On the long-term dynamics of solutions of water wave models

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On the long-term dynamics of solutions of water wave models
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I will discuss some recent work on two main questions: the existence of long-term solutions in certain water wave models, and the dynamical formation of singularities. The talk will be based on recent work with several collaborators, F. Pusateri, B. Pausader, Y. Deng, V. Lie, and C. Fefferman.
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3rd and on the the I think you also led to thank you organizers for the kind invitation to To this conference it's a pleasure for me In Paris for a month in the June it's it's really nice and in the conference also organize and now I'm going to talk about what a waste um most of my talk is going to be "quotation mark singularities and for some formation of 4 the similarities in these models are to start of some more general some more general description of the problem in order to get to that I understand better what the problem is also fabulous area so he had those I'm going to repeat a few things that 5 discuss still In 2 days ago because of the model year looking at the simplest level but the model is a model that 2 of them it's a model that has residents including picture several related person like news on interface and the food is described by the interface the indivisible being surfing 0 of and if is discovered that laws against civilians and we have equations and inside the fluid sits in a deserted in the fear of the bombing Congress boiling inside the fluid we have the order equations but the module derivative of these miners gradient of the pressure and they also have the gravity terminology answers to breed them pointing down I now this is inside the fluids the equation for days of work the city however but also an equation for the movie everything disease itself is moving and the interface is moving with the fluid which says that details the so the particles on this interface if the velocity of the body appointed some direction and interface was moved in the same direction and no 1 can write this simply as saying that the museum minus the velocity of the would have to be engines to the rest the graph of the interface I know In order to make this into a system and do to close a system we need to prescribe something only a wanton prescribed of the pressure on the interface and the pressure on the interface be prescribed sympathies to prescribe proportional to the meeting curvature of the interface soapy of and deceive marked times a couple of Oxendine where couplers and encourages and singers a Boston Brahmin if 1 does not have the system this way even just making sense of was the equator what evolves the civil things that you've also making sense of the system it takes a bit of for our son has to think for a minute that the way to look for naivete is to take the divergent send of the 1st question would say that to 1 has an equation for those of the pressurization of the velocity at some point it sometime over there we get an equation for those of the pressure that time and the press also prescribed in the modest increase beleaguered the pressure at the time of the museum formation and NRC have depression the velocity and the interface of the city at 1 time then we get the uh the embezzlement increments we get from the equation Vt vote from the Wanderer questioning about the increment but there is a slight through imprecision in the equation foresee because I'm getting to the bottom advising the president would call you parameterize This is the man with the benefit the the same surface and that is a slight imprecision there as I decided that this is the system now 1 generates light for any level as other than it does not apply to make sense of it as a well pose pollution system and the went there is that there is the basic questions like the system of this type 1 and have the local regularity question which is skinny constructive solutions locally in time for once that's from Nice initial data cannot construct solutions opening time but what can also serve as the question of global regulator can construct solutions that extend for a long time were you long-term to regulate goes beyond the local existence time and I can raise the question of dynamical formation of singularities many starting with a it's nice at times 0 and at some point Beyond the local regularity dined at 1 would form something that looked like a singularity in the floor and the I'm going to do between family mastermind of is going to be about the last fall into the you show them and then ankle formation of singularities but I went to describe quickly and the order of PCs as well and finally also described them but 2 days ago another altering that is well understood and it's taken a long time but the job to the point that has emerged is that 1 has operated 1 has a workable system even the surface tension is positive or when it's equal to 0 1 has to to make a certain condition the rate conscience satisfied and that I have existence depends on the the match local natural the future of the same features of the system 1 of these is the smoothness of the parameters so we 1 of the have seemed to be simultaneously to be small letting some and the other 1 is what's called but they are caught constant of the interface meaning that this picture is nice but it would start from a picture of despite the initial time and not expect no matter how small things are the dynamics of have to record affected if it advance for prolongs it could have created sort of intersection the demolitions has somehow have to know that 1 cannot expect to advanced this is it's depends on it depends on the fate the what's called Opera concentrated constantly simply fraction between tendulkar accords between distant points and there are and that goes to 0 that would be needed to be a parameter that small but no 1 can answer staple this problem so that these problems are stated that I and just for the picturing them into dimensions and make some difference in 3 dimensions incorrect exact disintegration in 3 dimensions of the White House about pay unique conditions nobody ordered various ways Michigan would regularly conditions that um moved simplest way with the closest to this picture to assume the periodic relative to saying that the world is a sin so while it would be better yawning the excesses but Infiniti in the vertical axis savings in this direction and periodic immediately in the horizontal direction by House the the finite brought them here the but and can also pursue model went consulate later which is the 2 free remodeling which will be 1 of the expenses not just 1 flight and vacuum but honored have won for lead in the 2nd open because of this reason is not just when extension is not just an extension is a reason to consider 2 freed more than the same time as the 1 or another local as I said is well understood and it goes back to the earlier work of Nottingham obviously
