Nonlinear stability of expanding stars in the mass-critical Euler-Poisson system

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Video in TIB AV-Portal: Nonlinear stability of expanding stars in the mass-critical Euler-Poisson system

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Nonlinear stability of expanding stars in the mass-critical Euler-Poisson system
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The gravitational Euler-Poisson system is a fundamental astrophysics model of a Newtonian star. We first give a brief overview of the existing results on the free-boundary compressible Euler-Poisson system. We then study the question of nonlinear stability of homogeneous expanding star-solutions discovered by Goldreich and Weber in 1980's in the mass-critical gravitational Euler-Poisson system. We show that these solutions are nonlinearly stable with respect to small perturbations. We thus construct a new class of global-in-time solutions, which are not homogeneous and therefore not encompassed by the existing works.The problem is mass-critical with respect to an invariant rescaling and the nonlinear analysis is carried out in suitably chosen similarity coordinates. We present some interesting open questions at the end. This is a joint work with Juhi Jang
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In what I'm my I would 1st like to thank the organisers for this wonderful conference it's the 1st time in here so on before I say anything about math everything I will talk about is a joint work made Jewish Agen who is based at the University of Southern California and this this year and the IRS so the good thing about this long title there is that basically tells you the statement of the main here their certain special solutions to the system of beauty is to look at the or the possible system it comes from astrophysics it's a model of a stock and that we will be studying the nonlinear stability of that class of solutions so let me give you an outline will 1st ii sustained tend to spend quite a bit of time introducing the system in talking about general properties of perhaps at the expense of giving you some technical details in in the 2nd half of the talk on but I can promise that there will be 2 different stability results pertaining to 2 different types of special solutions that we will look at the end of 1 of them the first one is easier this is the 1 where I can I will probably give you a pretty complete media horde approved works and hopefully there will be time to talk about 2nd 2nd type of expansion so before any of that money just selling the gravitational system is the model you would find in a typical astrophysics book describing time so it is a Newtonian model the idea is that you get this far is a dancer kept together by gravitational forces so let me tell you what the stock is not about it's not about the relativistic models of stars but nevertheless part of the motivation for studying this small indeed comes from In a related and I mentioned that hopefully at the end of the talks and so as I said disparity delays as a complex fluid body and the word complex is crucial because it suggests that there is a boundary between the fluid and a vacuum and if you want to model such a such a problem if you want to track the behavior of the boundary because allow allowed for the boundary to move to this cause this gives us all moving Starbucks direct human interface so what is the model
the 1st October the unknowns the 1st 3 are unsurprising and you use it and the fluid density rho the pressure p the velocity field you and the gravitational field feet but you need to coupled to the ordinary questions the force noted that the further unknown is the support of the floor because you're trying to treat boundary problem OK and the creations again use so you have seen some fluid models this time around on the stairs we will rule soon will soon forget about this like the 1st equation is the continuity equations the 2nd 1 is that expresses the conservation of momentum and on the right hand side you will see the term caused by the by the gravitational field fee and the gravitational field fees self-consistent it's are generated by the by the mass institution so so what last fees for pyrolysis apostle immigration and these are the accretions of compressible fluid dynamics on top of that we have to supply this problem at some boundary conditions and remember this boundaries moving so the key nomadic boundary condition is the is there is the natural statement that the boundary is moved by the fluid particles so the normal velocity of the boundary is affect the normal component of the velocity field at the at the freebie the pressure vanishes at about 3 OK now you if you count the unknowns you will notice that there is 1 superfluous unknown here namely the pressure there is no evolution equation for pressure and you need to prescribe further equation to the horror of relation to actually calls system there's money creation 403 for you 1 4 5 but there is nothing for pressure and this is typically done by prescribing the so-called accretion of state now it creation of state re-creations of state something is something that in in the physics literature you it's sort of would look at a particular model of a star and you would hope to have a good understanding of what the question state is for that particular stock what is commonly done people starting to soak up all the Tropic of the creation of state where the overall is wrote to a certain power again and this Denmark is the so-called media about that age embedded in the US and it's allowed to vary between 1 and 2 so the way I was sort of would like you to think about this model I would like to think of this nonlinear problem as parametrized by again OK and as a very gamma the properties the qualitative properties of the solutions of this system will change what so let me comment
