Global smooth solutions for the inviscid SQG equations

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Global smooth solutions for the inviscid SQG equations
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In this lecture I will discuss some recent work, joint with Angel Castro and Javier Gomez-Serrano, on the existence of a non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation (SQG).
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around a top 2 an 1st of all thank the organisers for this invitation to requested me here and yes and no I'm going to talk about the surface was digestif equation in my collaborators are unfilled passthrough quiz in Madrid as well and have a moment 7 was Princeton so what is the surface was reduced traffic which was sister an equation in dimension to we're in the plane during all boundaries fade-out is now known as a scanner functions and is transported by air in compressible flow since the velocity field is incompatible with can define a string functions such that the blasted the sequel to the gradient perpendicular to the string functions and then the relation between the string functions and the skater faded is given by the square root of apply if we write down the velocity in terms of greater than what we have is that the velocity on the restraint friends of faith so the outline of my talk is In since I'm the only 1 speaking on executing and I will give them an introduction more international what is known properties then I will in interference I want to and then I will give a brief sketches of the proof and during the proof will then it talk about it patches solutions of the fact that that solutions because their motivation of the prove come from the point expected OK so it was traffic it's a model that the models the dynamics of the masses of air in the middle latitudes where they acquire really force plays an important role and then this is adequate atop equations of three-dimensional equations and denying and that happens at the surface is what is known as the surface was used to frequent which is so this model in accuses several references and it's had been in boasted and studied in several of In papers in the geophysical context but it was not until 1994 that's it Peter Constantine undermined if they want Iraq to realize that this was a very good model was a two-dimensional model of the 3 year .period so let me describe why here we have created a Florida including year so I'm going to look at the treaty incompatible either equations in the parties before the border city is the current velocity and equation for the bombing would disappear there is where it will make us a better in this divergent free and the velocity is as well and then great interview this in terms of the border city a single interoperate acting on so we look at we compare this with this surface but is equation the what plays the role of the participate will be degraded perpendicular flavor of which gives the derivatives with respect to access to "quotation mark data part of literary sector explore and equation for this sector is equivalent to the equation that work this city satisfies but this is better in the dimensions button to dimensions daily it Of course raid and they remember the new phase as a region free the loss of life is a virgin pre as well as the gradient of you are single and available for procedures years Greece transference of the Great so but there is also in a relations by with the conserved quantities in the case appointed in what excites who would have thought about his lines pretending to the border city in North Point and they actually move which is and failed to pull the velocity manager is concerned while in the case of a skinny then there is the level sets will fade plays the role of the boss declines to attend the great impotent in face of instances of transport equation the leveled sets move with the flow and then it is Was it that way there's since there is such a transport equation and 4 Escajeda more quantities concern is that you actually can balance the LP nor of any of the velocity P we on the 1 hand strictly swollen and infinity and this is bounded by mentioning and this is because the velocity on the restraint from each of all the time and they'll be norms of that are conserved and because of the which so is what do we know that here we have local existence In certain age norm and in the case of as Fiji b similar local existence meant there came a very because we're 3 to them and the problem of the formation of similarities
are open for both no criterion formation similarity following Willetta mentioned to and that he had the minor criteria which says that gave you control the interval in times of linking norm on the port this then there's no blowup at times capital the well similar thing it's true for stg if you control prevailed and I don't think a lot of the gradient and favor we perpendicular 1 compared with the other yes sir and there is the cost of the minor because something that from reminded resolved that if you control the direction field Of the waters at the so if it remains slipped then there's no singularities so in the case of stg the identical it is true for the direction field also degraded pinnacle and so on then so I have to do it if there's a similarity in their petition on stop OK so it there's goes and the similarities With 1 equation on the other hand now here is the single largest forest duty and the answer is we have no idea numerics and don't show any evidence for small smoking Jamaica I think there's no evidence of and it but in order to get a flavor all that many in play around and and and look at someone the models so we are in it so we have a transport equation and the velocity is the restraint form so in 1 dimension let's take the velocity to be the transformed all all of this case Leonard Cohen Esq In Our unknown the usual treated his friend form and we look at this I think this model where this is transported by this stroke and and the stake the initial the initial may not have 0 to be accurate we're equal to 0 business In this true initially since the greater will maintain best and then and the energy actually will decay Wednesday at his spot so for this we have we can prove to local existed in some solid in space and theories by NYTimes intelligence for this month for all