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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1

Number of Parts 
21

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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Release Date 
2016

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English

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Abstract 
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 nonexistence of weak solutions.

00:00
Complex (psychology)
Logical constant
Model theory
Dimensional analysis
Dimensional analysis
Oscillation
Population density
Electric field
Nichtlineares Gleichungssystem
Summierbarkeit
Turbulence
Physical system
Spacetime
Physical system
01:48
Axiom of choice
Threedimensional space
Free group
Existence
State of matter
Model theory
Connectivity (graph theory)
Limit (category theory)
Inequality (mathematics)
Food energy
Dimensional analysis
Oscillation
Musical ensemble
Modulform
Spacetime
Nichtlineares Gleichungssystem
Turbulence
Physical system
Area
Logical constant
Statistical dispersion
Model theory
Exponentiation
Dimensional analysis
Physicalism
Mass
Food energy
Existence
Asymptote
Flow separation
Wave
Shooting method
Computer animation
Nichtlineares Gleichungssystem
Funktionenalgebra
Physical system
Resultant
Spacetime
04:58
Existence
Functional (mathematics)
Model theory
Connectivity (graph theory)
Multiplication sign
Correspondence (mathematics)
Wave equation
Limit (category theory)
Regular graph
Mereology
Inequality (mathematics)
Food energy
Dimensional analysis
Product (business)
Oscillation
Goodness of fit
Estimator
Regular graph
Resonance
Forest
Spacetime
Nichtlineares Gleichungssystem
Turbulence
Linear map
Physical system
Potenz <Mathematik>
Logical constant
Statistical dispersion
Weight
Exponentiation
Mass
Food energy
Lattice (order)
Asymptote
Existence
Shooting method
Wave
Arithmetic mean
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Estimation
Nichtlineares Gleichungssystem
Cubic graph
Right angle
Physical system
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Directed graph
09:24
Computer programming
Scattering theory
State of matter
Connectivity (graph theory)
Statistical dispersion
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Maxima and minima
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Power (physics)
Frequency
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Term (mathematics)
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12:21
Point (geometry)
Potenz <Mathematik>
Distribution (mathematics)
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Point (geometry)
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Different (Kate Ryan album)
Theorem
Summierbarkeit
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15:09
Point (geometry)
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Finitismus
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Range (statistics)
Resonator
Solid geometry
Infinity
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Regular graph
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Explosion
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Regular graph
Operator (mathematics)
Modulform
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Nichtlineares Gleichungssystem
Weight function
Condition number
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Addition
Potenz <Mathematik>
Statistical dispersion
Forcing (mathematics)
Uniqueness quantification
Mathematical analysis
Infinity
Range (statistics)
Food energy
Line (geometry)
Category of being
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Computer animation
Symmetry (physics)
Condition number
Asymptotic analysis
Funktionenalgebra
Arithmetic progression
Resultant
Local ring
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21:31
Axiom of choice
Group action
State of matter
INTEGRAL
Multiplication sign
Wave equation
Equaliser (mathematics)
Direction (geometry)
Range (statistics)
1 (number)
Parameter (computer programming)
Solid geometry
Mereology
Explosion
Derivation (linguistics)
Estimator
Resonance
Phase transition
Square number
Cubic graph
Series (mathematics)
Physical system
Proof theory
Potenz <Mathematik>
Process (computing)
Sampling (statistics)
Infinity
Range (statistics)
String theory
Term (mathematics)
Oscillation
Wave
Shooting method
Order (biology)
Normal (geometry)
Right angle
Angular momentum
Resultant
Spacetime
Point (geometry)
Computer programming
Functional (mathematics)
Connectivity (graph theory)
Modulform
Maxima and minima
Fourier series
Regular graph
Inequality (mathematics)
Product (business)
Oscillation
Frequency
Goodness of fit
Regular graph
Iteration
Wellformed formula
Spacetime
Nichtlineares Gleichungssystem
Normal (geometry)
Free group
INTEGRAL
Statistical dispersion
Consistency
Expression
Exponentiation
Mathematical analysis
Basis <Mathematik>
Fourier transform
Line (geometry)
Evolute
Film editing
Algebra
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Function (mathematics)
Normed vector space
Integration by parts
Game theory
31:11
Classical physics
Multiplication sign
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Approximation
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Analytic continuation
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34:24
Point (geometry)
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Nichtlineares Gleichungssystem
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Cartesian coordinate system
Affine space
Proof theory
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Charge carrier
Maß <Mathematik>
Spacetime
00:03
as time and and you have to have a to during war and so I reckon and the organizers for inviting me to the next conference and please cancel and willing to talk about where was necessary and a scattering question for the 115 in space the full dimensions and this is a joint walkways that you wanted to know the 1 under the bus to her he said but so let me steadily the question this is the kind of Glasgow and London and a specific question which is a system of those trading equation on the way the question and the it's the kind all the more the equation for the plasma which is called the nonunion tug announced essentially generated by the nonlinear interaction between electrical stations and the hydrodynamical sedition so here you is that somehow describing the lasted almost of the electric field and that and it's describing the motion although I on density so it is spacetime functional units complex and innings Rio here under fire it up on it but which would be fixed throughout the picked this talk that it's as a kind of it's a positive constant for the trial speed a sounder speed so this is what we questioned the main reason
01:50