a little bit on the on the literature of the vacuum interface itself is known to cause Severe analytical difficulties in improving local or even local locals so the basic basic resulted in this in a fury of compressible flow it's been a vacuum interfaces are basically these 2 results In the past they arrived became independently in 2010 by components cola and Zhang in and so what the what they have to contend with his assistant the generously I will explained as we go through the talk that is caused by the presence of the vacuum but 1 of the
key insights is the use of the Lagrangian course it's so if you use Lagrangian cordons and if you rephrase the compressible on the equations surrounded by a vacuum you will discover I that there is a very creation of Quanzhou innovative creation lurking in the background market so that somehow the don't during there 1 of the big revelations it is a generates very equation degenerate in the sense that specifies a later but and this the word degenerate has to do with the fact that some of the coefficients of the regulation will vanish at at the time vacuum and another key assumptions so another key assumption in In in these works is that of a physical vacuum by under conditions so Our here I'm speaking what it's a statement that the Florida and Philby which is this quantity wrote to the power again on minus 1 there has to have its normal derivative and the vacuum boundary has to have a this is perhaps not of a well-motivated as a condition of so physically discrete expresses the fact that the particles have to accelerate of accelerating the normal direction at the boundary but of the ideal soon actually and and was soon moderators conditioned by finding particular steady-state solutions of the on the possible system that satisfied indeed if you open physics textbook this the star models the steady-state star models and look at all satisfied this type of condition the compactly supported ones the Office of these so there's a little bit of a discrepancy here here and citing works that pertain to compressible whether question you could ask me what about all the possible because I'm adding a gravitational field from the point of view will pose in this theory this is a lower order perturbation so if you understand how to how to provoke was in is here you will understand how to prove in for personal systems but nevertheless age there's a reference that I'm missing where's the chalk he often tested think so old sofa prior to this work there is 1 result which deals only with a priori estimates of Italy and led coupon mainland scholars so this is 1 . 1 million missing sewn up during commands could tones and land scholar and the on postal system this came much later entirely based on based on and on and on these ideas these American shamed and but this is the 1st time not much for the betterment of so for the for the Likud case when the density is not mentioned bonded so this is a significant West Virginia home of salt instead of I idled now try to give you have to do 3 things for you as it did in example all this famous classes that states known as lean and stars then I will I will inspecting a sigh of relief I will approve produce centenarians scale rescaling for this problem and that they explain certain of the notion of criticality that enters and add that to moderate the main main results so most famous classes special solutions are the state and reproduce them you set to be the only a function of X and YouTube easier to plug it in and the whole accretion reduces the statement here gradient of roast start again not to a strong start in the star equals 0 so if you use the fact that the sentence was a plus on immigration and if you call this object here this is the fluid and Volpi few call a W then the spherical symmetry and this is a very classical stuff you see this sinister physics textbooks you reduce the problems and to solving this ordinary differential equation book and the kid the question is for what powers of gamma can I find a compactly supported solution would find mess this is your stuff so far the answer is it began analyzing in this magical arranged than confine profiles indeed compact supporters at the same support 0 1 and they look exactly like the once gamma hit 6 5th in world under such solutions exist it's just the audience and moreover these special solutions in fact satisfied the physical work commissions of W prime at 1 is strictly less than 0 OK so the W the density profile is not move across the across the vacuum interfacing and this is 1 of the critical points that are works of coupons scholar Janice Moodie had to address the fact that you do not know if this is a generic part of this year so you have to find a way what it even if it all doesn't even need to differentiate the creation in a proper you can ask however and this is a physicist David hardly solution stable it's another numerology gets even more interesting if gamma lies in this range here the land stars early nearly unstable In deadline so this is wrong In