On global regularity for the 2D Muskat equations with finite slope
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Title 
On global regularity for the 2D Muskat equations with finite slope

Title of Series  
Part Number 
18

Number of Parts 
23

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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  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2016

Language 
English

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Abstract 
We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.

00:00
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03:26
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05:06
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19:21
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26:28
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33:08
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00:07
In can and in the world and in the end I think thinking the organizers for having us here it's really been a pleasure and also very productive and because of that change the type of so I'm sorry for the change this is all joint work with Respondent and there both here and so there will be some overlap with fur communal stock tomorrow but I hope not too much and you should relieve you of the spokesman said complimentary of each other home and so we're going to deal 1st of all with the way the equation following compressible density 1 that said there are signs the order for the velocity field you piece the pressure the 1 although about what turbulence and turbulence is generated at the boundaries of the statistical fears of turbulence deals with the behavior away from the boundaries so for all practical purposes might as well consider domain without boundaries and 4 drop talked about 63 and look at those conservation law that we have a 3 is the energy the kinetic energy at an allwhite awakened with force if you want if you want and so on if you can compute the rate of change of energy in the Treaty or either you get the budget deal well is equal to minors you are you before the you minus Grand dropped you fuel and if you want to include the forests you can and because of an compressibility Nunavut refuse the Virgin Street you can move spout units of virgins here curiosity about virgins and if you're multiple disintegration the parts together either the energy balance or if there's no forest energy conservation this is you so the also conjecture deals with House moves or you have to be in order to do this competition and it's OK it's not very complicated but we are dealing with weak solutions so we solutions just means you're solving this in the prime of times across the porous so I'm particularly impressed with some of objects and particularly confessed SJ squared off you sjs the projection opinion frequencies that to a J and if you know that and then you can do about it he also
03:27
from the SGA you of squared 1 this equal to what
03:36
is called the energy flux the flux to the boundary of his fouryear shell and that will be equal after integration by parts to utensil you
03:48
gradient square you plus
03:55
the pressure if you want but the pressure this appears between because used the registry SJ square used evidence 3 as well so this disappears and if you want you can put this in the form of a lot of the talk will ignored for so the question is when it's this 0 well you can put 1 of the SJC here and you can also and that's why a lot of tractors because this is 0 Knollwood 0 because as you use a very small function so I have no problems in showing that this object is 0 and this is called plus the 1st when this is called energy flux been altered in some papers might have felt on subject this is the energy flux through the show tutelage scenario may ask what is the bomb and I'm not doing this
05:08
injury chronological order but it was proven in the paper of Constantine just Cordova told a freedom that interest In 2008 following an earlier ideals constant Indian Dickey from 94 I Inc 94 at the same time the dishonorable but what they proved is that by Vijay and by the way this has nothing to do with oil this bond I'm showing here is for any vector few is less than the summation over I figured that 1 to 2 go miners twothirds J. minus or tool to go and here that little pay built up by you 0 3 for the values just as 5 plus 1 month you you can't just leave flowers very clear when does this guy was 0 so you can ask when does this S. Jacobson affinity that correspond to energy conservation and the flux of energy and while the answers here this you recognize of course is the best of states be 3 L 3 times and if you just have a supreme over the little painted pieces to be bound by the right if you had just have this the it at the limit is fine you will market this year but should get that it's 0 you need that this little created pieces actually and this is the result of the main result of just just Constantine automatically replaced him so it became just of constant and freedom that fit going this is the result of that if the weak social order belongs to this and energy conservation but this is really the reason was because of this flux as and the flux is really the important thing because that's really what connects on soda furor to come over here is the only reason introduced no of course this is the last in a sequence of papers and I have mentioned I 94 to former Bear Constantinian duty who