Mass-subcritical NLS

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Mass-subcritical NLS
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The talk will address two main topics: (i) Well-posedness in the weak topology on L^2, as well as its role in proving symplectic non-squeezing in this setting. This is joint work with Monica Visan and Xiaoyi Zhang. (ii) New large-data scattering results at critical regularity; this is joint work with Satoshi Masaki, Jason Murphy, and Monica Visan.
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the the EU and if you can find a way to do that "quotation mark actually 1st the final words in the title role suggestions to use imitation so that's the assuring equation far greater devotion the plus focusing the mind sign which will become more surrounded the title of which was which proposes the focus my offenders that evolution equation for field each time clique of maps on the income numbers no 1 can serve quality for this is the mask to the massive you'd integral scoring conserve the floor and what this massive critical Maine what it really means is the conservation of mass precludes interesting behaviors that interesting only behavior that small-scale because of you solve the equation about Minnesota the land on the pop but if you look at how the massive Yolanda varies it's going to be this program factor which most of this and then we have this so if you land that is going to have buy you some solution some SOEs which is an interesting nonlinear behavior if on a solution that had simmered behavior at small scales that you need to take landed To be big but I don't want to be bigger than the massive losses and please is smaller and of so massive critical is complex OK so that is the topic of talk these men these definitions will last year but now may be the most important announcements about the theory of massive critical analysts is that it's not being gay you name it talks about it but there's a lot of things that we don't understand tents at the end you know if you want to get a flavorful what's known won't do not known as the place to start is not what told of workers into Nakanishi who probably quite deeply building on the work of his references even find different but so it's not difficult in this repairs when we reasons you neglected is that it's actually trivial nowadays right now and is to prove that its globally closed in Alto warfare matter into simple applications for practice before adjustments it was torture but not this time on the
other hand 1 of the peculiarities if I can build the political firmament innocence the lineaments of critical you feel a bit like Alice's looking-glass but when looking glass operation with the mayor if you will the like Alice the walking looking Glass in that all characters are the same but they don't behave like they used to amass critical supporters In particular this is ill posed in Spain for example the the real problem and I just say Oh by the way agency is seen as the scaling directory gladly which is due to 1 2 of paper which right held his estate was his name so rather than explain what mainly proved following unpublished papers :colon interests Intel the meaning of ill posed volunteer agreed so to sketch habit of what goes on so I imagine take initial bout was also introduced when he enemies in this region but many take initial about which is hired about 1 but is very fast I think the plight of the Amistad is living truly its scale and so that sort of flat in the middle including truly scale you can and if you calculate the Zimmerman is about he 0 and recalculate you 0 in vertical space the final despite the minister announced is this can the about held to the the program to minus please it's pregnancies negatives was the huge Farrell if elevators notes effects of slow-moving what is a not much is going to happen a 4 times so you want birds and the Times said but look so that means only here it could have an effect of this timescale enough I'm agent was consider so that I knew what was going to be as 1 plus Allende quarterfinals limitation of 1 please number thank you 0 and you can see the new 1 is relatively close to 0 well it's about to the minus 2 times the moment about which admittedly is future but analysts look at times near the end of this interval so the solutions just a about 1 so 1 of the equations going to have basically a 1 here To the copy knows that have a wonderful seldom minus 2 To the copy now over time interval of late L to them L to the minors to that's going to add up to something is going to on up to a phase shift "quotation mark size 1 drink by the time derivative of an interlinked elements to take this do the face that the human dude the size of this when his size 1 plus 1 is such that 1 Minnesota 1 ones too right otherwise to small something peak power it's still 1 government peter out of most given that but so over the course of this extremely long time they phase the here until 1 minute is more not panel there may be some parts of it some pattern and so was the so but this yeah that explains samples of his like a lot but all the of resumption of Miss right and that's thing time his dimensions of links with the sea on economics but so that means that at later times a week cities around the back of the my these things American life of a plus sign between them we've been consistent at this very much like the time at the podium cancelations but were plastered bromides so this time there OK so this is the main 1 Huge solution starts very close but over an enormous amount of time is a very close together course the smell like that all right take for granted that something might the foreground but we could would use critical rescaling the 200 of the year so you make this time by the time it television's happens as small as you OK that's not terribly dramatic that's sort of like scaling critical questions do that produce anything