galleries bigger equalling 4th and its exact following nearly a stable so it's just some basic following their analysis and of course begs a question about people took questions are these results can they be upgraded to a nonlinear statement so the 1st result can it is not terribly surprising wants to have a growing mold which is indeed the main source of instability in this region and once you have a good developers in this theory then you can prove nonlinear instability in this was shown by Jang in 2014 but nonlinear stability is a much much more contentious points so there is a there is a conditional non-winner stability result in a variety of this sort of a fairly weak apology generated by the by basically just the engine by arising from 273 which says if the solution exists it will be nominally stable however there is absolutely no reason why the solution should exist why should small perturbations of of these statistics should any but for the reason for this is that they right so the result of Ryan interviewed told apologies so weak that it doesn't really see the the 3
of them the freebie In particular the type of stability a result if it were To Holecz if you think about it will actually allow you to however supports splitting up and things like that in the topology the so it's really you really need something else and fact I will talk about it at the very end ital highlighted this is 1 of them important open problems and this this figure developed OK so let me try to perhaps explained exponents the chalk in a different way and this hopefully will I will help you appreciate that there is an invariant rescaling of this problem to each gamma To each day you can associate the this invariance rescaling we see where you just work it out and you realize that if throwing your solutions and wrote till then you told also solutions and the way re scale the time depends on gas so really the value of Gannett has a lot to say about the structure of the solutions of the creation in particular the pressure and the potential you can work it out from these to scale OK the 2nd ingredient that insurance to appreciate what conserved quantities while the masses this is easily the interior of the density and the total energy let's just briefly talk about the total energy this here is the kinetic energy this object here remember we're always look last fees so if you intended by pirates this becomes negative agreed in the square so this is just the the gravitational energy which gives a negative contribution In this stay here at root of the power of some sort of thermal thermal engines OK and that the crew the the question that is quite natural and in this context is is what is the behavior under the rescaling the new discovered the masses remains invariant Inc exactly when they're Michaels for 3rd and energy remains invariant exactly when gunmen equal 650 and these are the 2 members the popped up here OK so this is to suggest that these numbers this this numerology has a special meaning all the results for the remainder of the article of this piece presentation real pertain to the mass critical phase Jan-Michael for things and said article because the the actual article was supposed to appear tomorrow on the on the trip consider America and and that there several spots where the truck the reset and erase all the results for the remaining of the remainder of this article pertaining to this case anyway but opted this is that this is the key .period remembers Michael for thirds is the most critical cases of till wife physicists lot like this particular the value of gamma missing it's a rescaling that keeps the mass preserved so it's a very appealing to think or gravitational collapse for expansion that somehow some crosses that sort of cascades through the use of scales wonder but it preserves the Mets it's a natural so that the beautiful thing about this is that in fact there is a result of golden conveyed who physicist in 1980 when they construct they found explicit examples of collapsing and expanding solutions of examples of stars in this mess critical cases and the but now that do some some computation so the government will support 3rd and if my so similar rescuing it goes so it can work out what this coefficient is here said to minus 4 thirds 2 thirds 1 over that has 3 caps so this is so this becomes a PE over land that apparently OK so To have itself similar collapse results in rescaling would correspond to having a collapsing solution that scales like like all the others whose radius skills like T minus the 2 the power two-thirds on expanding solutions that that that expands at the same rate this would be that the corresponding coefficient if you had if you want to call this a sell similar :colon however if you fall through the paper Waldron and Babar already implicitly there and then explicitly in the directs works of McKinnon Lane you will find the 2nd family called expanding and collapsing solutions that indicate that expand and collapse of the OK and this is sort of this was part of the mystery for us to understand 1st of all the the day the reasons why this is so yes which will cost the lives we have this 1 I did this is not exploit this is just the sea I'm putting this sense here I look for this informal interviews slides and will give you the exact form of that call for this guy there's no correction that this is perfect this is this is corrected of not money is just is just something we used use no no no logs