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Global regularity for water waves
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Title  Global regularity for water waves 
Title of Series  Trimestre Ondes Non linéaires  June Conference 
Part Number  8 
Number of Parts  23 
Author 
Pusateri, Fabio

License 
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. 
DOI  10.5446/20818 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2016 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  We will begin by introducing the water waves equations which are a system of evolution equations modeling the motion of waves (like those in the surface of the ocean), and discuss some of the works done in recent years on the question of longtime regularity. We will then present a recent result, joint with Deng, Ionescu and Pausader, about global existence of smooth solutions for the 3d gravitycapillary water waves system in infinite depth. The main difficulties in this problem are the slow decay of linear solutions and the presence of a large set of resonant interactions 
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00:02
more among more than 1 In the old and the generosity of the direction of commenting on list saying that it brings in a formal reply :colon writer and focus on like a bank organizers of the march with indignation to the conference saying that the speak and also the stadium I had a very good time actually in Europe although people discouraged it but said it was good having a veteran's benefits assertion that there was a great but the all right so and I apologize to me and those of you are seeing if not a new 1 of development stock look OK and that somehow to do this is also a diverse audience and the 1st part and I need some sort of more general introduction to the problem so that could be boring for the experts and then the 2nd part of top mourning the death about animals isn't result is jointly and then post at the grant Vermont there and they didn't get the names and Madison good because this is a menace on the 2nd part of books are about the most visitors on the gravity of weights but before they give you a list of merger and election spoke about the models and disaster of the new law approved recently so the general goal is to understand the fluid motions for a food feudal complies region that it's moving evolving with time for Astana example is on the surface of the ocean and there this is a question
01:56
posed by the loud blasts sometime ago where these labels the problem for what we are asking that given any the initial disturbances also localizing ability surface what will happen next pretzels and musical beaches of some 2 different animals when you say perturbed the drop on the surface of the water and you have created the ripples will interact president samples of known introduction of wage and you see some sort of phenomena which millions more twodimensional
02:29
pocket knives the shares which I want municipal talk about the models at least study integration of introduce also described a minimum tools of makers understand more computers phenomenon that just what the job losses such as for example lively form in Pentagon whether these are reported and so this would be some sort of questions about something of world rights may be connected to the
02:56
information somehow of for example you can think of models were off fluids right here the solution would be more unstable 1 type of instability that once users for example a theft of that she past each other creating some sort of formation they go the variable and assaulting a phenomenon Of course observed in the results of experiments and they can be modeled by the the 1 with the original show introduced at this and some approximation all right so this could be a nice of future research research but selfanalysis and rates on now so what is the money OK so interested in in a fluid this industry and so want underscores and it's incomprehensible and the floor it will provide a region the changes with time so it was only at the end of our and indeed an article in the which is a function of space and time the velocity messages on find frighteningly 18 pieces skill of functions the pressure and was the best we shall which is constant for me in this world basic and the operations of motions will be inside the region of the flow in when orders equation of the just the Newton's law here and on the right you have the internal force given by the pressure I accepted by each party Bayou particles on each bottle but is running ones and then you have you might assume and for most users will notice the presence of gravity OK so acting in would be the last quarter of space and the food is in compressible provided evidence preconditions and I will look at the most of the bushy problem so specifying initial that I will go that of the specified is the logo which has to be In compressible anyone else sources provide initial demand OK I know the system is not closed yet but some also the submissions articles is recovered from the at this inside the domain but it is the condition now and 1 it's a specified is how does the domain the ball because of the city for example does the boundary moved and it is as follows so was the fluid moves the domain is morning and is operating consistently Seoul in the sense that particles that the boundary always saying this is the 1st major I have a party on the Bundy at the initial dominated time despite the listed beyond the boundary and the food just evolved consistent with his boundaries and moreover the way that this would move is In the direction given by some is proportional to the number employment of the velocity so losses tangential there would be no motion otherwise the velocity of the particles inside would push to food to move program so this is just expressed by this 1 this time gentlemen universe Baston boundaries and there's 1 more so these are city and conditions than you need 1 more a condition of the boundary about the pressures suppression said is determined you need to know what it is the boundary and either said that the pressure is constant pursuit animosity was able and willing to match the the atmospheric pressure or is just so that so you take into account the surface tensions effects on the surface of the liquid 1 will that this is the pressure is proportional to the mean curvature of the I have to the OK so this is so either you could have a lot of as a deal might want included in your model depending on what to the goddess accompanied with itself soul and what there is actually 1 of the leaders with the highest of essential and this is the worst surface tension even think of it as restoring forests because if the lead with tends 2 Ben right this would push up the name covered with which pushed of the value of the pressure of the boundary and so create a larger gradient which would then decelerate the my soul was then make the tend to relax when some sense this could be a fault of restoring force at least on the the uncertainty scales
