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On special regularity properties of a class of dispersive equations

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On special regularity properties of a class of dispersive equations
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This talk is concerned with special properties of solutions to the IVP for the k-generalized KdV equation \pa tu+\pa x^3u+u^k \pa xu=0 on the line. We shall discuss results concerning well posedness, unique continuation and propagaion of regularity, and compare them with similar results for other nonlinear dispersive models (in particular Benjamin Ono equation). This is joint work with P. Isaza and F. Linares
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talk about what both for the KDB unless he signs that say something where baldness EU Gambro baldness accusing contraction principle then you get automatically that demand that the solution is smooth even ordinarily of that is by free you use contractual principle but you'll use contraction principal menu including emotional for example in the questioning of cases that is very hard to prove the minute sometimes you have to be welcome approved increasing the message space and we would like to argue that at the end of the book you will see that there some examples that you give up but this still you have existence uniqueness and OK what is known is that the SEZ entity have the 2000 local wood bowls by born in Medan capital In directing all of these medal that Warner's the is the 1 that you use for questioning role given that this is not a question of fraudulent use he anymore is the 1 that you use when you want to prove the system and continue the pair underwhelmed by God below is the 1 that use indicted for question rewards phonetic system however for the KGB since Gold Medal and everything here is was the contraction principle you get below .period 0 2 miles to the border in a letter saying that everything there then you can go here but you can go on to the border with contractual and this is the best you can do you want complexion you cannot have a map this is that the solution is will be low-key and this is song we've got lost no reason which more than that even the vocal line of 1 this is the a sum of money next they you have no hope whatsoever to happen dean of the pending the that with the amount that the solution is not revealed the finding meaning that if amusement center "quotation mark then from here down you cannot have contractual from here don't you can have no hope to With that affiliated with it however you go for ballot tool you have that 1 quarter Romania right this numbers the United Evening there is 1 and 2 and 3 then the Bears Ears we to Claudia here was the one-quarter In the power you have the resolved little locally mine of 160 and the city is getting by the way is getting here is mine and then you have an example that you don't resist getting there all day in the power of the you have these situation that look at the life of once again then you have a gap between local and grow and if you lose you might indeed be in this power of any power when you have the extra information is therefore a small they couldn't have lowered solutions and yet I can't accomplish value because there is no conservation a collaborative gays louder than a new regional unity and this is something that we've proven always come and raised Blavatnik a sequel to 4 this summer when he went into and tool and more recently the give us 1 it that is nation of Gibraltar in that case and 2 weeks ago we saw that the you don't believe with intended to extend that to we have you can go beyond the power finalist in hospitals and service Olivia thank OK what is known for their winter on question OK for the
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talk about the case and this is also the mentioned several times during the KGB you send the Detroit class in Detroit class 25 is afforded arrived KTB and you want you can please the truck blast using this as space this is a new varieties indicate a new models this space into himself the only if you have these conditions you don't have the condition is not true you knew that they were latitude to surpass the Nikkei to for the prosecution to do otherwise is OK do was saved up for the event and the normal you don't have anything like these students and truck traffic from this this isn't a case of cement under unique in the non-signatory you had the following some of that goal the say so bother Ewing you have already about this and at that time that the role that the meat the good that you have exponential decay to the right then you breast said financial decay for that possibility you because of romantically that means that the song cannot but I believe the while you have a but city because if you consider these operators in these estates this ability to become the Swan and his wife have the heat equation inside all the author would save the Sumerian run fast enough get suppressed OK we will get there and that this statement is not true you don't fight OK long alone so not want the weird it's OK you can work with much of the faults this OK you don't do well and you will see that you can have a single letter did at the end of the year for them event yeah we wouldn't it OK what I would say that Will they saying we will call it but
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you need never sold and I just don't kind of union Arizona is if you have 2 solutions I do have some conditions that would prove that is the solution all you have something which we can ease you have 1 Funston something happened the the difference condition is much better because you have any 2 solution for the 2nd assuming that the 2nd solutions to you and will see store in the I chose this result you know that the law would be located society covers all say the following you have to get and solutions for their somewhere here and then an open so they have to be equally and they still solution buried in it something like that I don't know this is true for the original content but in the 2nd such the 2nd the uniqueness is something that only applies the the KGB and