can happen happen Obdulia quickly but it's still a very large solution a large solution which men the safety of his and other things differentiates the supercritical a subcritical setting is the action of Galilee boats rights over news about the galley boost as simply a translation in Farida so I think my totally harmless function and modulated very highly emotive but behind the selling character the agency known romance season negative sinister up to those less time for this is needed by highly modulate my initial bout the longer the 0 rate but course the nominee of the symmetry of the equation distant correspondent shifting to moving reference frame the nonlinear dynamics the loneliness of the dynamics will changes the same dynamics it just wasn't a threat to the New Delhi burst To give small that the so it's godly well posed in the mass space it is not globally results local while post this not uniformly opposed even a neighborhood of the in the critical suffers but and serve that's 1 of the enemy imprint In the light in the mass super grandmaster case Kelly versus using the in the massive replaced injured energy-conservation basically killed and officials that had been in the mass critical of saying this is this is the message to the daily has taken serious if that is the sort through the US also continues to be adequately and that so it only if I if that's why put "quotation mark because I know that some with an objective of a series of what it does show you is that you can't expect to write down the same arguments that you do and you know we 1 of the but we so something has to change the that light the looking glass funnel effect on me the continuum therefore 1 step too far there is explicit Looking here which is suitable transformation which some sent intertwined subcritical disagree about putting a coefficient of 1 so all talk about 2 1 of which well was collected in the 1st 1 on proposition however let's take a stand equation and now it's asked so if you and so sequence functions you in and new infinity solutions in areas and then they fired massive redefined Messman acceptance of the initial about him all the sequence of solutions can reduce weekly To ensure that this last solutions
then on a fairly in a timetable courageous weekly to infinity which engenders a number of questions of the form why why is it true why
would you care and maybe why wasn't done before in fact it was done before the effective and not the statement that the proof see the
approaches the proves that for and acquistion of me trying to answer the question why what when why would you even ask this question we know well poisonous the north apology me I should say at this stage is the United States and well poses no apologize and which good business the end of the book will think of this as well because it is not unknown topology within the week college this is was statements here in a week apology so what well the 1st thing is a little bit about this that our understanding because right well Posada's me for a linear map of standard exercise in income from Elvis for linear map week pollen continuity from the political due to the weak topology is the same as boundedness which was the same as computed from the moment appalled at the moment all but for ammonia they are the same in particular Norman condemning strong on all continuity does not imply we continue we squeeze is simple example you will administer lady think about a simple example falling raised example of a matter which is the norm continues but not we continue be summarized as close to the said while some like Altos tossed characters converge weekly 0 but the images of the courage of but is the other direction troops I don't know right does the week-to-week continuity guarantee strong strong continuity I don't know Of course in terms of saying this is an interesting statement it's not relevant but I don't believe that the history of millenium the the Fed books that I the but so why why my thinking that this woman coming in but this well because it plays a role in joint work with the Commission and show John John number 1st which is about not squeezing so when it is taught in a couple of places here in the 1st conference in Wilsonburg about effective proved the most cynical squeezing for the Mets critical cases and he tried To my mind she explained that you she tried to explain why it's not it's not very easy problem and part of the owners say is that don't try and don't apply the mythology they're here there is a moderate like consortia which serve uses some of the ideas in that paper but cuts out some of the horrors that come from so what is known squeezing so I'm inside the lid state a boss In Hilbert space standards have a radius of and season out of everything in this 1st puzzled about but that they will raise up on alto around some point and this is where my initial doctors don't belong mission that is going to be and then you can ask at some later time does the solution you obtain missile and I know what I mean by cylinder I'm a cylinder with a cross sections to mention so however I pluck out too Real coordinates from a complex function like does is taking a product with union this is a unit that In just plucking out to real court and leaving others along I'm asking is this studies to real coordinates in some balked saying a complex number Alpha covering it up so now the real