horror Craig and then the local wasn't in this shape that I wrote here which Soboleva norms and not what what the latest standards natural local Bosnian soberly spaces goes back to the work procedure from the late nineties and not a lot of models and uh there's been a lot of work well proving his local board provincial but was this theory you know all of these models and by I wrote some names probably not all of them and I apologize for his people by again at the opera civilian blood on Baros Ambros muscle on the line you blood could could punish chain could tarnish scholar Christensen course stuffy 9 others are virtually shut buzzing so that there is a lot of work on this models but most of the 2011 is more work after 2011 In any case the picture is 1 of the most of the work of the 2011 has to do with the reducing the regularities instead of thinking of villages being each day and they would be let's see loss of live long enough to try to reduce the regularity relative to how long can go there and I have a worshipping that Issac was an e-mail program in 1 of these problems it's and I can get to the critical regularity by depositing the problem so the probably not possible to to prove that poisonous the critical regular but not the global regularity results there are much fewer and the they're much more technically much fewer but much more restricted wanted time we know how to prove global regularity easy if we have a small that it's the same kind of picture that 1 has seen the that prompting me to know that you smaller or close to some solution did you know In this case you also know that they have to be a rotation also cannot have any and they also need to be in the entire space and cannot be appealed the case so all of the global regularity results they require these features smallness Europe nationality and the domain itself has to be Of the entire space because that's the only 1 that would allow to do all the way to the mechanism would be used to dispersion and 1 can only have dispersion 1 using the from space and and I just say it's also pretty results of the 1st result of this fact was and always will result of the jewel from 2009 in which prove global almost lower regularity for putting gravity waterways reading is that the gravity of the gravity coefficients posted by the Service Station coefficient of 0 and this was followed by German must when you shut down the damage the same problem the dimensions of what you look at your wasn't from Sean has the capillary waterways a wooden Jews "quotation mark was arrested my sports is also workers most modern other than that was I was at there and I and about the same time I was under law would look at it to the problem but to the gravity problem and we prove lower regulators of passing from almost doubled to global during approves of this and also sieges result behind the fermented Tareq and if he meant that our global regularity my student Wang see we revisited the problem of global regularity and he showed that unlike in the global regularity in an infinite energy class and in that class again remove 1 if anybody was too small you know know not another Norman because it can remove 1 moment and conditions now the opposite problem when news the this is this work if he meant that I was singing 1 momentum condition handling and variables and also found it was sectarian myself but without that condition and there's this loss results that of five-year poked the ball through a couple of days ago mutually had we look at the forward problem the G the foot problem bold rhetoric and service station the dimension that's worked with during the course of the samples of so this is all that this is all about I think it's all very well understood and I only want to make 1 point about global regulator work because I think it useful for other problems so if you don't care about what awaits based in 1 point that I think is not about what it was but it's something that we understood very well in the context of what awaits some of these all of them all of the the fall of the results in the dimensions of the the mechanisms to prove the control of the highest order agendas in Pembroke decay and it turns out that in this case went into 1 of which featured in the cases Wenger's equal to 0 or my equal to 0 1 can prove 1 of which for the interest on the hardest part the 1 with the case for the loneliness of and that's the argument that to this that the ports in the three-dimensional crumbs all of them except for the last 1 the 1 that from you talked about a couple of days ago now in the two-dimensional problems this 1 idea that developed very well and which got very well understood which is and which appears in all of the results starting with the Jews result which is so far the so-called Tulip were the so-called aquatic energy inequality which is an appointed 1 electable is if the operation was duly elected proved that you have an antigen .period an increment in equation was given because in 2 dimensions becomes a one-year interface and the best you could hope for is 1 of a score of the game and as we know it was good that indicate cannot get very far but if I would do if 1 could be dented the equation was still being then Of that could have led to the new principal activities that and almost always are and so the mechanism that I think it's kind of a it's well