and people will come to that at the end of the talk of the food from the other 2 findings in animal beneficial similar I mean that's what this case but the 2 families of collapsing in expanding solutions the the soul it a I for whatever reason they in these works to the Library this often never talk about the rates they sort of the Analyze This and they say their existing solutions but if you really look at the rates this is what will happen I don't give a give a more precise statement so this is an informal statement of informal statement so don't ask me for any any norms and the and so on at this moment and this will also come at the basic result is that this cell similar expanding pro expanding profiles noticable collapse the ones behave this about two-thirds are "quotation mark dimension 1 stable and doing their own expanding guys will be yes stable In this case we can actually talk about asymptotic stability in the school mention 1 of a set of perturbations Where is here but we don't we do not uh characterized the asymptotic detractors there just some objects that do not necessarily belong to up some sort of nearby linearly expanding profile OK this is the news that this is just of the results and get into the technical sticking slightly later In particular desire To my knowledge of placed 1st nontrivial examples of global solutions for this rebounded problem and what is very important for us we can actually say we can actually say that the support of the of the stars of the nearby perturb started actually grows out of approximately the same rate as the underlying OK so nominee let me take you
back to what what ideas going to develop this theory and this will bring us than and then to the to the proofs of these results but as I mentioned fundamentally again this in this work is to use the Lagrangian course it's OK so you've seen the 8 about its severe obviate they simply that the flow particle trajectories and because because ideal I don't know make an assumption spherical symmetry it is actually not fundamental to adjust these results but I want to make it because it will display in a very concisely the fundamental structure that that we need to worry about a case so I assume everything is right with symmetric now the the the alive made it's slightly far pompous to say structural miracle but it is a beautiful structure of these equations if you if you use Lagrangian coordinates In spherical the symmetry you discover that Kite which is now my Lagrangian that satisfies the 2nd or the equation ITT FW kite equals 0 where this W is some nonlinear operator which is of 2nd order OK and this is an effective cause Illinois evaded questions on a Compaq domain but now you there's this thing here I don't want to display after me because it's a beautiful but I'll tell you later will clearances and you will be convinced that it is it is a spare 2nd order it's a 2nd order a joint suffered joint operator Indosat and year a correct attitude is so so I can so that jury raises questions so if it were not the statement symmetrical or irritation if it were the purity of the for the system the the Mott the call metadata votes at this satisfying effectively meant that equation the divergence of the foam which roughly corresponds which captures the irritation part would satisfy any any creation like this so their regions of the Jacobo their versions of the of the of a of the velocity vector field whereas the colonel would satisfy the them but because of Castle explained because remember in the Lagrangian caught in its role the overall the kids you very use now pullback respectively keys novel my Lagrangian time use my Lagrangian velocity plus the bill the pullback after pressure would be something and you solve this guy's mentions so this will give you a few mission transport equation for curled in the treatment of sort of estimates of kernel you for free so the idea is that new deal with the full problem here estimated they were using the current into separate ways what is the cost absolutely so this is in some sense of the heart of the approach of tones Koehler agendas the framework the functional trimmed slightly different but this is the the main idea not so there is a there is a media creation that's what that's like .period and it's exposed on a Compaq domain because you pulled it back wanted India to fix that the free boundary had to pull it back onto something fakes since of maybe creation accompanied so let me know repeatedly
arrive there I really those only those solutions of gold right in labor and other people I don't call them homogeneous solution because of looming the UN's that's the kind just Orlando team doesn't depend on our now this feels this will tell you that this this and that's really reduce the previous equation too To all the hero lambaste double bought plus something which does independent equals 0 so you can separate variables and if you do that discovery
there is an Audi satisfied the land and there is another will In our this time set aside the W if you're interested in finding compactly supported solutions because that's what qualifies as a as physical as starving the vacuum interface and you can't do that there is a magical valuable to stars such that for any dealt a bigger than those built a star you can find compactly supported solutions to this equation and this is just an you can really prepare for a range of values of Delta and initial conditions you can really classify see what are the rates of collapse expansion and so on so I will do that for you in a 2nd as I said the console
