the way treating the major streak testimony and solely for it's just on the border line if you use that I believe if you shift connected into the way they knew immediately reduce the prospects that's why we have to speak to the endpoints ticket system if that's the best regularity for this reading of the system however if we have a more regularly before this reading of all moment and we can do that so we can somehow sacrificed as part of regularity by Stieglitz shooting I put this really nasty cuts and instead we use system in before the wave equation that is why we can somehow boss about the 2 them exponents where trading of component is better although in this case we need to use week we have to restrict the range Overstreet cuts long analysis today I don't get into much detail about the in particular we cannot prove that the solution satisfies the same mostly 3 cats estimated as the 3 question and I if I suspect that they indeed and not that is satisfied all of them and so this somehow suggests what I had in the previous results in this case the rear misreading of component has too much of a unity was not covered and because if feels for you sample the broadens its is the norm then as it implies that the solution has therefore a range of the streak testimony and automatically as long as you choose best Eugen exponent so therefore if you want to close all of the estimated just by using a program space there and that if the solution does not satisfy such streak testimony there is of course difficult and other results concerning the federal government poisonous or it's using this kind of space so this may be a friend of the difficulty of butter we don't know if the solution reality fades to have them for a range of tickets OK so what about
the energy space there so In this case falls the previous arguments because of the Greek someone if and bidding for insensitivity have long known the next inning however that this is a very critical case is not like that creating a case of the skater equation in the sense that we have cities solutions is a bull in a smaller functional space and also in Niger functions space so this is where the energy space where we have the problem but if we don't care if you raise that regularly before the wave upon born in it's more and then of course we have our results but also if you the lower the regulator for the trading of employment and end the season I just space but where we have also was necessary as well as the scattering this by the previous simple argument so some of this Greece ,comma space is somehow in between her and the destructor adults as to
use the following the argument but it's also kind of classical argument by using what week compactness together with the conservation Law and it's a very simple argument so we stopped we cannot estimate in this space but we just start with a smaller space and tried to to approximate the solution by the canceled solution from there and then by using energy and mass conservation law assuming that the solution is small so this is only for small solutions and the energy and the must conservation gives us operated by armed robbery attempt on how to end to their fault that we can't take we weekly meet using the equation world body time and the Internet at their weekly meeting may have more energy but anyway it has a decent inequality and this week by using this digest space we can say that this a as he cannot happen and because of that we know that that he made the solution is unique in this and I just space and therefore if that energy decreases the EU Council of the question arguing about what 2 times the and then you get the contradiction because of the energy always decreases and then you cannot conduct 2000 zeal and therefore this dose of energy cannot happen and that means the convergence has to be strong and therefore under under the same argument works for the muscle and thus we get stronger solution and also strong continuity in the energy space and the same argument walks for last time together scattering in the strong so just buy cheap trick we can close the spot although we have no Monday nearest however he should note that this structure are also does not theory you that the it will oppose hold In that intermediate space and
indeed there honor that corner . 2 and 3 as the decimal structure but the where we have a strong in Bosnia's 1 1 user is that we don't have the consolation goal at this point and that proof here is also a pretty simple after applying the misnomer arguments we can extract somehow this is a transformation of the solution is actually the lost part there in the space case and then that grows problem is reduced to get to find any certainty that a for which it is buying up all is lost to and that it is mediated arguing that because this is like you have a product and shake who wondered about it up Bloomberg it gives them related to this basic estimated and also it is related to the ridiculous ridiculousness of of invading it still is not embedded in the 1880 therefore it's not the knowledge abroad and that the product of them not entered into the same space and it's essentially the same as for this biting at thrust formation and therefore had to be essentially juice company example to this inequality with this model modification to avoid some consultations between similar problems but this something like this very simple gives us the companies so the stop and if that is I think this is the right thing .period you through 3 insurance would play retains the full confidence beached stranded in Hong Kong but not with the virus carriers not quite review was for only through and it's only the 1st of these is the reverse fixture at news of the freedom on the history behind the plate improved greatly through is the the woman in the street or control the result of stressful was aware of the world used for the unit was sold for 1 kind of students of strewn with at city it's not always together with the streak so we have this the time writing a transfer for which we have to use to direct the destination for that this kind of biting expression without any some of when the applicant in my head but it's not clear if you can the questions in the community stations to use human crucial swing I use that's what we want to do and yet who we think probably in area of the natural conjecture is that in the years the fast-growing states and that that this difference from the critical shooting a question is that we don't even have any non-unionized tuning in space and the idea support saying what we should do is to get a nice estimated for the shooting equation but including the waiver component as a potential and maybe we can get a nice instrument for such a trading commission with a potential assuming that the label component has this into than the grandest and that's what we are trying not but that is his the use of force the question of how and when the operation would be directions over you have to try to we should move solution violence it is because we don't even have the look polls this fall data faster we need to have a did you that we may have a similar strains the statement that we don't have either we don't have the solution them the to head the good news is that it used to be that but you know that go across to the public we need the existence of a solution is what is always the creation of hulking don't we need a new part of the settlement we don't have any affinity for you have to do have few