that is the kind of grass that question and the simple we consider this the specific undertaking space dimensional for and approved by some people ,comma on the of existence of resuscitated wearables mysteries that together with that scattering the Free State established linking the attack so this is the mental for me for that kind of weird for this model because this season of physics in the area of threedimensional space so you Jerry people considered this on the in threedimensional moral about that somehow we find it an interesting mathematical questions and in the space dimension because it brings kind severe Sokolovic criticality organizers so just call simplicity of the location it is better to transform the way the question into the forestalled just by decomposing confusion to the incoming and outgoing wave and so on so that we can regard it's kind of trading apparently and then we have this title system forestalled equations and so kept them and is just that a company's expired awaits components and the smaller and is the real boss they said it may be viewed by some basic facts
03:22
facts over there the the heart of the system .period although it is 100 billion system and 100 billion is given in this form and I must that albeit along for the shooting of component is also quick result and therefore as a solution space although functional space for the initial than most natural choice is to choose this trading no components from age 1 and that the wave components from him do so that we can make sense both the quantities at that is that know what dimensions and then you immediately see already here but for the majority of the kind of critical cases and because the makes sense energies in that in this age of crossing to space in the control than on and then just by using the here there and the support of inequality this can be controlled with and only if the demand to its borders and the demands is that critical cases where this was essentially the same however since allied with the it's not the skating lean body and that it is not the they could hear what is the skating great exponent of the book but and if you take the socalled a subsonic pretty neat and where you regarded that it's not sit on them the sound of speed along the monthtomonth accounting it corresponds to taking that any of us want to the then the cold spell the of book because we have this
05:01
equation for the weight component if you stand for money artifact impunity they you see that in the should be equal to use square then bragging this into the 1st equation you get simply the standard focusing crossings cubic nonunion shooting equations and the all
05:18
this equation it is skating invariant under a little that this is critical in the energy space for 4 dimensions so in this case it is clearly a skating Greek counterparts analysts we want other solutions in the sort of in the United States and the fact that the path to think however for but that is the from technical viewpoint that the heart of the system is even more critical down there shredding the equation under more the through the use of a streak of and the here at the somehow because it is the system although wave and equations and that exponent exponents which we can use for this ticket estimate is somehow the meeting and we are forced to use them in the . 3 it's involve including time and is porting X and under this is the exponent if you consider include photos shows if you consider the chancellor something you get 1 regularity in this space so you consider therefore forbid about giving it a fall in space and then finding in this space is a critical and it's even more critical in the sense that the support of embedding fades because of this "quotation mark corresponds to the case where it should be embedded in the inning people that this is 1 of the forest and this causes a lot of trouble at the goal in straightforwardly but so far let me just the recorded of the difficulties that which exterior somehow Prescott and the essentially started before however it is not only for the for the maintenance case but the Internet of the heart of the system and foster good problem is therefore that Okah existence of solutions and even that operate existence it is slightly more difficult than the shooting the equation in the sense that all just what the wave equation in the sense that the lonely legality has said that about and sold for example you consider the simplest estimated that just the estimated that he made the right hand side by the the deal In quantity then you lose the regularity because of this they about on the monument at the end of this can be easily seen because for the 1st although I question if you apply that energy inequality then it means that the right hand side or the source that should have the same regularity as the solution then and for example you assume that no solution using engaged space for it's not enough then you square in ages and therefore would be and edited the right people use questionably age is minus 1 and that implies that wave upon on and should have won this regularity however if you bloody thing to hear their intimacy you will have its it's minus regularity and that you cannot go back to status for the setting up of about this is the problem and that the heart of system on Internet and all this kind of aggression and in particular for the the heart of system this can be easily solved that by a few different methods and taking account countered that the nonlinear part they interaction is not really resonant meaning and these 2 products of 2 functions the lottery Eddie interact if you take account of those situation and there are easy ways to remove this problem and the
09:26
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 threedimensional 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
10:48
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 onehalf 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
12:24
smaller the powerful large data we can construct and local solutions and this the exponent plays in the energy space so that experts except the this special case for the for the other Rangel exponents and we have a local solutions for the same intended for the initial in the same space but look this the small for this is simple wasn't resented and we have also been
12:55
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 nonexistence 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
15:10
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 threedimensional 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
16:43
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 highdose 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 nongeneric 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 spacebased 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 threedimensional 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
21:32
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
24:19
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
31:13
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
32:17
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
34:25
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 fastgrowing states and that that this difference from the critical shooting a question is that we don't even have any nonunionized 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