initially this is strictly because of but of course in this Mother's Day encompasses the hasn't played in a rock so let's do the following that's strange take the diverges outside and only near-term sold SKG that's right so this will be another model this right that as ask the divergent all you faded so we just took the divergent outside and writer won the model in this case and what we have it is this deployment and for this and the stakes for the coaching problem at 0 .period would so what do we have local existence the next game 2 If the initial data is a strictly positive vendors coexistence 3 if there exists it's not such that of excellent is 0 and then finally singer and appoint except it was is a the nuisance to the meeting this was aid of the Denver beat Of this business model so this is this to justify taking the I'm thinking the divergent free outside so the model will be the conviction for inside In the during the disaster so this Mother's Day is a version of them Bloomberg based this 1 and this 1 not the derivative is heeding his call to yeah this is a transport equation thank you the Duracell just to see that can compressibility may change and this is what I said him just to get the flavor the have thought 1st is that the Western in it this is if the initial data yet no evidence of so that I know of more than 3 although the yeti no there used to be time for at the time but ended without the was the and numerically as well and then but that's 20 years ago OK so but right now but I don't know of any In anyone claiming the radicals simulation where their single-largest for smoking the whole the soul in this framework I have to say that Resnick in these pieces in Chacao he proved that it
has a weak solution is in the existence in and the problems of uniqueness is be called for this part of history OK so it where allow me to add the parameter to the system because it I'm going to mention several references and in order to to "quotation mark those references is good that I had a problem with the experimenter is in the power of the question when it had enough of so international this system from to the owner to Sgt how thankful to 0 used to the idea with is the of end of cycle to 1 is executed so inform the global existence and to the owner is critical so what global existence for officals 0 5 when officers particularly given it's not a problem and a and radio from a book and so on for the purpose of my talk let me point out that the rate of functions that status radio then it's a stationary solutions and that has to do with power supply Russian surface this incident and them only terms you said that's for the introduction and the theory 1 2 cent is that we proof that there exists a nontrivial Lowell solutions for the attitude you quake In Howard the solutions cultivated well OK so let me take the history of the planet at this is for people to 1 stg John lately that the modified 1 them when the it calling it stg OK and it in time I don't mention this to be I will say that some of so it is prestige and so the solutions are going to be a population over a range of functions they within is going be 1 inside the circle outside of the circle I think there's going to be 0 ending between go from 0 to 1 in a small way actually it is going to be it is seeking now it's going to be a part direction this and the perturbation is hedged going to happen to and Paulson that 3 Paulson so we rotate by 120 which would get the same In a structure N A and the solutions and they rotate with the constant will now the thickness In our solutions this thinking is very relevant for the proof and the thickness is 0 . 0 5 late arrival explain why so these are the solutions while we were working on this problem Hey we learn from doing so and then he published a paper where he actually a In computer displays solutions of this kg "quotation mark won't have familiarity and these are also these are the same thing but there an Enders blizzard so is it an ellipse outside ellipses 0 and inside in faded small and the way that he doesn't know is that he goes to the 3 in the equation and he called good busily and that the ellipsoid so are repeating solutions for the treaty so when you look at the kids in surface it it gives you this a ellipses but it's not the patch inside this move no worried as they see what have come from when you cross the OK so let me it I'm going to show you how Watanabe maintained he has and the tools in order to preserve the so 1st of all we're going to In between the to 150 with 0 we're going to take the level set fate we want ,comma trends and by Zee Althoff rolled role tells me which leveled the tie and outlet tells me what place of levels 6 then in the function as is going to tell me how I might function faded that changes from 1 level to so we but this in in and in the equations in the transport equations and this is how it looks like we are going to show that the solutions are rotating solutions with constant angular velocity so we impose that in award and so now z it Our knows going to be X X of Alfaro the function as it is going to be given and when this decision told you what it looks like is independent of they are at their unknowns here are going to be X about the role and the angular velocity lands and since my pictures like this I only have to show that the solution then the equations that the solution solves this equation and support of of therefore the derivative so how does that look like it's going to be a function of C-4 function and has this a structure where a remembering is going to be 0 . 0 5 and is model to the decreasing and actually then it will be very important for the proof that in the middle so winter will be another Bader but when you add 1 wonders say that the Beta news to struck the beta from 1 in those regions in between we're going to ask for a linear decay then when any tends to 2 0 if we take a 2 0 we recover the problem for the bench so that this solution will be patch so we thought this and sat in the equation we use the fact that the velocity on the racetrack forms and we have had time to review the disappears and we This is the equations for land that an X of from and we have to solve for this we have some freedom to