understood in the concepts of waterways and the things that a joint mechanism by now used to prove this quarter contagion according to the the Energy is controlled by we need a growth of Ford terms and the point is not to lose the derivatives of that appointing this 1 of them is not lose the derivatives so too have the same number of the Bahamas number that is not to the different you go in the right and sending the left-hand side and the other 1 is to get the 4 times have formally 1 can think about is the shuttle is a normal form but ostensibly carefully and that wasn't a promise estimation of and this is as I said this is inequality or a form of this accord is present in all the global results in 2 dimensions it goes back to the 1st were procedure and the way he wanted would construct that won't solve the problem of course is still growing it quite like that have won only 1 was with the 50 point Isaac and that alone was give the 2 did almost global results if it's of selected thank you organizers for the kind I have to the number of cases so far that so basically what they're saying is that too but was so other centers that serve them or front rows the despotic energy inequality and 110 couple this with proving 1 was for the figure given the 2 almost true but little-known most global solution gestalt of this just out of the speech in order to get the global solution they some more hammered on the summer structure the system the thousand 101 instruction that's what variables of the United Farm for example for the gravity world but is not having promises I said finding a quality goes back to the work of seed
wish should trigger the logarithmic lost time in the scenic quality did not affect what she was proving which was almost global existence but the she did have along with needles from me and I we went to her proved friendly it was a removal of written it was it was not a significant rule across the decades ago so what improvements to all this is the reason why does that well understood and starting with the body financial and adjustments and learned from the work of fertilizer and loss of old the day had a certain way very nice way to decouple the creation and use 1 of them that you wanna have you want to be abandoning the equation to same a one-dimensional was that the rebound and like the couple distributions to address them to address them then what and I did was still use what they call the comparable rectify structure which means that the cost of the certain not all that but those were the same weight as all that 1 cannot quite implement the decline and that the field method that all the victims respond in the same way as was the modified energy method of if he meant to borrow the to clarified its position thing the world but I really like about it is that it says that if you do normal form for the purpose of being an energy estimates availability to them together because the normal form is basically a division and adjustments to its assessment Asian Flanders and together they worked very well be forgiven the nominee for example which is which is the real concern here and then go what what we did so at the point of view that energy estimate is always better than in four-year space than the physical space before it's more flexible in the way spacesuit put everything in the forest the would in the spirit of the methods coming from the same too and in any case of this so there was not wanted to bring it up is because this kind of thing that I think is really problems we will find a way to make this work will 0 4 waterways but I kind of feel that this general it's based in general pictured a despotic energy of despotic energy estimate is very robust netting the only thing that's needed there is not to have small devices so in principle even problem that formerly expands in a way that there no small devices than 1 should expect to you're able to true do online art form while not losing the derivatives of the probe dissuaded connection it failed aquatic engineer wanted you felt you've got small investment advisers not possible to ruin the quality of the Cyprus exactly the usual that's Eurofighter discussing this model that probably discussed the 2 days of all of that's exactly the central issue the use of force the residents 1 cannot prove that what a Canadian court convicted and also I prove take case of his this is the word probably described as some kind of a partial normal form that we did it depends on on and on the juridical image Marcus and went about Holly debate about the main topics when Shandling to discuss samples a which is formation singularities Knifley think of the local bosses did not as this is that they have a time of existence depends on the smoothness Of the objects and on the on the Arkport parameter which has to do with how far the surfaces from self so that 2 possible scenarios could say that you're going to create a singularity at 1 of them is to do the room find something that leads to loss of regularity and the other 1 is to find something that creates a self intersection note In the most regular desired also several things could think of was in regulating the middle of the fluid saying that inside of which appears to be very hard because of that this is the 1 has less control despondent on the on request there's also the moving the morning in the face of appears to be there very hard problem that 1 could also see the big 1 would also try to I understand a lot of American the level of the interface as of the 1st what I mean to say is that the loss of regulated problem can be the copilot into something inside the fluid or something on the interface now that might I know you don't expect them decide that however did say that his side also there is no word consider these nor the city could you could block there it would be in the 1 aggression was because of what because workers another place comparable I'm not saying that the world and can really solve this inside and you should wonder went into the water some In any case so that the 2 things that are that there is only 1 solution beyond the mechanism that's known that loses regularity in it's proved a man for which is a mathematical proof is this mechanism display