have slightly weaker assumptions in particular In the state of constant Indian did can onethird plus epsilon 300 1st consider conjecture about it said basically about the sharpness of this bound isn't true the solutions 1st a little bitty pieces 133 bungled 2 0 so for instance 133 Infinity is it true that the energies conserved so we could the states this is definitely not how 1949 the of the conjecture as income you will discuss the subtle differences so we can know rephrase the on side 49 conjecture as In it has a rigidity .period and the region party is exactly this if the weak solutions are as smooth or smoother and then they conserve energy that's rigidity so long small whatever that means the L 3 in tiny vessels from 3rd 3 C 0 in place of rigidity just to be clear it does not imply uniqueness it complies energy conservation In the opposite of rigidity is that there are only given of space and less smooth than that X witches what the religious most means then of the 3 do you want the 3 affinity that exists with solutions of offered you either the next which not conserve energy so that can be said distance and conjecture and I there has been in the past years tremendous progress on this question it is currently still open but there has been tremendous object especially by the program started by the police and security here so I will mention that in a minute but before I do that let me raise oil from the Boston right now use folks necessary a few words about connections to the common bond of 1941 .period so obvious both 3 I would not have raised the question it's very important that we put into force now is enforceable in the secure about turbulence about the longtime behavior of solutions to this equation and of course if you remove energy have to put energy back so think of efforts of stationary in time objects stationery and smooth it supported it let's say supported and in the frequency for all we care about the war with the it's a means circuit OK so if you checked with the energy of NavierStokes what happens to winter a divided key on the 1 hand you knew to square what exactly the same computation so actually should do SJ here but that's not the way you will get also the limits and Jerry goes to infinity of pie Jane
12:18
and you will also get don't you knew and you will also get new with With the plot my energy is definitely going to think you're I suppose the Weiler the official Fernandez folks that this goes to 0 on much much weaker assumptions you can actually do it for little leery Hall is not enough he a the better the miracle of innovation brought has the best result at I don't really don't want to insist on this but I think the best results is this appeal Q with 1 over people Swan of ridicule hold less than a halfinch should be given for this institution brought in the seventies so I think this is the weakest results saying this is 0 so but they can learn if you wait for the solution to become statistically steady or at least 2 to reach them a place on the back or because you're putting the force which is very small with the social which would inherit smoothness form the forests in particular it will make this and therefore this is not here so if you're looking for a solution which has lived forever whatever statistical that means then NavierStokes you have divided the of energy energy dissipation rate and of course what you put in look at what is the problem fury about the fact that so ,comma the main assumption of this religion is that if you called it knew to be the energy dissipation rate due to the operation the local option where they can average against what well if you knew you had the unique ergodic invariant measure and strong flatten well then it would be just the leverage against the need to vary you need good invariant match if additional you would know how will you know strongwilled large numbers the couple's theory that use this is just a longtime management and so whatever this means this is average encoding statistical station if you want to think of longtime averages that's OK and the assumption of Kolmogorov is that the men in new goes to 0 in knew it's on number types in which a strict policy so this is 1 of the fundamental assumptions fall the ,comma fury of homogeneous isotropic turbulence homogeneous meaning it happens everywhere isotropic meaning it happens in the same way in all directions so in particular should keep this in mind it's not some kind of spot I think it's really feeling everything up but if so the question is is related on several years is of course yes and the relation is that so you do you can show you that you can go for my argument on statistically stationary solutions is that if you have a statistical study social phenomena books and you could prove Khawaja's weekly email to the statistically stationary studies Wilfred of order then he could prove that actually epsilon these are a lot of it so board approved with a lot of gives I'm not sure what the value of that is this should be preserved I feel the think so this is 1 of the main connection between the 2 years is that the epsilon predict assumed by Kolmogorov is exactly what makes this not conserve energy and again this is derived to becoming physics textbooks under the assumption of statistical stationary teens look at are the other corrections other collections in the answer's yes of course much finer in some sense for testifying in the Seattle was structure functions