question is is it true that for every initial in some ball that is later times every solution studying there case announces this cannot happen so every business in the cylinder announces know if little or no small thing progress so far indeed the generality of general simply the transformations to fit them off and in finite dimensional Hilbert spaces this is true this famous their McGraw the grumbled moments squeezing there and that is no analog sufficiently general analog In incident and prices to cover the Pyrenees but but starting with books in the great heroes can improve this decision offered general so that the more prisons but for the follows the Favor Hamilton impedes the flammable Hamilton previous examples but as a the only something you care about week and the answer it has been proven to be not following this thing doesn't work I various cases I should say the fact that we're working on alto is because this is the simplistic Hilbert space in which the dynamics Hamilton noted changed invite this is this is a given and for the PD case it's also given effect records it has some examples which show that if you fiddle with that you can get squeezed and if don't work promising but expects to wouldn't allow space that we so you have to work it out some of which by so doing we quickly overloaded history without running thing so as I said that some of the ideas that it was a pity was initiated by cops talked about this for a relatively general theorem with strong compact consumption which cover various peonies health and beauty circle and emboldened extended the Palestinians PT is unsettled which the success of tossed which is 6 applied as well as proving it the CU analysts on Torres the lighting :colon accused of light that broke in town proved squeezing for the 20 degrees equation on the Taurus effect right there very long paper proving regularly regularity global posted this in the simplistic spies and at the end of wallpaper to promote squeezing and that's 1 of the reasons but Willis various scientific reasons In the paper but maybe the bitter emotional reason is you don't understand what poses global poses fully the account squeeze it sort of a new level of challenge local poisonous from of and so Hong Kong gave a simpler proof Moseley's indicated on the Taurus Rugova was not called for durable really show the Benjamin behind Benjamin by all I thank you equation it is not squeezing on tourists and minerals prove somewhat initially that the CU Klein Gordon 3 tours mostly right now if you listen to what is passed the word porous appeared in all statements of result in effect right we're at Berkeley and listen to talk the missiles until the Mendelssohn and 2 people the new audience said but were about and so we were well aware that ideally with criticality right of part of the innovation Wilson and a paper about the cubicle listen to and as Monica hopefully explained to you it's long complicated and triggered a massive criticality we can avoid some of complexities come from prequel as the 2 of them on skids where were you would my like to know why the soldiers who go along with the Coleman news of this heart if you think that you can't squeeze in the convention to sell the insurers expected cuts Wiesenthal convention down 20 France but but it is important that this could be due to Sunday's syntactic bombings is that conjugate pair of course but so early if we were on the Taurus so was the basic idea of all these things starting with Coke's if on that you take your information BTU approximated by find mission .period which is already a pity so sodium but being approximated by fund mission Pt you invoke growth although we pray for 1 another the protocol the original grumbled statement tho some separate press the arguments and you say Well I have witnesses demands action is not done but rephrasing the language that works for us find witnesses demand squeezing in those 0 5 additional context you need to show that witness persists in the original .period so that's our approach is not held previously the phrase that is just investments and but for this for the whole nuclear space at the this is the language of but so if a on the finding additional it from the toss we could find the means to lies the problem purely by introducing furry cup on the Ukrainian space right you need to further we need to both truncated frequency and in space and it turns out the best in was the prevailing frequency deal with that problem in on itself and in a posterior space but the sequestering agent infrequent so how we I frequency we replaced by the following Hamilton PGA of its Rivesville clean you can plus some money on say myself border light in color gave him was the small narrative we take a function you projected onto frequencies listed in Figures quickly particularly all the predicted functions but that's not all and then right was the helpline at you right in the whole time with a projected you when you perform the equation difference between developing left over by the piece so by Ross theorem there exist from the stores there exist solutions that PTE which starred in a bowl and In that outside the and we would like to do is
take innocent infants now if we send into infinity the ball weekly called the we could call the initial limiting initial about a possible subsidies with quality little battling up inside the ball so we take our initial bastard and we can take a week limits and make better our limiting emissions that they will be amenable now if you're outside the cell and I mean the exterior the stillness of it's on the exterior of the cylinder is also weekly calls so why take these things at a later time and take a limit at this point outside the cell the week limit is still gonna be of service you in 20 villages weekly to something which we pray Is you infinite Yukon the vagaries Week is a relatively trivial well that's the proposition for the question is is this function actually this this function Is it the solution and that's exactly the question we propose and so this maybe on the tosses be complete solution is sort of half of this but it's all in the master cases we have a problem which arises from the domestic Kelty of this equation it looks like you begin alias will predictions which should make it more harmless the domestically analysts but that you need a major opera converged in the middle of a beautiful about right to judge a domestically-made approved the sequence of solutions UN I dutifully