07:29
and for example you see that right so at the interface of and wore the Pacific Center Station results from the greater action attraction is Montague us with the once inside the fluid rather than the ones of their which and now as the fluid moves 10 right so you can have a crystal festival now these are the equations wrestled is the system we have wishes inside and the binding conditions made enough several possible scenarios depending on what type of physical phenomena want to model OK so you can have your domain only that it to be entered areas and waiting adjust for sums of curve and the food on 1 side of the net and now you can have the model with gravity and or surface tension could usually call revenue is among a growing and most of Ascension Wankel Kabila with the wounds of Ascension roads go now the fluid that in addition to you my there might not want to assume that Tesco's 0 In which case recorded in additional of 0 program analysis said Anderson who also wanted a more general models we off to fuel its program and that will would create mean old source of possible scenarios you can have the food that is lighter than in either as they can be on top of each other that much here and so on Net but this would be essentially felt avoids you just have the same set of equations on the other side dissension and there's just some condition about but that the biologist move consistently with the with the reporters so there is a and many possible things 1 can consider but so that 1 of the main questions for for this evolution equation Russell Train the main question we want to 1 led to study our principal the level Boston so given an initial velocity field which is nice that is free and initial domain which is also there are also featured regular can construct just solution local environments is the streusel question and was it's already known the for submission now I there's some configurations that after you start the new console maybe for a short time and lead to some breakdown and what does break down you since although we think also up in this model and then another question is can you so once you know that it will come construct solutions for a short time can you avoid locked so candid with some conditions may be initially does so that solutions will be for a long time to maybe forever and was there and believe view of the therefore there are many more questions you can ask OK but the soldiers of the basic ones and the ones that have been started about the bushy problem mostly so far
10:16
but those so I stopped at the suggestion of Rome 1 to start to and saying that things can go wrong will go so do try to understand why the model is sort of completed the study we have
10:30
examples when things actually can go wrong so the first one is by Gustav 1 Lefevre an incident was remanded which is Afonso so if you think maybe just over the border From the situation where you have a wavelike on the surface of the ocean right so maybe something like this and this is a graph of a function then you could have solutions where if you if your description of the Ford is a gruff OK that might fail to be so ugly that binds women a gravity very steep and maybe at the time it will return right sold some sense this description was told by using nothing wrong is happening With the problem with the fluid and then the equations also will make sense just a description of the graph 1 not but the end and more importantly the result about splashed instead of Custer Lefevre an answer settled surrounding International which where the constructive solutions where what you have is what is in the picture than it would be rescinded digitally of fluid inside here where there water and vacuum outside and you can have solicitors the interface between the food and air and that nothing is wrong you start such a big and after a short time there are a few was the velocity and in a certain way this will serve in the 6th agreement so now your description although fluid as being inside the region bounded by Ismael Kurtz OK that sort of stops making sense right and even up to continue the solution of this sense this is the issue changes and this happens all this today was his disaffection .period here but the interface has a promise to remain small because the example that has been constructed as is something is large right here for example is large is the inverse of the distance right but nothing this is wrong then naturally the model division can stop making sense but for example if you had had fluid inside but that such a touching invisible to directory of ponderous creditors cannot happen again and it was collected from minus and couldn't score so this example of some type of breakdown of relations with the itself he said of so it's not so if you just ascension facile resolution to even just for local existence you need to have the surface tension otherwise sensible that's correct and then for short times I didn't solve it doesn't really matter if the pretzel so you could have the model with 2 few accepted this cannot happen isolated exclusively other was Wilson compressed the fund which is with city's 0 on the other suits and there's no way that 1 component the political quite forced to the ground and wanted to those that can In the union the above this village residents and mobilization the site where the offices taxi rides at that point where in 1 interface historic Forest known before touches here and there no way we're 1 of the flight was that it was going to lose all of the injured that is the view that another prominent either by such a force that prevents the admissible resident as most of the money managers is the OK so this is what can go wrong now on the other hand admissions for Nice and