and this is of course related to stock more than In anything so the structure functions in this corner of the area they're just you take up Victor L. look at increments Of the velocity fueled by if you want in a lot of books let's Frazier's book you're seeing you only to the longitudinal structure function when you look at things peril to divert few Elvis scalar function because raise this to the poet P you can integrate this and you can take a sufficient because statistically stationary average and control number S P 1 new all I can can that structure functions that peace structure function of common and what is observed experimentally as far as I understand with tremendous error absolutely not and too much on timeliness you observed that this is proportional exactly employment for fall and I hope I didn't screw this up what does this mean uniformly as new goes 0 for L in Paris in the initial regime Ellen versus a frequency scale and an inverse has to live between the frequency where putting energy and the frequency with dissipation kills everything so this is supposed to hold in this region and this is called the ,comma golf fourfifths OK so not what Why did I put this on the ball you should look at this and should
19:23
look at this space efficiency are it smells a lot like the the same thing we have always 3 and you have to divide and l into 3 pieces so if you divide by L you exactly have a velocity increment with exactly L to the onethird raced to the powers 3 L 2 1 3rd raced to power 3 and this is bounded there's not going to 0 becomes the exit I think so this is a very very deep correction between these 2 I will not discuss the number for over 5 at all proving this proving the onset of conjecture are in some sense the except the your uncertain about why there is a statement about NavierStokes intervention is costing them so from that point of view currently we have no hope of this would have some hope will not we in the in the Geremi will mention the joint work with respect and their actually not able to prove this we are able to prove something about this to so again the ,comma theory predicts that this should behave like a constant that's not to the twothirds Bell took 2 thirds as far as I understand the experimentally they are deviations from this law this is called intermittency this is sober lost this is as far as understand not as robust and this is of course exactly the same if you translate this but finite difference thing 248 which of course can it's exactly the same clean energy flocks to the wave number K which is exactly nearby and please all from the projectionist in case you have to square L I'm sorry behaves like that's not to the 2 services can demands so the famous 530 spectrum that we here in all the physics books is the same as the 2nd and this is related to what we're going to talk about because this has stationed at the ability to not 3 so with this introduction mind let me state that I I constitution in the site I should tell you that this is what is known as the sole there has been a lot of progress on part B of news and conjecture if status hearing in absence of the complex that will not make any sense ,comma launcher will review this open to give a lot more detailed than I am so I'm going to go on purpose of that fast so there having worked so Schaeffer 90 3 I think chairman 2010 when we proved that there are weak solutions of the order in the space of to which the importance conserve energy there many of them see him thank you and these constructions are in some sense wild missing out of the the blue and the 1st time this was put in a rigorous framework and in the framework of comics integration is in the work of the land I should mention that to spilled this but this one's for you although since a KGB in 2009 where the use complex integration to construct we sort of knew it would have bounded into conserving and the beauty of this was not just OK you improve expert finally put this into context related to the much personages combating problem and so on it doesn't seem out of the blue in now there is a serious obstruction that infinity to go past that there is a very serious obstruction because of high high interactions and that is incident hitting a couple years later were able to successfully use special solutions of your there's this special status solution :colon from inflows proportion this to see 110 minus and for a while this was the "quotation mark and called record and since this work there have been 3 more significant improvements by using more structure of the equation in particular using a slightly more carefully material derivatives at the same time almost I said and book was this the the team so again by keeping more of the transport structure of the equations they have this pursuant to the morning studies and a lot of fun it seemed like this required a completely new idea to go forward and indeed it does require a new idea the Osaka conjecture the common spectra not about where this places the vote of 3 L 3 and with this realization but Buckmaster In its appears diseases when is the the 1 close to the same time deconstructed solutions which I see 150 which ended with the addition of property that almost everywhere in time the C 1 and almost every time notice almost every time is not so nice if nothing big ability but this group will also worse so I will just draw on their own have improved this time and 1 time he was minus and to date this is the record