choose the prime privatization of excess of Rome is going to be a part of nature of the inner-circle somebody they find X role to be a function of art that depends all times the vector "quotation mark of sign and no already knows becomes part of a row alone so it like that in the absence in the equation evolved and what we get it this functional In order to be a solution is to OK so 1 observation is that if I thought it's independent also that a R is independent of then what we're doing all we have is a real solution then it becomes you and what we're looking for is is artists that Dr. depend on tho so if
we take our elderly people to roll or any bring solutions that it's 0 OK so we
are and they did to what we do that we're going to use is the Crandon arriving on each so far In our parameter meal was going to be very angular velocities London and function artists are and so they goal if To find and allow so if we look at the the plains of solutions where the this is London In this this far role in it this line these are people to roll maybe this line is a solution also the name of the fungal then we have to find a London Stock on this basis X and why such that the scores and nation satisfied so the couple derivatives around the arising around dissolution broke the 2 roll so around solutions and we we have to find XL why such that they got all derivative exist under continuous and we have to find the London In together with the states is that the dimension of the kernel of the lunar ice an operator is something mentioned 1 there "quotation mark I mention that the image is also dimension once and then the 4th In a condition is the testers so if those conditions are satisfied squandering of it says that you can buy bifurcate there's a solution parting starting from London stock where as the solution to all of this is the best so that's the cell In order to there and introduce day-to-day ideas that are behind this is worse than than used for force for departed staff so I'm going to go and review revise their bosses past who would dispatch problem is now and we're dealing with a solution that workers at the is let's say 1 clear and zeros outside it satisfies all the equation the curve this parametrize quarter on the seals he observed like this then it's possible To have when you look at the evolution of See you can close equation in terms of sees itself the once you know In a policy looks like that you can recover the velocity everywhere instead satisfies all your wishes to the question and so this and this is the transport equation the values of amenable they remain In 1 inside a curve and 0 outside so this is I'm very well understood in the West for their equity problem in Schumann and was the 1st approved that his prominence globally will pose 1 sulfur and later there was another proof of different provide 102 temperature Constantine a of the same result and the result and then 1 can apply you always theory to guarantee that this is actually an additional ones which is what about the while 1 can ask the same question Prince kg and 1 can close again in terms of the evolution of the curve and the curve itself and this was studied the 1st by hysteria of the people in 2005 where he proved that is locally will post In and in the space Symphony why because this operator it's actually Yelena arise you lose a lot of derivatives so he and his proof is using the match King yeah "quotation mark then this could unsettle the 2008 show that 1 can actually it's obstruct anything then gentle direction In a component that will not change the shape of the interface and you can but I component that gives you at and the special cancelation and with this cancelation you're able to ship to close the gap in various teammates in nature so he proved existence in a cage every silly this year we showed that among all the patch solutions this 1 is unique and the global existence 4 this problem is open of course all of this thing works for bigger than 0 and question of existing global existence is open also part of the but in this case
we do have examples of finding time Sinclair so briefly give you what is known as so in 2014 Francis Collins head on Bob strength they show that most class singularities Cundiff or so if you suspect that a self intersection between 2 of the boundary then the curvature must go recently this year in subsidies less linear Resnick Galeão and in Flatbush examples include and finally painting merits but in they add a boundary so instead of considering the problems in their the plane because of the problem and the happily and they take tool a patches that's opened dissent with about 1 . Anderson so under that already exists it's touching the boundary so the card condition and local existence theorems is fundamental that you have a control of the quarterback condition but their political existence where the corner contigent test and they managed to prove local existence but for a small range of profit and arranged and also 1 over 12 1 over 24 but this moral lesson that there is a fraction of it and political local existence and moreover they are able since it there's a lot of symmetry in that the patches attract and they made assumption OK let's original existence and the show a contradiction at the closest existence in both parties .period intersected by if they felt intersect confidence and strings Theron as the curvature has to block as well OK but it's a contradiction so is really is telling you that there's a similarity but we don't know how it the people think this is with them and this is for Alstom between 0 and 1 over the government wants to do it is a different problem but it from that point of view all of and local existence the point of view of 6 the dissimilarity alter yes because I mean it's if there's no boundaries do you have a problem of controlling how the patches on the page's move and you don't have any control over it so maybe the singularity cannot form because the patches are moving so much that in this off this singularity yes but this theorem saying if you fix the boundary you but this then the dispatchers do not move From there but the detract each other then it has to intersect and something think is very interesting it is true that it doesn't tell us anything about the they come in without them OK so a numerical as insulation so there no it's showing evidence of formations similarities for the patch problems and in there said so and it together with the team in the driven