more danger there that can be sold locally and there's been many words missile early works from the seventies and eighties and nominal through the holidays but the efforts was also something the admission that I think of evidence more and small data that there's been lots of very contributions by what credit costs and Shannon weighing in water that is also relating to the relevant solutions constructed .period models such as the publishing Escamilla trying the sense that the more the behavior beyond the level of balls and this is for a small data now there is a big breakthrough came at the end of the 90's when seizure which showed that no any so what the dark but it's an insidious Sobolev the initial velocity and American Lieberman nobody knew about the witches of solar regularity it can be sold for short times in 20 images so this was that if you think that for 4 most of the wishes that we know usually the level of witnesses in reality or something simple like this isn't a very well is not it's not for this problem and this is why there were many more words that try to win but 1 is also give us a better understanding and color many more models for example this case the case trade was irritation at this money but had been with the city and many works and that the said the very generous also even at the bottom with the very general conditions and then more worshippers ankle that fall and then people started thinking OK what is how long can we go in terms of regularity right on the on the surface and decison introduce and is in the words of prisons warrants Flanagan as our brokers will have a series of really great works where they treat their lots of Monticello ravages surface tension and have the best results in terms of regularity in the OK so there's also were voice but I mean beyond the list of people that the but I'm missing someone I worked on this I would say that the lovable and is a very well understood but not that well in the sense that there's is no unified theory I think that we can treat mothers injury legend riding some kind of simple proved that it can be spread easily or satisfactory the military rank
16:50
of they have to make some efforts on all to do with other theory OK now the 1st question was about and that's 1 of the most interesting the question of a long time ago but would poisonous for crystal what is a set up "quotation mark before I started writing them down a bit more about admissions set the following 1st of all you want to assume that the fluid is irritation so you have orders equations here then you will have the kernel of the is 0 congressional if you think of whistling through mentions it's obvious that you won't believe otherwise you'll be freezing waters equations of Anderson's this would be harder than always in 3 D and the 1 the nonplaying engines what is it is transported Rice principle it was the something that grows but but it will be difficult 1 and as in the program and get the idea would be that if you want to control solution for all times things the Chadian army given what does not could be the problem but sold so far this has been assumed but maybe In the future Edsel but so long as they're so inclination something can be done right but now if you have a focus of sentencing is is 1 thing so the thing is you want you didn't abysmal mostly because of the examples that his before that if something is large right then that could be a breakdown the fact that also but the very reasonable assumptions for what is it was the question so you wonder of a solution to be small and stay close possibly into 1 of the great rooms and the most simple won't be getting women which is a flat interface which has given the expense ends for example if you have insisted that that would be a model for wastes in the middle of the deep ocean but it will also potentially have a bottle and there's been some work on this sort of about so now In this case the level equations inside OK you think you have a glacier of Europe following with each of the next my answer is actually being on way or are still debating it you're talking about the plan a showcase of the dimension the interface will be either a curve or surface and in the summer quarters equations another bomber conditions if you have such a rough conditions were simply by saying that the the mediation is V. dollar censure the normal times the surface the peso these are the questions now if so besides the reason I gave for the group being 0 the win something fundamental the a there's also like some advantage a description of the equations so if the facilitation of the indeed would be given Bible which is unwanted and so you can reduce everything to studying the trace of this potential so you take the ball better from inside restricted to the boundaries which is given by his curved solutions to discriminates each of its and then right down the equations OK so the equations of the previous submission this 1 since he has available might function beyond what the people memory and and this is Audio equation 160 rested for the then shot is essentially been was long restricted to the to the surface of the presidential already inventory materials also designed equations endit the proposed on the on just on the bond so Xia just are at all and this is a system of 2 equations for the evolution of the interface and a bold action the please now that's up to think much about the form of this the 1 thing you see here that the the gravity and the surface tension and the only so continued lenient terms so this is deviations in missing a lot and adding that there would be about innovation sold you really see it having or not having them as already being prepped for the on the the Lena on the solutions what is it then reversible system which by that analysis it everywhere open this is where the United States .period arguments with global solutions because of the loss of derivatives and 1 thing is there if you write in this way but also and we it somehow the equation is known in the local just because you something about submission from the date of his return from abroad and refused no 1 can deal with the issue of the domain moving in a different way than the way did here which is why fixing saying graph of course for the Minnesota grafted and something else but it wasn't important as Martha Court in Sweden greatest problem in various ways but this decision was so constructing Lovington solutions it's very challenging as we said before and now if you want to go beyond the level existence ties exist in time and medical solutions you will have tools of the basic mechanism would be to use dispersion .period of the new directions and now in In every morsel where you have a president migration and would that ignoring ideas right which potentially the data is loaded program so in this set up in italy freeing some Olander online and extend the classes matters such as energy matters than that of its members of normal forms and all the freedom and of using similar equations you have to the reading of this and is different context now let me make a few other submissions about about the system so 1st of all it is an unintended structure has upon by several in the and Oh by the way