bounded as any runs of 2 2 no this is a pretty nontrivial coupling comes in particular as coming was a free multiplied his assigned as conclusion :colon doesn't have a OK so it's not because is not clear had approved the you could say amid the simplest version be well doesn't prove that the PEN so was just the tip of an emphatic claim is not opportunity because you know bring my son's opted to use customized funds to the blastoff right and this thing as both sides not played up from and a lot of the meat Of the paper will not talking about reducing to let lot is precisely dealing with equation if try and read proved me device called the at Bolton paper never time you see the nominee but this and that it would lead to the fundamental modernistic folded breaks on the back the nowhere so let's prove proposition those problems still exist "quotation mark so you win at time t is by the Duma formula but so what do I do now I don't think we'd limits what I would like to think we limit everything in sight so are possible here it is right as a linear translates weak convergence of the right wing convincing inquest guarantees we convince the but there know about about this so the galley boosters is they give you a mean if you H 1 theory they coming into Jordan H 1 H 1 gives you space regularity whizzing gives you tempering guarantee uniformly in the direction we agree continuous functions but there is no truth in time that makes sense but it would give you half of combatants but the Delta regularity you don't have the property right Delhi Bruce so you can have no embrace imagine a highly ocelot awaited whizzing by I have 1 moment and then moments legal because you have no uniform continuity not to you could try to go negative regularity but ill-co-ordinated irregularities in actually done substance but if you think about what I just said these galley wave packets whizzing past they don't hang around that they go very fast had we prove that they should have don't hang around local smooth the local smoothing is enough to show him after passing to a subsequent that this will convince some function space-time locally in the about locally in space once you have you can get hot it's locally in Surkhet spaces right so you're from 204 . 1 Ikena from to all where the 6 but you can't it's 6 because you don't have any compact 6 because the scaling some so against crucial crucial here this is separate but so by 2 compactness here you can extract limit once you have the convergence of these in civil LP space-time spaces you can get that this spring but the PNA ought to be harmless and after pages this turns into a hint of the click the buy Wednesday as the class in the but now are these 2 things people well maybe maybe not but it almost every time I Convergencia anything with and we reduce here but it almost every time sorry the weekly this is going to be there so this is true at least in almost every day but I mean after all this apology to the nothing about Major but but the right-hand side is continuous right why some regarding stricken section dispersed accounts so but I changed the announced it I wouldn't change the right-hand side given how appears and make continuous so we can believe this just changing the unannounced and have the souls of Dudamel equation that you find itself for now just because to solutions they to continue Seltzer solution in a massive critical case is not enough to say it's dust solution he needs to cut but basically become a Cincinnati 4 out of but it was human rights cut down because unlike domestic of affairs you can prove the solutions of this book equation of based but the exact same curfews in some prepare in the when predictions missing so the yuan's about a strikeouts nouns Sofia-based accounts balance of the year is truly vast solutions which makes you but With this state dealt with I think agree much more swiftly than the master a case dealing with space truncation we we basically no apparent that the Metroplex there's some kitchens but his fears OK so that was topic 1 I hope so how do not disappear for change be 91 million most of to if warranted warning to future because this is the same as last time but I would say there won't talk about it is scattering at the Gregory direct flights status scattering great regularity of their own but when I entered the 1st thing is to have a word accept sacking national soccer Jason Ready :colon Berkeley In the Commission telling the audience he had
a couple of conditions 1 way was massacred go this 1 talk about the 2nd listed scattering and you know linearity has some has a big enough power this simply looking at you need a little bit of nonlinear meaning that the power in the moment and the other was is gathering I the technical condition which is not the thing we most EU the change in the fields of it's going to stay for the foreseeable future policies negative just talking about clandestine 1 month now I suppose ever solution to analysts what With the interaction variable against Fulham in the beginning ,comma mechanics the interaction variables has the following property rights or if you with solution this wouldn't change so we ask that this interaction durable basis following ground rising pursued 4 so lifespan solutions with the property writing accounting standard gifts stock you can assure them that the fun you matter less than solution with peculiar property the school staff members didn't talk about it a lot then you step the agreement apologies so Britain so what did did was use status means it means it looks likely revolution if you look like a linear evolution that if looks like a constant so scarce in the states that if a case times it's best city in converges to exclude the time some function to meet some function externality called I'm so the 1st thing the Unabomber fair times in remarks the 0 4 mark start I mean scattering does imply stopped right it does do is gonna converge as to deposit and really sold matters here at this thing happens near infinity all possible 1 where is this thing if if policy will assume is bounded his apology Saskatchewan clearly apartments scattering lifestyle it that was my 1st sexual remarks but for small that if I think my initial thought at time 0 into various reasons in the paper that section about idea but for conversations with do every time I got time 0 it's in some way spell and so meticulously like I'm saying such solutions exist licensing program and so has been around for a while in particular nothing knows our approved of this problem has global solutions is loosely well closed 4 . the promise OK 4 . 