the following a service all of that and that state's so balls that you are in this situation that cattle had there you have exponential decay good about side was one-half another assist from this capital then the solution become analytic for the positive and the next that the system in distress forceful why you will only apply this to 1 solution is because using the cat Arizona late this month of fasting not descend on the view about what you're not indicate for power warrants OK this is also some kind of penalties like that OK this is you have some kind that is at the end of the war "quotation mark then what kind of results can you the rules in the nation and you wonder assault that apply to 2 solutions then you will like this we have find that would prove with With carols and that is the following you support that you have to get a solution of With some property the KGB was from power and they have enough they were not enough the case fortuitous in charge but I they satisfied times even close and that with his weight I'm not dying while the in a the with this way then that 1 you're not assuming any band equal when you have only assuming that they if they together to grow OK and then it went to say Well this is are these business to those go you taken a question remains what why these things happen catalyst for wide at the half here this is the decay of the 84 then you say OK that a simple they any dispersed if question I if you the game more than the fundamental solution will die then you have to be equally the translator said the churning equation I decided happy kid that is the 20 this is not the result of what he said Q appears this is any issue is that the fundamental solution the but in general before the churning aggression the fundamental solution does indicate that you have a completely different business is if you ask too much for supper for the churning equations about the you asked this ain't got and we have had during that we proved Manila delivered them about it for a simple for the during the questioning you ask that they have combat support at 9 0 they said that you conclude is that the defendants either they don't explanation is quite the the opposite the U.S. too much immediately you have to have early and the question is that you have some kind of uncertainty principle and insisting it has to be OK then they should be Is this job when the men were coming to you about at
before this constant administered the shot but this
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normally go with the professor from the University of Maryland in Colombia and the United said he received his followers of all that you have to have been the case initially then what happens you know that you are not going to presenting the case what is what you expect it that is the gate is stuck with the day-to-day times "quotation mark song and that their day at this level and you can see that when the sequel to 1 of these constant use just a number it's not beak then in that sense you have to devise getting I'm sorry
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write to both the couple pension that scene Infinity's report that you have a solution of that but it's what you expect so bothered to these function is at any time at any time 1 time the Commission is a one-time bonded dangerously and this is sharp but you can check and that is not normal site conditions this at all good
potential that is the Commission can be as big as you wish then his son said that's the problem with that you need to die for the KGB is because I've ever right and in the end it's in the heat the question you 1 time and the door will be "quotation mark "quotation mark then after all this the this OK what can you say for their Benjamin OK for
German or not you have the following situation is supposed to consider this a space that about everyone power the case in too as a bother that you intersect these forms on behalf you I remember that the band jamming on but you don't make sense you can integrate the dead at the scene and why you will be able to say that the 1st then what is known for the reader monopoly something that you would prove His mother 2nd that is also from Colombia but it is a following as respect is OK decides not to chart if you are the gave less than fly-half therefore all solution is the press and time with fights not shot at this you have he view on October 9 during fly-half then it's because you have me it if you have a lot CEO and reach 7 have it's OK everything up to 7 have you will present no you reserving half the diving then we passed from Dubai KGB 1 he the question to 3 times Benjamin on and then you say Well maybe this is grown knowledge that fact that the U.S. done with these data that have became more than 7 at this moment but a dividend insulin you will have the same became only 1 more time effectively its own logic cancelation between the nonlinear time under the new if you and never never Of course and then all of the nothing obvious known it is it even of solutions not of the is well some of that can be extended for all the powers from power more than OK OK now we're going to talk about the best said that the propagation of land "quotation mark then we are going to war with energy estimates and when you were with energy estimate for the KGB the head of the KGB is this In any case the bothered you went to war with America and you don't want to see is an using medical care then you need to control the use To keep going but this is a quantity that for many and and this is what we need because we are going to use the and then where would the war with these kind of