23:08
this is usually called the Sakharov desire and coordinates the exponentially each which and fight and this is called the initiative Zakharov of great formulation and
23:19
affirmed that isn't structure will be fully this is considered his also Tonya and sleazy here that you have granted the Englishman and last and this would be the pollution of surface tension here and this is a comedic energy OK which was possible after the receipt of the potential because GE is a surfer joined the firstorder operator now she the only known of course if Kosovo .period vessel you don't get much control on your solution just by the conservation and now since raises another said loss staples to a flat interface leafleting arise around 0 what you get is the deal
24:04
radiation is the house space the additional is just that a lot of Brad and it would give miners GH and c mon publishing as this would be the
24:14
immunization at the close the flat solutions and so they have a special relations is rightfully it is given by the number juicy right of gene Cincinnati secure OK and also sold solutions of their brother of the former ROC leading ways of the formula from 1 site to another Of the shortly so now as they go toward explaining that the main result in sitting in the 1st and see what has been done on the question of Wilbur regularity and commiserating one's life so but the 1st result was seizure result and she showed that if you are into limitations With gravity waves the surface tension they can construct solutions which he adopted time he 1 already on possible reaction would have to the size of the data but then later than this invention members which advises you will prove that there you can actually found a higher dimensions of was solutions and this makes sense because in 1 dimensions your conserving the muscle ways but they're more directions whispers and so your solution is supposedly faster and his gives you more control luminary open for the rest of this is for just exactly so all of his life for either Jewish 0 and despite its forcing a 0 0 jeepers and the same result is also to interventions Solters remembers for them but that with that implemention Sir Alex and I wouldn't want to 1 would think of this result untouchable solutions but and there is a wide it's going be done before is that there is something normally that happens syllabuses time by me the 1 squared and wants to capture that doesn't the bid 1 of the main things that makes you this the solutions and enclosed argument for time going infinity 2 many independently at the same time as the law you different prove anything and the about and nice proof of removal seizure was result almost Lublin and complemented the to approval of the global result the solution 1 the violence what it is we rebuild this result by improving the space is innocent incense which is a good thing today that they have uh infiniband opponents leafy branch of Islam particularly removing a moment condition simplicity in 2 dimensions if you assume that the candidate is fine for you have of the people so all of this so far is that it is just that G generally all of this is Fuji positive and thing missing all of this the right and now I do the opposite so now the results was 0 and seoul doesn't let and Apple didn't read the indeed so there is a lot of problems tho intended something dramatic is different and so it's a different problem some but it will be at the of noble solutions 9 police also had the problem because the reason is that so you see further and you have to the onehalf of gravity waves only up to the treehouse no good console this is a special relations which means that the waves with more frequency travel much slower here and so they tend to hang around this creates lots of problems soul the 0 frequencies in the issue here and there that's what able to solve the problem and also if meant that naval proof of global solutions of Woodside in different assumptions so this is all for lowering now so what I want to talk about is the problem where you have both marriages of a sentence soul and tried to explain why this is really a different problems OK so 1st of all so mistake on the beaches and we said that we can describe the the solution by the profile of the way then the potential restricted to the interface fight and now it's useless just calling you miscommunication which is essentially the viable there to buy them Estonian OK so nice it's efficient and honest nice initially docile Finkel was arrested and others are the more precise assumptions so you want to be the agency of summons a large of sufficiently angular regularity soulmate here is the addition of the field dimensions and gospel the case 2 0 at infinity is only going to device also closed it hasn't played a lot of thought solutions None this is true then you can have a global solution for the problem where you have bowls gravity and surface tension In 3 dimensions so in the case of twodimensional interfaces open so now what I tried to do in the next at least for the next 10 15 minutes is the citywide problem is this different than the others and 1 of them many fields and any fivetime and some some ideas of the proof of this of some parts of the world the soul that a major difficulties 1 is that long .period specialization as an inflection point if felt both ginseng wine this gives you less did even for the new solutions and yet the difficulty is that if you look at the wedding parties in the way they interact them that much more strongly than in the other cases have been infiltrated with me and now I