2 yuan urbane meant that conditions to its clear seat and you know what it takes control look at this scaling critical spice many of the predecessors had done analysis will it's former Vermes scandal by itself he said if I write correctly right taking a derivative is is reciprocal length so there'd be city has the the same dimensions is linked to the absolutist city of course I made the mistake of you well it so this is a critical hypothesis and solutions exist and what they mean by this local solutions exist for big got some locally for large and globally small In particular small doctors solutions of a stop these global solutions the lighter right down the local close contact and tools is but a small bottle and the solutions to these events and lovers strong stations but at least we have no reason to doubt the staff holds for all solutions into the Forbes Inc but if you have a reason to doubt him please tell because I have no reason but in the focusing case the head of course it doesn't hold for all solutions so style doesn't always hold for example you have himself Of course there might be number 0 it's scary if solution happens scatter style will hold but an overall initialed after use of cellphones there sultan's break well for assault on right this is a limiting Procol near enough right this thing is we're going around the complex which me in functions for which reason I propagated by a large amount of getting its spread out a strong to lose diseases spread out of his prison a lot in this thing going so the service doesn't hold salt on different half of all that the 1st case but not a scattering 1 peculiarity again about the Mass subcritical crack Is there now salt on the stable right police GM takes off from the stable therefore GM they sold homes cannot be minimal structures the scattering any topology right because they haven't neighborhood a short space there was now you made to the right name that's just because the Caprice example but progress was so now run when the general market while you making such an assumption well if you know about the right state critical equations at non-sensitive considered Gregory guaranteed but the moment we don't really know what the except to assume that operate and there is no cancel conservation electrical curricula so with Group there is something that smells like a conservation law at the irregularity which is uniformly energy so the circumpolar ranging which is the energy of the solution for the mass critical current event comes in master equation is sort of performing variance with the solution the domestic equation and transformed by another solution which has an energy that's a number that energy Is the plate is the sort foliage of the 1st solution so falling energy does give something but you need more we need the full weight is a mail to and in general they will only give you something at the critical level if he is big the
fact that for certain falling into lead was something useful where we can critical bands it requires that the critical regularity the thing has to be smaller than the current curricula but because after all they get something that's even more negative than SAU Candice interpolate immunity clear ,comma scope already you hold OK with me with conservation of mass and bring this up 0 because so great it being too negative is not is a good thing being monitored enough that and as well as you look at that gives you a quadratic equation for P and the solution comical distress export so if P isn't too negative as defined by radical right "quotation mark unidirectional there things they maintained after advanced from the conservation from the foliage injured in the for this small class "quotation mark and this state this is a much narrower set this is a strictly no sitting discussed in here but now dirty secret my hypothesis actually embodies operate K so if we define dysfunction Jack which is the solution of the Heisenberg equations the the position operator get some interest they're easy to think about it but I'm not thinking this particularly possess a solution the Heisman equations to differentiate and following claim for the answer to that is the answer and the French about things in times using which you can also right USA no many signs of begin again if you think for a moment you will variety and particularly of the from if I'm right just by differentiating but so this is the center of mass intercept operator on right if you look at it is a considered quality for equation it's so tautological was it represents well claiming was also considered so the center of mass goes in a straight line and time 0 where is it trade houses the same domestic times here is a conserved quantities so the time-dependent intercourse for by and by this operators unitary conjugate this the unit this it unilaterally constitute favor operator Woodside location by itself appreciation so they get no I think functions of this operator which I do always see power of this operator and so on the stock news equivalent the statement that Judge this operator Victor operate heresy explained that the functions of this this thing times you to Tuesday right now if you use say this representation use of eliminating this will actually tell you but you have some opery