from their lows munch on which wind 2005 in we found this season numerical simulations In we took a pass this test but the status were attracting each other and at the same time it there will be but the attract each other and playing time the silt intersect and the curvature pulls up at the same time the intersect in indistinct .period and what was I am really interesting is that it was a self and itself similar Labrador locally cells and this is the claim and this is a contract then so let's take a minute no 1 ellipses and and see how it evolved while interesting if the collapse which is very close the circle then the lips what happens is that starts rotating it changes the shape changes its shape but it remains the comics and thing if you take a longer it lips then it starts rotating you lose complexity but you gain that again and you keep going and seemed like the school exist but when you take a longer 1 then it uses and I forms appointed supreme singularity at the same the same type as we we got for the 2 patch OK so this is sold to a and Scott ever and predicted another type of singularity and this is another example from lips in In this was so but the 2 Scott took this alleged at some point before the bloc before the singularity and this is the holy looks like being the sooner OK and orange is when the song they had this then the again this is a these Oregon this instability Oregon the same thing Soon Jung he it's a cascade of instability this abuse OK so
M In some more .period variation so let me hurry up and sell some explicit examples for the year so state aid for the past problems in this circle or speeches solutions for a lot of fun the Hitchcock In approved that ellipsis fought the auditor are repeating solutions the known also by the states OK so it's only let's all the time and we showed that for Alphonse bigger than 0 the ellipses are not providing solutions and we were concerned about the change of complexity and we prove rudely that day a patch can actually changed comics back-and-forth and
then in 1978 the demons of whiskey In predicted 4 to the order the existence also taking solutions with ample symmetry 3 0 3 4 but any In and symmetry and this was a prediction from the new
and it was in 1982 that will be proved the existence of such a solution and in 2013 to think me the John McKay went over there the fact another another proof using the credit rating of and they proved the existence but building also proved that the solutions there actually seem together without Finland honey we actually prove that their real In 2000 Frankel shows that if you take them to be 0 the only solution is sought and to think in the same direction showed that if the other velocities negative the only solution is the search attendance and you can bifurcate from many the all the areas for for example a pattern inside another patch and that's what they did or you can buy for bifurcate from the lips so it to the conjoined but there be proved and the other half several allowed the bifurcated from the lips and we have to prove that the real and a OK so how did I started working on this sale is subject wire did I get interested while it when it is Fig me joy and the dryness then it To improve the series and the 1st year so I invited you to give a talk and Madrid and I got him interested in this subject and I talked with terrific end ital him more like don't you try to prove this for sq Jianli a we know there's global existence of rescued you we have no in St any a solution that there exist globally Bowling that are non-trivial so the year after 2 and it's in the paper and that you work with his former student housing and they proved that 4 out bigger than 0 and smaller than 1 these solutions exist and the are but there was a problem they couldn't reach outside a of equal to 1 basic Yuji which so I invited to fit to Madrid he gave a talk we chat and in and he is playing as what was then the the White the couldn't do it actually were not even not even to prove seem affinity between 0 1 and then there are fears highly amid myself we we we study the problems with the different formulations and and then we realized that the funding the problem going from far smaller 1 lovable to want is this local last of novelty derivative they were working on spaces to implement this and from the spaces is quite hard but to implement it and the soul of the spaces History so we changed our him in framework and instead of working on the compass plane which will work on Muriel world and and in it's all space so 1st we proved that the existence fold symmetry for in the age came it's 3 plus lot the bulletin and they would prove that the solutions are real analytic now 1 it is true that it from Rabinowitz tells you that is unique the branch so worried overlaps it's the solutions Israel analytic but had RBI doesn't tell you how far you can go into branch to maybe you have a real political 1st in the beginning of the branch and was then and is becomes intense I don't know and so M this 1023 yeah look at a little more thank you Frank and Hadden of itself excellence that let's look at this and they poured in for the patch so now and we're going to bifurcate from the circle right and we assume is repeating solution will the London there is still no our and set is ah althought also in know about the thing about flat with blood that being in the it's cutie equation and we get this function has to be equal to 0 when are of fights a constant This is out dramatically 0 so we can apply to crime that we can go and try to provide the kind of