29:53
feel I think of the weather was equation another theory and with multiple think about is just didn't take just the linear equation and then add it without acknowledging OK so now here uh alter the way we we had in in such a way that the enormous preserved but OK this is if I have time I'll discuss why this is a bit more relevant but if you don't that didn't solve this problem but then differently than some of problems sold and here you can think of this model and it also restricts the city the abuse and functioned as restrictive frequencies was just on radio now so that it loses amenities and so these Samina part with adequate poisoning in only 90 and in this regard all interactions that are not so relevant eventually terms and also is a good model to to work with it considers the mask that issue with poignant and I sure reduced to this you often work debate with the weather was equations about ization embodied in the way that others are Britain's leaders did with the white position that something that is different you to double as the equation usable unknown overnight but so long as people of the but some also you can concentrate on all such
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integration so this is a leaner parked just like it from the equation which is very different than would different to dispersion relation and some part of the morning and effect of the form the OK so now is the basic strategy in general on this is that the following 4 opposite the problems that if you will you don't do it that a solution can be contained as long as it as finalization right and the basic you will try to positions and further it is by constructing energy function which is educational arm blasted into the storm's nativism that the fields which if you have imbalances again but for example in that case none of additional environs but not stealing but wishes that have other to equations that inevitable some kind of run on by the annexes initialed right and daytoday so solution that is fast enough right Europe there enough then you them approve something like this but now it turns out that Ford was such a fan was after he ordered the dissolution the case OK so you'll want control the energy by assuming the sources of dispersing and the way you do it is by some estimate of getting the norm or some other way norm can get to the summit later In the 2nd part of the argument you try to prove that assuming that different from the energy you will we proved that the digital assume the went on the intricacies of the wire will stop argument because of this is the basic strategy now what happens In the waterways cases of so this is the integration now if you have 1 of the 2 signaled G was able then the special editions on the genitals and munching edition and you get the full gates of the is the dimension of the of the physical space and but the minus 1 is the nature of the right was he was also not as the secondary contends the same but the data up this is more frequencies would be different but safe if it is 1 the prime rate for that so important if you by the time it is cancel fully game the foreign news us the now and then and if 1 of them is that it will hold them and 0 right then you have an inflection point in school this fear of infection .period demo 0 OK and you see that so this is the dispersion relation director of the broker business the velocity which has a minimal at the dam was 0 which is this value here event in Seoul and therefore you will lose 1 receipts over the but that is like any competition but then I hear you see 1 important thing wasn't as dumb as usual there's another 1 which is demolishing get too hopefully at the end we should go unnoticed for which was anything particular on the in solution rights something that will appear later in Monday nearly nothing special here now there are 2 major issues with got problems which all the news is not someone any either that I know where the is presently equations have 1 of the 2 following features for either during the day was to the minus 1 allowing you to close underestimate more easily and this is the case when you have 1 of them being 0 into dimensions yet the from the game or if you have a full decade you could remove the part of the 1990 that decays slowly opened for this the procedure that can be done the last Mama form that the moves as the leading the and this is the case of the 2 models both gravity most of Ascension and sophistication of proposal either 1 of the 2 things was necessary so important now here others said a fails because of the over 6 and also the field and the various is a brief explanation so if you want to remove what it from an equation you should look at how 2 waves interact so as did away with figures in adopt with with a different things Wixom as he does with the see and this isn't a special relations now is 2 of them never combined to give you the dispersion relation of the sum of the frequencies then so this expression is not the than at least of the former level you are able to to remove the bullet terms OK now so if you have gravitywave sort of the Kabila ways then you have some solutions exhibiting ones for civil example readable 0 KSE was here with the right side OK so that could be treated although in some models harder now if you are both gravity and capillary waves young lots of solutions of this proposal there is no way that you can remove the verdict and it is just a here seems fixed To be 100 0 just the America and is the tracing the doubling of all the solutions but then there's nothing special but this is some sort of relief but but for this expression of this onslaught of solutions in place so there's no way to do that and so the mechanism must be different capabilities said the image but the amount ,comma which is the frequencies that into go into such residents relations or forcibly about children as those out such cover here but dominion holding you don't remove them and even if you the from the day that the number of the Fox said something to remove result ,comma police and popular property but they haven't opened brought you open it you open so if you removed just enable them then Simon inevitable some old bonds on the stressful and they have to it as fast as if it was the focus of the meeting and explain right so he 2nd energy that sort of looks like the of the renewal the solution of you did it