docility press prescribed initial conditions there are 3 parameters in the problem the delta that I mentioned 1 the 1 initial loss at the end when the 0 the initial radius of the but there is a concern of energy associated with this it's trivial it's it's this quantity here landed .period squared plus 2 adult being the Eulerian descriptions it can always go back and recover the original solution in delirium description the density and the velocity will look like this OK so the key points the best things like Lambert in the minus-3 Strickland is shrinking his warning that if 1 expanding the density the the flawed also terror itself apart it's it's it's the and Amanda as I mentioned self-similarity would correspond to these these these numbers here two-thirds so it turns out that sell similar solutions exist but only when the energy 0 if energy on 0 they're not going to obey the two-thirds of all and instead of sort of solving the DOD I that all I also
couldn't resist not putting this on the on slides this is the a kind of of bifurcation diagrams that explains what's going on so this is a Delta Delta Delta exits this is the land the 1 axis command fixing the initial initial reviews to be exactly 1 of so what you see here the libel region this is the range of parameters of who give you always only nearly expanding profiles so a profile that behaves asymptotic we like some constant times to these guys here and there the thick blue line this is exactly the self similar expansion and there is an explicit formula will give you for them in the next slide so but they you have to insist that in the energy 0 so they obviously unstable because if you can go either way from the surface the question is if you look at perturbations of energy 0 or are they going to remain stable and then you have the collapsing but some always you can see most of the collapsing solutions in this strange here are Our and heavy itself the exhibits of similar collapsed and then there is a there is this a lower lower lower dimension along the line where the "quotation mark would collapse is in fact it happens Italy so this is the structure all this of the space in 2 two-dimensional parameters OK and the basically the basic results is ah this liable region is stable and there Our this is thinkable region coordination on stable so this line this line have an explicit formula which is just a statement he you have a question and you argument here I well I can tell you very weak statement I can prove that these are not only stable in this in itself similar frames but this is not a good statement you really need to I don't know whether such phrases I don't know if you Britain and Ireland here whether is collapsed because this isn't a problem I was thinking of his Italian dailies such that the so-called writing neighbor completely discarded for whatever reason and there they were they were interested in coal and building collapsed and expansion self-similarity but if you look at other physics literature they actually mostly don't want to deal with the with the 3 boundary they aware of the difficulties of countered that sitting there in their literature and a single goal is the only example there have a sharp abound in most cases they're worried about constructing some type of solution that exhibits the earlier collapse supernova expansion are where you have infinite support but the indicates some sort of Decatur figured this would be for them good enough case and this and this is this also leads to good problems because most of it is also known to so part too hot to study the stability of the expanding guys this is the formula that Franca lead asked me about this is that explicit formula for for only the similar expansion whereas you don't you don't have explicit formulas for billionaire expansion there sort of proxy but they are leading with justice so I will from now on focus only on the expanding profiles and perhaps pompously call supernova but let's just say expansion and I wanted to recall that they satisfy redecoration of this that and FW some 2nd operator the to study the subject well you adapt your unknown to what you believe to be stable by the by by dividing by basically you define CSI To be Key divided by land about and then he expects site to be up a small perturbation 1 which happens to be a steady state of this equation in this case so if you maybe sunsets and plug it back in here you get this equation 2nd order of the now I know how to deal with its well sell
similar expansion really behaves well respected scaling and variances of the problem so the natural thing to do is to time With respect to the corresponding of corresponding of similar rescaling as you will see the billionaire expansion doesn't that the probably creatures than the adjusted very well so it will cause a certain difficulties that will not be present themselves in the case OK so let's move on To this got .period any questions about this so is is it is it is it absolutely
clear to what was going on so let me let me tell you the basic structure
that will allow you to to
understands that so let's now focus purely on the cell similar profiles to remember when you write at down so similar profile it's something and that it sounds like the 2 kids so this is the radius of the of the start 4 K C it's a steady state of the big questioned
on the slide so I will go right CSE is 1 plus a perturbation table 1 suspects that you do that