singularity which is drawn here which is that if we start one-stop it's possible the statement interface that's very close to being the self intersecting and a small ferreting out is smooth except for the fact that it has I would spend a pinching at the point like that and all of them as possible to predict that like that which advances for a short period of time in a way that decrees that the self section and that it is a very stable phenomena assessing the discovered a few years ago by cost stroke order Botha Vermont Macedon Gomes Serrano underwear it was a new proof of the punished for their it's a very robust phenomena in certain ways In the sense that filled the in the sense that door origin for example was in the gravity problem but likable service station without changing the conclusion we put through 1 can put Bautista without changing the conclusion and go from to B to the EU without changing the conclusions it's a robust in that sense but the only thing that prevents this mechanism used what we found a few years later in joint work which only fair from Victoria Road which is that if 1 puts a fluid in the middle so if 1 has this feature would want to read outside and the 2nd from the PLO I have no matter how well the how like the fluid in the middle is it's not possible to create a special breed preserving things so this is decided to exhibit that within the band already singularities memorandum described wise that I think that's relevant so In the 2nd thing is relevant to the question of producing a singularity through Milosevic led the interface but let me introduce as the model to make it the exact models of the model as a said for to free the interface is very similar to those very similar to the model for 0 1 footage is but 1 has to fluids would 24 investigators and the separates that have to be separated this interface interface and the freezer separated and they both involve according to order questions in their respective domains so 1 has for 0 1 hazardous equations this much of the realities of his your Jason at the figure to use these we want you to and they don't want evolve according to death according to the letter requesting them in their respective domains the interface
itself is to slice of this some compatibility quantitative analyst there not independent of each other because the interface itself is to move relative to both ways simultaneously and the condition of the interfaces that dictators -minus it's velocity History to them but this to be Denton interest of the coalition shoes written dimension 1 in which it's adult product with of the derivative of the offices and also the condition of the pressure on the interface becomes the difference of the 2 precious and that the difference of the depressions would have to to be I want them to be proportionate to did today the curvature of the interface so this is this is systematic harder to understand that this is a workable system it takes a little bit more effort are there for this particular model at the local opposes theory was proved by a double life to prove that indeed this model is what always think descended if 1 starts with the initial data that's nice 1 can extend the solution for a period of time and so this is the model 9 later described
1st Oh there's 1 more than I have 1 more slide described what I would say that this 1 remaining it's like imprecision in these situations which is the fact that the death its it's the choice of the coordinates of the fact that the exact conditions that DTC miners there was some velocity it has to be dungeon but has to be tangent through the interface but exactly what the engineers it would mean 1 1 can be like a specified went to make a more precise question by saying that DTC is accorded to you but of course sometimes justifies its elusive because victor in the attention space another sponsor can be made to depend on everything except where parameterization constantly interface anyone can take it in our 1 can also not specified in the sense that they small tho many smooth function were the same way but that that could because corn is the way area on which we don't use here order Lagrangian point should should be more useful each 1 of say DTC is equal to you it's exactly equal to the cost of the deal be taken it would pursue
sentiment led the weekend described the construction of the the singularity of focus recorded 150 fishermen resident ,comma ceramic um Justo uh just a contrast to the case of the book but the point is to contrast their case of 1 fluid was to flee problem and try to understand what is it in the 2 freedom although that figure the prevents this and the exact definition so that the definition of efficacy and this is what is the exact definition of the we are caught constant is the picture by droll here that 1 looks set the uh the worst that it 1 just by looking at 2 points off and then taking the court and dividing by the length of the arts and the was that 1 gets ready the boys is called opera constant and there so the the exact figure ran of 4 the exact same of this city of 4 of these people that their mission is the following that if 1 can always use a solution so visibly additional forgotten the but makes no difference there is a small periodic solution In the gravity Of the refugee problem so this is what the international at I have here .period the equations written altogether so the 1st equations you want the equation the 2nd equation is the equation for the interface the pressure the case when there's no surface tension is said to be 0 and then uh billion compressibility and Europe nationality conditions in the domain and there is a solution that I need a starts out from so the weights straighten starts out from Dhaka condition being equal to 0 6 starts from a contact point and it develops time to 2 curve which she doesn't have any more consequential Council agenda was the more the more natural would be good to go the other way starts with the requisite about reversible anyways 1 starting pitcher that's a separated and 1 creates a contact end of I'm going to give a very quick idea about the probe but the proof is that the profit is using government is on local while I can reduce the problem