will show and try Wilson Energy estimates now if you really want to see how these BA you want look at but it's convenient at least look at it and Frisbees just read the products out and this becomes convolution free space and so you have 2 functions with the maximum of derivatives and then 1 more from now as we said if you try to learn adjustment disproving sign of and this we're not decay at least for the refusal of the quality so you can adjust the depth of what's inside and an estimate but operation the differences in Yemen not you should be fine so many restrict when your abdominals so I can assume that this criticism are not in this for a very large doing against so this would be like large left frequencies now I was studied affiliations with me for just whatever was inside this is not going the interval so there and they cannot abide originally wanted the now so try to win 1st approximation described using a solution service let it be the profiles the ones that were meddling a flop and the subsidy for U.S. and get out executives expression from before which sums up all those situations and also what do you do now you can look at 2 bases soul when is function is away from 0 OK then what was little further worsened duties libraries Ali from signals was but there were some Parliament's affiliates of about what that is accepted the knowledgeable if you are From the residencies Yukon average was this will send a very fast and predicted in practice interviewed by in time OK using this was led by something that is vanishing but hopefully you gain more bearing ruling by parts in time so for example if this wasn't a solution the of it'll just be if he wasn't a solution is just show that the Soviets would be 0 but then cleared the lighting by something but supposedly gaining because the Iselin it is aquatic expression but for as many issues associated with this but this can be done so you're dividing by something that it's both small but it is not Morrison and did most of 6 Dexter function gain the right to at least make you think that this is this is right no if that are close to residents said what can you do right sold innocent about this week if you wear closed about 1 of the right and this was a functionally and 1 that is the sound level said and the measure would be 1 of the right for you will be able to interview them now forces wannabe that those otherwise living in their side of the vessel be something
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a special summit association that you have to know the will to extract and discounts from the following landmark focus sold for this type of free and over so this is exactly the same expression
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that I have here so I think of this diet in a bowl and inevitably rapping on mental function with such the way have innercity factor In
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this deepest and fairly and honestly will situation those function feet due to be less than some bombs and moreover you can restrict tool being tossed abdominal for that for the for
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this function now the 2nd is the
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following it has a simplified version of the statement in you are close to the minors to fluids said 0 said To the residences then this overrated it is a quality which is what you want to put hours of it and a centreforward seemed again if this is just a regular Oswald function in looking at some level said over so here Is there so here is a resident said which is just some sort of lips anyone got close to it prices of withdrew to the beach then if you just have to look function right receive would with minimum 0 then the measure of all of this it would just be Libya right but this saying that actually wanted it is more than open so the main thing now that policies based on interviews arguments of seem not to valuable over the standard bound for for now the generative Friedel greatest whichever nomination on the June edition and by cannot be done that way because Soboroff function but this is a kind of the cuts very narrow OK and what 1 finds is that 1 has to study not just the action by the action all of this facility this action is assimilation additional 5 restricted tool some specific directions 1 1 restricted his direction is that if you want to integrate by parts of the lined with new so you won't mind doing in the direction that yeah where the function when this function is those is not who eventually went to it in a direction that is tangential which is the action of the Grand of and this was already someone earlier version appeared in the world and to the it amounts by then is composed of the homeless people the squalor I hope to do the of alternative 6 months no I'm fine in the sensible so that people to people unable to hear so 6 minutes someone that's OK so 1st of all we're . then mail things that you can closed the energy estimates of Isabel was 1 of the media's as other structural things that are important but the pollution which I which here so it but if enough time was Cecil but obviously examples in Tajikistan and this is the 1st part of the argument 2nd part would be mine laughter we prove everything that you assume the ball the function decaying in Belo lies in space and so on and innovative it's enough that officials say that the energy of the battlefield it's to be right focusing particularly want to believe that the solution to gays and you want to control some legal norm to guarantee them now the very general basic approaches the following again you look at an evolution of this form you are added to read formula might unduly wouldn't care about the loss of derivatives any writing free space so this is exactly the same as before the brother motion in Newark struggle situation
44:05