so this and this is what they meant by adding the lock of most of local derivative in a it's novelist spaces his testament to assume that that interval is bounded 2 and that's enough and our faces X is going to be if it is for them in the states for order is going to be the sum of and we're going to ask this to be in this space age capers well and then the space Wiseman signs that we have tionary for the kind of arriving at an earlier meeting just point out 1 that part of the Grand Army is to see that the dimension of the current familiarize part is actually finish 1 so we apart and we want to see an element in the colonel and we take the element H equal to "quotation mark of em outside we substitute that indeed the new part N what comes up is that this equation has to be with us a simple as that so is giving me the eigenvalues and they died Street item that they have room it will also be a lawyer velocity and has for each and fold you have this Of course now so the problem is reduced to sail through taking to bifurcate from this and undelivered and then of course when it has to check the condition of image and tempers but it's as hard as hazardous look sold their hot here was to choose the right spaces in which you can work there a reality may prove is that you choose in in and spaces that are real it properly in order to framework to work OK so let's go back and and In and see what happens in this smoking during this whole case In remember we are bifurcating from role go straight ahead of us find of Alfonso knows this is a small the profile and that is going to be there's no lost on Nov derivatives because it is small solutions such so we don't have to worry about your the the Knicks norms of this sort of politics stops it's OK and but again we take the call to be Our In the space X this is the In the new part and we have to look for the elements and element in the car so again we we do as before we substitute in an hour this concept this and what we get is not
a skater equation we get the function where I kill that looks like this the killer looks like this and to deal the is that elliptical OK so we need so we need to solve this perhaps the case the patch it just a skater equation if we take the B-2 B-1 and we could limit at April 2 0 will recover this we recorded the formula of the angular velocity for the best talking so it quickly and what is the problem keep land most of the dead a joint but this Compaq compact operator we have a smoothness and 90 lad this is because of using it using the properties and we're able to show that there exists a solution How is this well and briefly in the reason this that N this in they add today and that's a major part of the operator it gets smaller as we take this guy so the goal is to find to prove you will you boil out then the problem to to and up to 2 showing that the eigenvalues there's a gap between indict involved here you don't know what what any single item but for the symmetric parts you can have a guest and this is a numerical guests and you can really prove that ever human now taking this sufficiently small we can make died in value of the whole thing To be Close to this how however small has this has to be 0 . 0 5 it's enough OK and and how do we do that we'll there would play with the S in order for this to work out a M a so and once we showed that we can and there's a solution when it has to shoulder the regularity so there's a bootstrap argument where we take advantage of the Ramadi or fast and then we have to prove uniqueness
and we have to prove that the called dimension of the images 1 and that the Trevor somebody is also applies to its Is that also we managed to do that and let me conditions said 3 minutes in some remarks and some very recent resolves itself this is a L but solutions a separate nation right so we will procure with bit effort Everything should work for in that effort 6 days in the past that the profile is not there In From are required to get a perturbation that works and can we get there we proved that the solutions are C 2 yeah that's us because we chose F to be see if we make our at school there but the world will make everything maybe we have to take us thinner this distance to be smaller then and we can make a this solution to be of hiring melody yeah we since this in order to show that this is a small we use a computer system proved we have to do this for each and so is that the fearing for alliance with for 1 single and any of the 3 but we believe Dad did you put any other and you have to crept numbers and and maybe this Savoy's revival be modified but they will certainly work and this can be extended for any outside party good report to the UN and now very recent results of what they just talk about this and our and 1 of them with this slide will be in the archives In the a couple weeks and it is can we get the computer give proof of this aired exist of this solutions without the computer and and dances Jesse bifurcating from the patch and the way we bifurcate from the patch is that now is a is there is this distance we're going to and bifurcate from April 2 0 8 deserve the Pats and at the same time it's a moves to the branch it opens up the smoothness of the function and when then if it OK so if we do this and they Diaz that we re scale in and was killed in profile at and with surprising is that this work for any and and for any profile left can change signed it does not have to be 1 of the increasing he can't have any shape just have to go from 0 to 1 Clinton but In here we sense that Grand which we cannot tell you how commited opens up so we can reach a solution and the figure fell for the only solution year on the the company also said that the new the home the the report and read OK so back support makes it so I think so the question for example in a stationary solutions and so the result of both land and in Franken that proves effort to the what dispatch it is In is a stationary has to be a circle an but you can construct stages solutions of 20 0 yeah that .period are not totally surprised and complete supported and there circle OK so the fact that the scope of his report and it is already telling you that there is some restrictions vary in strongest should a restriction so To the problem but can you construct this from a guy ocean or something and the answer is yes OK so it yes but it gives restrictions for example in the station that I think of Frankel's result is also true for STD and it's also true 4 a duration of the a small perturbation of this since the OK so sold it so it is until been told to fix results says and and so this the solutions is their angular velocity is it's explicit and there'll and you know in order to prove do you think underwriting of each right I'm the same maybe there's another perturbations that has this destruction all the symmetry and and you indelible losses different missing but the norm the knowledge but to fix results for that to the honor says that it is negative there's the only solution is there answer I believe that can be extended to listed and also 2 the cases of the small case but that's something to be done the