I'm just dying to make the theorem sound was interesting but "quotation mark but the Securities and dealt a solid investment for signs 1 but so that I can tell you solutions in this space what's the harm this is story space detection this is the week week LP space with this is paid and inventive we break Greenspan of and as scaling critical think In the end it is associations the appearances it employs scattering but you know if I told you I had a skeleton critical moment of history of small right you'd expect us to prove scattering a page it doesn't work and the problem ultimately comes from infinity the fate we is the standard problem knowing that an elephant me knowing that a week LP norm is small does it mean effusive go goes far enough in time so is finite it doesn't need a far enough in time that same enormous small right it is the classic while posing as argument for critical equations right if if I helping always finite did this on on the small smaller Martin conversions but isn't working out in weeks space vendors with this problem reappears everywhere right trying to solve initially local closeness forgotten a week felt in critical weeks which spaces doesn't really work because you can't find anything small who had the same problem here for now small Data solutions a very this for why we small .period solutions scatter well 1 way of saying what we know small got a solution scat is small .period the fact it's actually in this part India the total arranged to does that mean it takes almost nothing the winner but it does take some time so but again this is not what I mean With Through the Looking Glass fibers massacred leavings of it fun rightful mass supercritical equations what you know you out have critical operate dams affinity and you won't get too bounds the continental Finnerty in time and in his 50 pages in between turning an infinity until infinity chemical where so how do we do it well in some sense of giving away the model which is like all the papers have been written on conditional well pose administrative assume operate bounce proves that scattering for a mass supercritical equations were like informally supercritical with you want to wear way of hazards 3 steps which is very real so the existence of a wall communal communal this and In 191 days and it's come and pick compactness almost produced the caller theater I I propped up only the sport for years the arrest of but it was so existence middle criminals and there was produced nematode is reduced to such similar and number 3 is we're gonna show that there exist compared to now many papers could be summarized in the following 3 sensors but of course everyone's different Indiana is of trying to assert with through the looking glass over the snow saying but when you talk to it it's different consolation prize linear profile composition in this peculiar at what you defects of compactness their little unusual their Galileo and scaling translation is broken it conversations broke missile-related UK but what only myself similar the normally sell similar means it's a function if I take you I can write it as a function all of the reminds 1 of the 3 some functions of its sovereignty so that's what so similar that "quotation mark "quotation mark means what I
myself similar with "quotation mark I mean the function of being on time as well but as you very this parameter this function varies over complexes OK so saving a single function for the truly South similar it varies over complex so behaves to some degree like a journeys similar except of course you can't use elliptic fear because of Woodland Park and then we approve this so why is the only 1 type of unit was on 1 enemy in a sense this is the only time in enemy that's consistent with what's written over there right if you look at the scale on the functional worked to to be in a week space it has to spread out exactly this race good cause easy to trivialize "quotation mark was perfect and why doesn't exist it doesn't exist because we can prove that has mess it was the solution rose up at 0 the 5 messages double 1 since you need to go on and on and on about you and you have use the new limit in the use the worried about didn't like this thing where the money is this a property solution all right there at the top of the list of that and at times Europe the rest of the guess is opposed to this well but there are many ways to motivate its right 1 is really to not the case at all right now even if you have the care Infinity which is slower than critical norms that means dedicated fatiguing continuously mess with you forever OK so but we found a scattering of right if you want more severe singularity the origin well then it's probably opposed this 2 answers the problem on a In this is the view of the world of work will good luck is not using the right there is a lot of factors of your reasoning that the city had about leaving you will be easier if the are things are so wondering this is it this is a space for which have lower pose the Baker made it a condition at a single time so I can sample the solution a single time that a little bit and still constructive solution into the future right this at a single time doesn't mean much of anything and if I do some hatches a time where it means the most thinking Maine if I made even a slight modification of the solution might not exist at all that does not mean you what you want is your bootstrap assumption your productive hypothesis right the union that the hypothesis that actually has the solutions right and that that's right that's why so taking critical LP bounced doesn't it wasn't going French international criticism the the on the were the Monday's meeting livestock is because the meat sold out at the literature is enormous lies dotted dishes quite scarce but there's me I have something to say think about I will have some said that this moment comes in and says I'm from on and on the few young my