now if it is not the role that would be OK but we said the business solutions on the other hand here you also can resort since the another level of the highest number of derivatives you could resort to integration with Western adopt Bulgarian soul it's playing the fact that the Arafat for example you don't think of them being Schwartz functions all beings will take and dignity images of them OK With respect to the freedoms and rights this the real fraction do not being able to prove that the solution disperses is if you have the Feltner lost a nation so if feasible to 0 and the granted the same time is equal to and this is the basic approach that was put forward by Jimenez mentioned I'm very successful in what ways from before now In this problem unlike in all the other what a waste problem this evaluation and if you look at the this phase he does the last set of zeros and if intersected with the gradient you'll also find that this is a good connection with Cecil would be the form they gossip exuberant goal for the wind to be 0 and that's another solution which has discarded but if you also want to face the music books he has to be of less 1 and this was their other special frequency the term this sort of Internet before now what happens said that we look at this expression and want understand how it used for very large fines the band and from there it some information on what is the space that we have to allow people the solution in and see if we can actually proved the dispersion between soon before the US his expression and let's today and the time parameter just as please note how this is being sold if this is the face of the grant an example you can supposedly get decay OK assume that for example father's was function OK so you can restrict by this stationary phase restate that even those additions to a settler fee is quite small like the 1 over the pine barometer and where it is more like 1 over the route of the time around and if you got just for money with this gives you a gives that the output of such an expression will be this what function that is localized on and analysts are exactly this residency just using beaches because of where I was to what limits and 1 arrest great so if you inputs was functions and the stationary phase Ristic you get such a function for whatever space level disillusioning hit us contained this right so you want all functions the lean years facing 1 functions that of form that is supported this thing on wheels and also want to alarm to be strong enough to be the case receptivity that would be for example that 1 oneonone work were forced to work in this proposal did you want to win some Jamaican at Longwood somewhere on top of it but awake for your function into the remedy for this guy and this guy is quite rough later taken the remedy is seeded actually behaves right indignant along India's like this crude of the distance from from this sphere of residents to safer take such a function would be in such a space model customers small losses and so another choice where put your your function prove Europe will strap would be To have the Exuma factors like X On the Real Side and that generates as the root of the distance to To the industry was that appears in only nary nearly as the output of all the responses of and this was 2 1 labeled the leaders norms I mean 1 way to actually implement this a good way to do it is to decompose in items that applies improvements in space and here in this case it also off to the decompose the depending on the distance from the sphere and then to control these norm as it would involve the argument because you have but on terms and for those who can for some of them even was the heuristics from the stationary phase and then when you are in nonresidents editions love loss of 13 interactions with the care of and yet but the visit them not when you have money navigate had gone a 1 way of knowing of localizations with the generates and during all of this is as for the moment that so can be done and this will close that will step of the way the on which later with introduce to the noble existence of no because Fulham you 4 the truth is the United
49:10
activities for each of the security sequence I don't know if you use so I think when it is the next May but so you're thinking that their rights will be the weapons for example is that if you look at the fact that you have then this would be this would be useful for some solutions right unless they will restrict something about the support improvements space you will have this rules that lowkey rent because see welcome may be out of his to the but saying this right so you if you look at the stationary facing a dark blue something like 1 of the this additive is an intrepid Stewart twins who too the question of who was the then as happened in the international leaders it's all the right if you're writes that before appeals for them moment it's not going on and they also want divided the sole depending on so exactly in in such a neighborhood of gonna warns of we've 1 already right sold you can quantify the acceptance of this would be the scale also for the also that long put a little on the generator for use and this is what the please NOTE were
00:00
Focus (optics)
Multiplication sign
Direction (geometry)
Surface
Model theory
Physical law
Mereology
Weight
Goodness of fit
Fluid
Manysorted logic
Lecture/Conference
Gravitation
Resultant
01:54
Surface
Musical ensemble
Process (computing)
Observational study
INTEGRAL
Surface
Model theory
Sampling (statistics)
Water vapor
Insertion loss
Drop (liquid)
Maxima and minima
Wave
Maxima and minima
Fluid
Manysorted logic
Meeting/Interview
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Right angle
Pretzel
Newton's law of universal gravitation
02:56
Logical constant
Principal ideal
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Direction (geometry)
1 (number)
Insertion loss
Weight
Rotation
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Manysorted logic
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Gravitation
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Area
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Time domain
Maxima and minima
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Arithmetic mean
Velocity
Time evolution
Equation
Differential equation
Configuration space
Right angle
Resultant
Directed graph
Spacetime
Randbedingung <Mathematik>
Surface energy
Pressure
Surface
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Modulform
Tangent
Number
Wave packet
Time domain
Force
Wave
Regular graph
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Equation
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Normal (geometry)
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Condition number
Addition
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Forcing (mathematics)
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Universe (mathematics)