solutions it is How much in order to be 2 people have been the bodies and for the KDE I need a quarter of the year and something unity quarter why is because this they got estimated with the rules give you that you going 1 quarter of the year and you gain that indicating the ongoing that this in the coming up with the weather could OK we have this solution OK now that Jordan is the following and I want to remind censored before
saying this you in this locker
during this time can be I'd be very large case equal to 1 2 3 and then you have to be all the
remark is the following evening debate that I have is that he weapons but doesn't have more than this Brian then you will always obey OK we would this state NHS has with you are and you know what you can get there French because it's all about war completely then we have the following results is the following and it is also was very sad sampling hence soup that
you have added that for which you can apply your local assistance during our role to both of these days for some banks the defended happened to have more than a year then this solution have the property therefore fall in time all the regularity moved to minus infinity then it is something that I want to say is that they statement the same the singularity of authorization of the KGB move was Venice speed to minus infinity right where I want to see that this is false you don't want the fight and whether you need for regularity this for look other the quantification is this 1 that should be the quantification moreover you have even that you
have already ended this being half of the line whenever you gain impressed
1 because you have to look at more then then this is it's a statement about that it relied the move with infinite speed to the left 20 and is of the order see that it is not true you have to have these concerns and of course you are
not saying that this solution this is false yeah you're better then you would be better than any make binding senator "quotation mark the servicing he took the pen of everything here are frantically looking after notification petering OK there's some questions the 1st question is something that can detect Nakanishi asked me Is the following OK you cannot be in too you
have a situation you don't start neck In each 1 of you will never be in each 1 but how how far are you from being In each 1 well if you held the 1st of B if power at minus infinity then will be mental with this 1 because if I don't go this way he had false 1 I have to put that pressure that way down a few good 1 here you have 1 minute would and this is inside approved he said that closely to work so that it in the book OK then instead of a instead of giving a Colorado in the picture more where you can say other solutions for both let me make it in each 1 to make everything that years support that you have that this image 1 so that you have an excessive and yet the use of cereal the local-based family it's absolutely fitting in very large suppose that you have a corner here the use of syrup is not image through here contains what can you say about a solution you can say that following 14 negative about that we have a lot of assistance 14 at you have that you cannot apply the previous during you cannot apply the previous John and then you have that this secondary matters or disillusioned never belonged to the boom doesn't matter how bad .period accordingly In the past you never know when here they stole because you were installed the new thank you this is the case In black they were let it get it is ever so local constantly accusing you have also then you have the situation there immediately you know something about what happened here they planted during the the you before your delicacy that everything that happened here for the KGB having then you have the same situation he was the boss at the ballot for X in but you're having been omitted the EU held constant then you have this is final and you will never see these gone In this in the future you may have seen in the past but maybe not you have all the information you don't have this when you have look at this as a coronary and then you have a question then if you will to the good
that going with they Philippe you're assuming that you have indicated that he would welcome you remove the fact that you have indicated that he would visit to both these conditions in the Union of fraction of solar space then the answer that you have thing sold Indian cities yes and this is an old would appeal 1 of the with viticultural OK is
this related with the KGB on the North is also you can also prove that for them in on a question of the proof is more complicated you have to do with the decommissioned .period because you advancing half but the you can live for mentioning is center for so long that complicated and then you can do it for them in Jenin unlawful for them KBB here the question is is this relate that wasn't OK then you will hear a disabled for these the question disintegration that we have here if I I think but the 1 you have to be really outfits equal to one-half German underwent these probationary the is not only for the stupid and unemployment that is not true for the other but I don't know what proof -minus then the question is the was when a dancer to be no and let me give you 1 example and this is an example consider that you have I was sitting here KTV of these types and then you have to put the hypotheses beyond this question I mean this coefficient the greatest good there seemed Finnerty about talents for 4 and this is the important hypothesis is that you have a solid lead the positive approach why these vessel not truly you have and then there another provider without additional tools and you have company town for example you will have something when they hear you're not but he you was extremely nice then you have the revision of a rarity that should something happen then you have to kill this possibility understand these hypotheses and this hypothesis and then on that basis 2 hypotheses then you have a sorrow Greek up analysts now say the following