of local bosons and the way to do that is to think of hot expands the the velocity in terms of a 100 president was seen in a way that the cost for the conditions at the several ways to express it .period can think of it as the gradient of the overly most people passionately think of it as a region perpendicular restrained function and that the way to do it in this problem to think of the velocity is being the best of what operator applied to a function or made and the functional make a spot on the TV the functional manganese on the interprets the interface of work disappeared and the new road operator it looks like the basement understand the like of variable coefficient Hilbert transform into it's looking diffraction is some sense the singularity which wearing 1 dimension is already in 1 dimension and has assumed that the issue would transform its integrated installment up and is therefore not for you ended no the system can be really done in terms of young that is store rebels you know make chapter everything have times of it and the only way you can have a splash the only way the concerned especially about the functional of going to infinity the panel display and idea that that goal that wasn't expected to make a change of variables it does out at the gym was watching durable suspension believes you over to that splits the plane the agenda of the meeting those after the change of variables 1 has no 1 has the grass basically the the graft change ability to look cool it's good to know nobody knows splashy be I need don't know kind of need to be like this would and there was available to To make the change of variables and then think about the problem as it was a local poisonous problems of this type of data that is of this type that to be consistent with the splash rewrite the equations in terms of the student variables which are you know in this change of coordinates and then on their systems show that the system is 1 also were for a period of time and the forbidden by me in such a way that that this species separated from the line about the 1 goes back 1 Ginger to what went into the separation so that's so I can think of all the problems in this way .period is still solvent appointees wanted still sold it by Energy estimates after this change of coordinates but now I'm going to describe now the problem In went to describe now the problem that has to do with the 2 free problems what we do and that was the differences so what was it problem that of Fadlallah show exactly this mechanism doesn't work the profitable this doesn't work because if there was some feeling in the legal and making change over most of the race to create the problems that change verbally itself will not be I would not be you wouldn't want wouldn't get this 1 that would translate this into a small problem but by by making discouraging injury but the question is is there something intrinsic or adjusted approved doesn't work and what we show this something intrinsic and by the union and also we started out of the way we didn't know where this was going to be possible to have a splash or not so when to stay another precise theorem that we pool In order to explain why is this related to the book was formations hundreds of a 2nd to the following if we're looking at a solution such as we have a solution of this before system and it here so it's all written of in this so brackets here so I have this to flow into the way equations the equation of motion of the interface the difference of the precious and there divergent spoke in compressibility conditions sojourn that we have but we have a solution of this and I thought it would discourage many of them the described the precise some so we assumed the disease small the universe in the 1st phase of the road you evolution and we assume
that the velocity on 1 side station with the just isn't it just as the problem of just as the theorem that lithium for the splash and then we assume that the bank is equal to the beaches attempt is equal to 0 and the snow this no contact in the interface and we
assumed that the bank is equal to 0 and the other fleet is also somewhat but did not issue any anything about the other fleet at any other time to 4 times the sequel but then the conclusion is that there is no longer is the equations makes sense which means that there is no mystery don't have as long as you don't have work that did that to the function f is as long as there is no sense intersection we have a lower bound for Doha accord constant which means that which means that the interface will not be able to touch which means that this is danceable argument is news that the interface will always have to to stay of bounded away from all forms of intersection In other words is not possible to develop first-place singularities while keeping the interface with and won the velocity small in the case of the 2 to fluid interface now in order to be able to say that is not possible to they attitude to develop stimulative is that is that we don't said that is not possible to create a special because we know it the sexually likely that if you start with the fruit flies to develop singularities but but but on the other the other part that's left is that is that the show the reason why this mechanism of picking the specialty being the sport is not possible because the reading of the bill doesn't have time to get out of the way as long as there uh this to curves smoothly would be approaching in a way that's quite directly because they to cross that approach that could be approaching and invest more than the distance that is between them to small In the freedom and doesn't have time to get out of the way a very likely scenario that think it's very possible is that as we approach so if we start with a picture this way but at a later time as we approach the people start creating a corner it temperature if starts creating the corner the leaves more time for done for the flinging them into get out of the way and I think that's quite possible if we prescribe Doctor did too but would want to push the foliage In the direction at worst of touching then uh then the only way that can keep advancing each gets sharper and sharper in order to allow for the