26:30
quoteunquote to achieve this they had to basically explored the full power of pine intermittent if you want to make a connection with the turbulence and then share a communal will mention of some of these works so what is the main here and that I'm going to talk about them and put it on this spot the against the joint work with Tristan and there that exists we solutions of new either which live in the space C and time aged onethird minus annexed by age I mean the soul of they'll to space which cannot conserve energy OK so the couple remarks are in order 1st this is precisely what lesson of course there's miners here this is exactly gunmen were expected to executives we accept this for a while there is nothing is the right which is a major and we go it's the 2nd and rather more curious reminder is that if you compare this result with this 1 and you freely use the power of interpolation so again so but monster Galanis sophisticated proved that onethird of a derivative belongs to an Infiniti in the onetime and infinity states but that's basically what it proves what we proved Is that the same thing but we have an affinity in time and to states nobody's interpolated spaces you get exactly a
28:44
303 these are not just some random spaces so if you just know interpolation because it does not mean that there is a need intersection between the 2 subsets of solutions that's what put adopted line just to be clear that to subsidize the infinitely many uncomfortably many solutions constructed there and uncomfortably many solutions that we construct may have the intersection ,comma the very likely do doped so this is not that line will be against the scope of another paper and then we can all put the cost of conjecture to rest so I will try to sketch a little bit the approved in the remaining 20 minutes it's going to be I would not be able to fully compare follower with the previous proofs because it would really take over many hours to describe the previous approach but is the advantage of course the communal will mention some of his previous proofs before so I will freely .period also differences which may seem a bit rather than the 1st OK so the proof is based on the same idea of using complex integration so what is the main ideal for comics integrations the approximates the solution of the order with some Victor few whatever you meet again so you make against the queue and you making also an error Part this is your guests this is is in error and these will not solve boiler and they will instead sold or the training after on depression also is a deliberate projected on the registry functions they're not solving 1 of solving order would mean putting 0 here you're not able to do that because you made a guests random guests so you don't know the goal of the game you know you have some information on this and in particular you know that this lives at certain frequencies for you so this lives frequency London Q I think the most neighborhoods or just there's no need right yeah the initiative written depression because I think these errors will propagate concession right PQ and these these approximate solutions that you have made certain frequencies on the Q which you should think also has a huge number to the Q they're not there she super exponent think but for the purposes of the talk I think it's fair to say we can think of them as being exponentially growing and Cuba dropped Infinity's so what is your goal you want added that the perturbation approximation we want to find the Q plus 1 is the to plus W Q cost 1 of in such a way that adding this objects makes this error that you have made you're adding a correction the higher frequency and if you will therefore obtain new solution approximate solutions like that which lives at a higher frequency I you see
33:09
1 but hopefully there's a certain norms is much less than that of a certain age you can continue to do this in the limit you have 0 and that more knowledge in this process he also managed to prove that the sequence of the accuse remains bounded then you will have right otherwise I you may corrected and end of each use you make them so lots of people converse to anything so you have to do 2 things decrease parental stress and make sure you're not adding to wild things OK so just for the purposes of notation let's say but buys the number less than a 3rd but that is exactly 1 whatever it minus means can this the regularity that we get and let's say that error we have made you know another 1 you should notice something this is an tool and all of a sudden writing 1 the Reynolds stresses quadratic in the velocity the like the intensively so in some sense you have to put in a lot you cannot afford a new space and said the error we have made has a certain size and because of the regularity that we want let's say it has size number cute and to win which instead of writing this object wooden right don't you plus 1 but so Our goal is that this 1 has size limit to bust the budget was too which is smaller right because this is negative power this increases soldiers or be our goal it that's what they clothing and that went skipping upon not what is the equation of major once you have made then we can write the was the pressure peace so here you have the old Reynolds stress With its you couple the introduction of W Cubist 1 W 2 plus 1 from the collection of and the pressure gradient if you want and then you have to more areas once when