Capillary action
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Pressure
Perfect group
Local ring
10:15
Spacetime
Greatest element
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Weight
Grothendieck topology
Rotation
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Forest
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Resultant
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Existence
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Wave
Regular graph
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Term (mathematics)
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Pretzel
Condition number
Series (mathematics)
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Einheitliche Feldtheorie
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16:48
Group action
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Equation
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Set (mathematics)
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Inclined plane
Infinity
Theory
Time domain
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Frequency
Regular graph
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Term (mathematics)
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Equation
Glattheit <Mathematik>
Stochastic kernel estimation
Homomorphismus
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Focus (optics)
Standard deviation
Cone penetration test
Graph (mathematics)
Sine
Surface
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Mathematical analysis
Operator (mathematics)
Algebraic structure
Evolute
Existence
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Irreversibler Prozess
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Freezing
23:19
Algebraic structure
Division (mathematics)
Insertion loss
Food energy
Time domain
Summation
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Insertion loss
Operator (mathematics)
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Glattheit <Mathematik>
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Conservation of energy
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Point (geometry)
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Parameter (computer programming)
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Derivation (linguistics)
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Relation <Mathematik>
Physical system
24:13
Receiver operating characteristic
Linear equation
Model theory
Multiplication sign
Direction (geometry)
1 (number)
Parameter (computer programming)
Mereology
Grothendieck topology
Gravitation
Cubic graph
Geometric quantization
Position operator
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Proof theory
Link (knot theory)
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Point (geometry)
Moment (mathematics)
Infinity
Mass
Food energy
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Twodimensional space
Term (mathematics)
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Inflection point
Wave
Hausdorff dimension
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Equation
Theorem
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Resultant
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Theory
Number
Wave
Frequency
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Profil (magazine)
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Field (mathematics)
Subtraction
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Surface
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Limit (category theory)
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Ranking
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Family
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31:14
Group action
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Building
Interior (topology)
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Hand fan
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Dispersionsrelation
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Number
Wave
Frequency
Term (mathematics)
Profil (magazine)
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Dean number
Focus (optics)
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INTEGRAL
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Expression
Transformation (genetics)
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Estimator
Plane (geometry)
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Doubling the cube
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Ranking
40:21
Group action
Direction (geometry)
Multiplication sign
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Parameter (computer programming)
Mereology
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Summation
Derivation (linguistics)
Resonance
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Interior (topology)
Parameter (computer programming)
Nominal number
Mereology
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Term (mathematics)
Functional (mathematics)
Measurement
Maxima and minima
Exterior algebra
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Set (mathematics)
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INTEGRAL
Lemma (mathematics)
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Algebraic structure
Evolute
Estimator
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Boundary value problem
Film editing
Family
44:04
Axiom of choice
Spacetime
Musical ensemble
INTEGRAL
Model theory
Logarithm
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Insertion loss
Function (mathematics)
Parameter (computer programming)
Bound state
Subset
Fraction (mathematics)
Summation
Derivation (linguistics)
Resonance
Manysorted logic
Phase transition
Heuristic
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Moment (mathematics)
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Functional (mathematics)
Sequence
Connected space
Maxima and minima
Category of being
Frequency
Quadratic equation
Phase transition
Normal (geometry)
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Physical system
Supremum
Existence
Twin prime
Divisor
Set (mathematics)
Routing
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Rule of inference
Number
Oscillation
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Frequency
Voting
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Root
Term (mathematics)
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