freeing them even get out of the way and that hopefully will but that that could potentially create a singularity coverage singular and now I'm going to describe quickly what's involved in the probe so as I said in the beginning we didn't know which way to go because there that this is a splash mechanism is to be used to be robust almost any other change and that is it's actually very very close its along it's close by along also went to describe the main ideas that we have in the proof of this theorem them and that there really 2 ideas 1 of these dynamic of the other 1 is at 1 point sometimes so long as it 1 constant but dynamical part is to do with his body of work disappeared so if we look at the equations in terms of the bounty of autistic which is the beautiful overwrought the operating can express it toward the cities on the 2 sides in terms of the public operator of dysfunction Omega and they have this formalized he won his beautiful front of Omega blossoming over 2 and give the to is spirit of Omega Minor so mingle with 2 and then if we write the that's true the equations so the true 1 it requests for them for the velocities and we
can make the difference of the papers in the main difference diminutive the 2 models is on the slide 0 which entered the difference of the questions and get the burgers like equation and the burgers which is something of the form up is equal to one-half of Diyala forming a square the blast some kind of multiples of young fighter for making and that some some Simba terms and then I can think about is a burglars equation for the bird legislation we can preserve the strongest summer camp reserved for Omega is that Infinity is to be thought of as local but what can preserve because they have this Burgers equation has shown that these functions if 1 and if 2 are small this part of the assumptions for the theorem that going there again that this word is it that this body of work is to have to stay bonded Over the time of death evolution and this is that so this is the way is that makes a difference relative to the 2 dead tissue when only 1 but then and now
once we know this information that the bond of autistic status of bothered about the the Bollywood reduced bonded and what you want to say is that if we have that give them if we have a situation in which we have 2 2 points the majority like that could have to Boynton Beach this should be the points of order for 1 and fight too so if we have 2 points the image the distance between them is small then we want to be able to say that fill the distance the difference between the normal velocities of those points would have to be smaller and smaller there with the same constant sorry this is smaller than action and the difference between that of workers in the villages have to be smaller than excellent does that it really was a load of 1 of which and that has to do with the fact that all of these cooperatives essentially a day session transform admits that it would transfer me infected on some kind of able provision Cuba transforms and we have to work on the Infiniti because as the best information we have about the bond workers so that the the loss of load of 1 excellent and on the progress in the following way so we are looking at and so it will what is the information in the problems of looking at these operators that
maybe you should start from here to the information a promise the smoothness and this movement but I can speeches and enjoy the speeches separately so he so you have to cross described the functions supposedly small and we have this operators the 1 favorable make up Musselman up losses with 1 side so this would be the thing should blossom he won hearts up Orlando I don't the things should minus the 1 side on the bottom so on that but we have this summer got lost and we have with the function and we have a so that the than have done the velocities of this so if blasts plays the role of the velocity of being the velocity of out of so the silver transforms this operators T-1 and T-3 they are of a region on the slide and they are essentially that the normalize versions of the overall believed they had you so the behalf operator if we write it in this picture in which we let In retrieves we spend the pictures we have these functions infringing that correspond to the sides of the curve and they will also have to localize improperly so the distance he's Epstein on the consultant have to take out a distance in the horizontal direction of such school eventually think of essential parts of the of the beer Holbrooke operators and they write them survived supporters that through hard the parts of the Beaverbrook operators will offer modernizations and the system that is so the information that we get is contained in the 1st lines so if blasts is 1 velocity and losses of 1 in terms of Omega plus in terms and in terms of woman behind and that minus also is the same as a similar formula and the gene is what measures .period that is what measures the difference between the 2 but the 2 velocities that you want to show the smaller than Exxon Mobil 1 which so the harmonic analysis promise you 1 becomes so often is a reductions becomes that we have we know that the functions only got lost and Omega minors but they're bonded to from the the dynamical part of the problem now he thinks about we were just as the process of expanding but they have this dynamic information coming up minus bonded and they know that the functions offend you are a small strip of small and we know that functions were more important than the functions have blossomed at minus this move as well and although wanton derived the fact that the dysfunction small serious small and there it's like will like to the problem and the difficulty is the fact that the information got bottleneck that is coming from the work music which went the Burgers equation you cannot get information about the derivative generally get information about the function something that that is not possible to get information about the recent don't have good information about the realities of the ballooning of anything in the normal bodies disappears so I have to run an argument according to this infant in arms and the use so