this material derivative will fall on your perturbation and once when your perturbation is stretching the previous effective OK so following the innovation introduced by community last long we call this the oscillation error you should think of this as high high interactions in frequency nothing more just like Of this week on the transport there for obvious reasons and as the national for reasons that India's major combating problem you have very similar OK so the goal is to rewrite this whole righthand side of the equation as but virgins 1 an estimate the size and hopefully it is of size the proposed to so think that it is not something that Bloomberg you and while soviet because it was a sum of many pieces lives involved all the size each of the corrections for leadership in a little piece Billy but are that's the sort of thing I will mention it in the press With shows up at the end the whole yes you may have to cut in all these previous constructions in fact is isolated spheres it has Norfolk In this construction is in fact 1 of the main ideas is that we have all which is basically related to to such Boston inequalities intermittently it's not really a shame on you 1 small pieces he said the smaller of the 2 lists of US markets elements of the previous year in the of which should we actually make a supporter because we had a movie projected me you know we're all we couldn't on top of the project just so that we can you have to wear here you thank you thank you look at itself you should already know something if WTO boss wanted Saddam because 1 has any hope of canceling part Q which in turn has that size they're looking plus 1 then you better have doubled futile swung the video 2 fears of signs off at the will to make this precise the was 1 of focus In other words it can possibly cancer and it lives at the next frequency lives as unlikable 1 as Alex pointed out in a shell in fact whose thickness is rather subtle OK so now I want to just point out very quickly 1 thing often very quickly because of this time you can actually cannot go with better more fair because of the Nash and you can very quickly see this on you can just computer How big divergent skiing various whatever that means false W2 proposed gradient Heakyu right the size of this to get the size of this meeting was In an infinitely 1 you can check with the sizes are you get 1 that you were limited lost 1 by that she was because you have a derivative falling on the lowfrequency guy this knowledgeable inverse it's because you're inverting a negative 1 operator of something which is at a higher frequency and you have the sizes of the objects which are just build pupils onetoone house the altitude we and you can object but this I will be of the desired size I like to cook so it's a computation you can very quickly do it if you assume John magic it takes a while before Super Six by widening the company OK so know what is the main idea I didn't mention anything about the proof so the remaining 10 minutes all these ideas by the way I all the previous construction than say anything useful now I will say something new while the previous constructions are getting stuck the regularity audit is 1 of the reasons is that the transporter is very look at that you're making highfrequency perturbations so the material derivatives can hit this highfrequency perturbation that completely Q I mean you get 1 of those exponents so what you do is that instead of having stationary Beltrami flows you let the flow with the velocity field of the previous with the too what would you do that 2nd property that you use that's where starter was built from falls many certain company under reconciliation it's destroyed because your your your your oscillations are not exactly the trunk so you can let the transport equation run for some short time How long time that is you need to put in time capsule when you material derivative hits that time couples you make an error which at 1st seems to have killed your skin because it's as big of a size as what you started with you have no improvement you think you're done you can play some games and that's how you get here by interpolating but basically this is the main so what do we do in fact hold corrections actually look like so again I will not write projectors and will erupt not trying to obtain the projectors but they're here this reduces some lower frequencies if someone could use and some objects W. exceed which are built high frequency oscillations I will tell you exactly what goes on in too much about what they are what is this index said looking at the sphere of Lydia slumped pupils on the natural sphere In the previous constructions this index that were couple .period all always including the diametrically opposite 1 a couple .period this is basically I'm exaggerating some proficiency London lost 1 both eggs composed with the fall of the field of cases would be at time 0 but of course you let this run and what happened in the previous constructions when W. exceed interacted with W. minus the face councils you produce something low frequency and that lowfrequency object is exactly what canceled this lowerfrequency object what can we do that that's this interaction hold all of this interaction this interaction is high Heinz bids Alzheimer's at the same frequency number post and that 1 you win because reinventing the divergent so here I want to emphasize it spits out something at very high frequency ,comma