the view Haas 7 that he would there's been these problems
strong for the questioning half an hour and unique solution that don't here hold categories but the fee if you want to believe that seventies chopped and you vote of approval and is very close the reach of many of them individually you would use its adamant OK you need a legally I improvement by you for a 2nd argument you
will prove this then you have you you are not only an infinity welcomed continues the news here and then you need some continues the benefit would you want to pass the right and he had his at the peak of seeing the losing their by white was in that that because you don't want to apply wants me .period because his Britain Bolton and that's not part of the nation's right but this isn't the because you don't want to let them that you have to and then you get the vessel and
this is where there is news and analysis last year would even tho I was certain that he but in the hopper letting you will have more than a year and is then insists what the KGB you have the propagation and you
haven't been and you're going quantities there is not a matter that you have constant coefficients nursing
OK then the all but let it and this is a will answer the question but if you make a statement that's for solution of the KGB this singularity move was seen fit its people their right is false you have to quantify OK this is a result of what it needed for Warner soul and soaring right is the following OK I don't sing the newspaper that this you follow the you get that you can't stop With stopped that a Funston bodies in each 1 therefore if the time is not letting the dude who would be the 1 Of course it will be for Everton H 1 but it will be the 1 and when the time is an integer you would have had corner at the origin because at the end disappeared appealing this appearance therefore you know the that the singularity property was inference there is some a of what located here who can focus upon people you say Well OK then there is maybe the Infiniti snuggled what about age being a factor in this is based where they there the guerrillas exactly said you can have they don't have a lot of regularity he was being more than 201 and dignity that even this and you don't stay there 1 million and a half lengths of Swanson OK and then finally
we have the question amazing I would finish ahead of time so that it is OK the question is do you have there's some model that people have proposed to data right the only have been mentioned here and a woman's right here the db In a question and then you have use the situation for the media question is then the Westgrove Dennise Lopez was also by In Nicolosi last Gary In to then what happened was that wanted me that question With obligation that whatever it is it what happen if you have a singularity here the similarity will be there all the time is you are a Seagate left the in this city the world you would be circuit left the they're afforded nothing president then due the hassles fanatics there is no that wasn't there is no influence anywhere on that OK and this is the result and then you can say well what happened a few go to the come as a whole with and then of course this is invisible you they become as a whole that is an integral assistance I need this case factor here is not there then you have something called the confirmation and most 1 and they by traveling then he said the result breast and content including the involvement including note that perfect system and reassignment Chen song room uniqueness of these kind of solution see your starting with something that is the major money Western insisted their resolve and then you are not ruling that this is continues you're not having them not having any country other than the other day and you have routine I'm not really there was there was however you restrict yourself leave here everything that happened With the median here the fuel they start with logistical OK and this is ever was the and we have right in this and funeral homes you To isn't what the villagers the source of the principles or in the middle of the year service life of users our agents along the urea and was earlier this summer on the right the priorities to lower our Virginia replaced their lives on the shape years and benefited from a group confirmed that his group was rude you traffic in the U.S. the thing 1 side and only for people to hold the vote here think this is critical from In the last year of the 1st one gold they said you can also OK they should is what they if you start with something that is for the come as a whole right Danny Willett you can prove that his estate leeches all time and after that time it may not be leeches anymore it would be at holders OK but it did you can extend all yes I mean that the similarities it would be that the age when Norton of muscle this was the decision of the heart of the visitors and my view of the when you can you have the floor he said I think of the this is a problem and that is where the final along that you were as good the initial data although some time after that hold will carry all the years of of rule of law in this area for those who have use of that for you so he didn't approve of what appears to be the case in which students associated with the city was here in Glossop invested in the C 1 nor but you can continue to get this the easier to use his people have identified songs they parts on condition that this happened said solitude and the president of even the morning and have from conditions all of year the only problem with this is is on but without it said that you would do well before that this is a step toward what kind of relationship with wasn't which is the 1 have because during each 1 4 and one-half the criterion more you and you and for which