subdued to prove this proposition he was but what it called as your method in which we define and norm where there's also another problems that 50 good global regulated but they have to define enormous debts but consistent with the problem and we have to analyze of the system in a butcher .period terms of this storm knowledge is the norm that we define is basically a were transformed it such as it is obvious in size actually Hubert transfers that we are looking at and that's there also them that they're not exactly the day their verbal provision Cuba transformed coefficients the depend on this other functions infringes on their function here he a certain type of Hilbert transform and they're looking so we measure of function so you measure of function a by testing against but by essentially regarding its Hilbert transform and then measuring the proper space it's written here is In body to it and then it turns out that it's important to get to the classes to get the bond on GE to be Ichilov 1 of which we cannot lose more than all of 1 over which it was like should to the 1 then that would not have prevented the splash has to be a it's in the most accuses Local 1 overexcited turns out it's quite clear we are quite surprised the effect it turns out jetted exactly so you just exactly of sewage close the argument to prove this to do to to prove and I'm not only have room for a moment I wanted to say a few words about why is you that we are looking at this model so my hope is that there no we don't have a theorem but do we have tried a few things the hope would be to try to create a likeness of the splash of despite so well and so is the 1 with the hope would be to try to look at a man evolution that creates both a splash and a simulated the same times what 1 would like to see the dynamic is that it is the interface starts approaching 2 words of splashing it would also have to create a corner and that serve them now that's something it seems to be pretty tough so we have some work in progress when 1 feature of this model and that the like which is better than other more wanted was the same which for the most innocent worshippers Fiji for example like weight you put any questions you as this question can you form with simulating the interface but 1 feature of this model is that it In some says there is no place you can go sell it if he does advances what what we should do this theorem is that is for the interface was it's also possible to create cement estimated the song can have everything cemented annexes and if we can get it to advance uh then you on your way during the way their evolution finishes by creating the curvature singularities because otherwise we would otherwise have biologist in other ways to create a special for singular so that's now was a difficult day with what's hard and haven't been able to do is to control the flow 1 is the 1 who wants to be able to say that this interface advances for long enough and you start to control the flow of AS you lose smoothness so the Mossman is used as a it's a pleasant a problems it's got for the flow 1 has to use energy is nice but at the same time control the flow teenagers so we don't have will we don't have a very clear we don't have the we don't have a fear of the style that too I think it should be something that this could be very interesting for this and it's the kind of thing that continuity 1 gently for this model 1 might not be able to do it for them for their modeling Yukon has only 1 fluid we know that especially created by without without losing some of this is not possible to create this mechanism there focus on them going to stop you thank you the question he was transferred back work from the day we could find a good answer to who will it's possible is not so easy to do it but it's 1 of those it's Howard that's how the theorem for display sure spilled solving backwards no 1 can now they're more probes but that origin proof was bristling backwards it's pretty important how you make hazardous it might think it's pretty delicate life is pretty delegates to do it this way but yeah it's certainly possible I think that's very nice and your pictures but the modified to make the sauce similar "quotation mark and that's all that so what would happen in subsequent periods various methods former touching and it's also that if I could that enough to be able to tell you that said the kind of uh there's several things and undermine my feeling is that the only way you can really do this is for some sort of monopoly City because it leaves his start if you tried to solve the equation is proposal will pose a problem I don't think that's I'm not sure this Ginnie made work it is Indiana the daily this week and said point is well bows in some quarters that the what sometimes words about his reason for the 1 Albany City which you are working on my understanding the question another question Is that a lot of very different but it does not always remind me of the according to the Fed works and that is used interfaces are approaching each other and their smooth you assume so it's pushing the interior of rapidly so singularities of avoid that situation by the time the role of 100 or what is the studies on Chelsea but is that due geometric configurations forcing cities to say just it's all procurement the interior of the lighter fluid has not moved that near boundary walls of media it's not doing so and happen so this my hope because I'd like to prove will also regulate it is not the only side effect Chalifour from and I really do have lots of research to see him fears that things will happen differently maybe according to what you say my feeling is that if you set up this velocity to be high enough you can set up the date of the virus to be hiding of these no time for it to develop something so complicated it just wants to form a quarter for the things in the media to get out of the way not as my feeling of was that this is not the pope's over there no posting its formation for the good which is singular but the just think Charlie is with so that I have a lot of discussion lots of discussion Chalifour bodies and we could never really see exactly 1 reason why 1 is more likely than the other question do