composed and you win by inverting coverage and this is what the oscillation and actually had a bit of rule that apparently not once you too much of the past costs so what is our construction instead Our vector always objects W let me start by doing the North and the South Pole there is far fighting many of them think of them as they have a selfappointed infrequency of Victoria they have addition :colon so basically here at the North Pole at the softball and at every other points Boston during the previous construction would be the biggest liquor what I mean by that we really take this next see party right which is also sidelined and we add to it project which is related to the Danish :colon it has certain features distribution colonel because certain separation between the 2 elements in the Mexican city elements it has certain number of OK so separation a number of elements call number are just so that we fix it use try to get will depend on cue so the higher the cued them or object to putting look at what happened what is the advantage of this what is the norm of this new object that we the latest dependent which is the center well like any Chicago we compute the other 1 1 user log in and to square root of the number of elements Infinity's a number of so what we're doing and given mind we want this relations and we already had selection for the coefficients right already this or the correct size so notified the Felder inequality may make use of any size more than 1 was Krumeich correction this is exactly what like that it is almost a forgone knowledge this is how much higher frequency than so when estimate the Oklahoma this it's like the comic book deal to moms and so if you don't want to destroy the size of your perturbation you'll normalize audaciously built on the colonel who have helped to 1 that's the quarter scaling if you want to maintain something like that this almost or somebody really comes if you have sufficient separation In a frequencies there if you don't have enough separation this will get screwed immediately built along 1 well by scaling the N 1 1 is no small what you want by a number before we just had 1 element it was on that 1 so now all of a sudden we're computing the transport error and this is something I have to very very quickly finish I hope "quotation mark when you know computing the transport air these cannot fall on these deletion planning because we followed them with the battlefield of the previous collections they can only follow the corrections I'm sorry yes there is a mixture of the thank you thank you so this is the removal of U.S. here because this it cannot fall here because is transported to it has so not only compute the other 1 1 this is what you have to compute the element to transport them again you need to use almost a formality this lives at much higher frequency if you have enough separation than this 1 due to compute the other 1 more you get the element of this times the with that this will give you exactly the same things and in the previous construction except you have known about you could cost you have improved your main enemy was exactly this option no of course this improvement does not come for free let me writes that involved this dish colonel or so these guys help you when this interacts with exactly the softball all you go and you know when this interacts with that your good because they always speak low but at pockets 1 1 1 the least that low but now what happens it is I just moved closer to look at what happens if this 1 doesn't interact with the South Pole with this 1 How High Is this interaction the answer is not very high it's that so now what you he your seems like a stock right in the area made from high high interactions a section of high it's not sufficiently low either there is in some intermediate and this seems to screw and then it doesn't it you now include more real stress collection you come with another velocity shield and you correct those interactions which are of medium size those which are of high society you're always controlled by inverting the virgins and this is where the thick and the thin at the thickness of the show comes in the thickness of the shell is how high do you have to push the error or stood to win from the beverage could you optimizing and now we basically have a chain of renal stressed errors in every single step every single 1 correcting the medium errors you have made From these high high interactions which don't spit fire the advantages in every single step of this chain you have slightly decreased the amplitude In the separation is slightly increasing also so you have fewer and fewer enemies and we can prove that after finding many steps independent of this fall and you did achieve the desired on and this was the 1st on the everybody it added the W is the depicting the what I should use only the initial date of these are flawed by the velocity fields these Havel economics teaches the correcting go stress which itself depend on XT the context any chance of running 2 were variations this theme the bailout solution but the 2 iterations at the same time maybe not but the To get stolen 303 but not by running them at the same time it seems that the stamp or intermittency that you guys build doesn't really good gelled with the station intermittency that we will that would build and that there's some idea we don't know yet if it gets you all the way it gets you a bit but with only home the fumed