it demanded the initial images the energy of perturbations 0 edges closer to the citizens of conserved .period and of course the key ideas they mentioned is to pass the south similar time if you do that so this roughly me that you then add the new time hit behaves like a lofty OK because land spent like two-thirds if you do that you discover that this coefficient here this constant cycling the Tour on the other side of the notation that's that's popular in the in the literature but this quantity is a constant logistically lessons here what about the nominee narratives to expand the nonlinear it's year-round around what OK and you discover and this is now at the point that perhaps clarify 1 of the questions you discover that in the end DeLeon areas ation of this of this all operator it is precisely this 2nd order elliptic operator here when I want you to stare at it for a 2nd notice that when when IRA 0 this terminations and when our 1 remember W. Delta is is that is the density profile is a density profile pitch roughly looks this and also manages that are 1 focus of this in this sense this operators the agenda because it it causes it has this vanishing awaits if you wish at 1 0 and at 1 but this is not surprising that this is the type of difficulty you will see in the in the study of just cure-all but it is suggested it tells you what sort of functional framework to used to infect a deal with this problem it tells you that you need to use some type awaited spaces but not the cool thing about this if he passes so similar time you he expressed new now try to find the crucial for 5 This is
what you discover so finances the 2nd secondary 205 plus strictly positive terms intensified as -minus something negative times 5 plus this leaders operator equals the right-hand side which is known OK and you immediately see you realize what should be the mechanism for stability it is precisely this bloated here which acts as an effective and this is as a consequence of the fact that we are expanding very expanding our profile around and expanding solution if I did it around the collapsing solutions that the signed here would be with change this would be sort of the opposite of that OK so it's a lady creation cause a literary creation of combat demand but the descent into so the correct statement is that there is a high the energy if it is issue sufficiently small and if the energy of the physical energy of initially is exactly 0 there there is a little global unique solution to the above equation and affected decays exponentially fast in the US In the variable this translates into some sort knowledge break decay How can I have the following question here is the result of this is where history of student days thinking for the question because I will now in next lighted the engine there the Alberta is itself a joint operator with respect to this lady remember this vanishes at 1 of this nation's 0 spectral get OK 1 is in the mouthpiece of this operators so so if you're if you're orthogonal to the to denounce space you're good you you you have a city of this operator of this preseason is obvious what I do want to pay what should pay attention to this the 1st term is just the L to norm in the corresponding rated space if you want to control the 1st derivative you have to raise the debt level of the generously in the that weight W. dealt with he is correct correct so this is a critical observations of what its own industry which at the time time of the
mansion for also is a cost is a constant B is a constant due to the because of the self similar structure because I'm expanding around so simple solutions so it's really it's really captures that this is the sort of the correct information about the self-similarity that you need to rest on their is obviously that would guide the term here this is a bad guy this will give you 1 exactly 1 unstable mold but this is not surprising I told you that it has to be unstable remember that the cell similar the South similar expansion was according mentioned 1 of phenomena so clearly this has to be the case but this unstable mauled grid which is caused by this this negative terms but it is the remainder of the areas starting 0 energy data and so that the unstable direction you can check this is transferred sold to this end 0 energy surface so you can't really control them you can't really control the the the instability which translates in this case in controlling the In the product of fight and the and and the generator of the null space so this allows you to 2 carrier carry on the essence so think of this as so this is a relationship that you get bailing erasing energy equals 0 this is the quantity that control just by doing some simple energy and this is the quality that you want to control together spectral now answer to
appears question indeed the energy is motivated In when this case for quite concretely in the 2nd case more philosophically in the case of expansion by the works of these people and that now we come to the 2 Europe Survation indeed those you I don't want you to gain any infusion from this all I wanted to see is that each time take us further normal derivatives I have to raise some indexing nite in my in my weight and this is this is how this works this is dealt the case pieces are basically mean you advocate the other full power of W Delta Two to Europe to and this is the only way you can do it so more spatial derivatives implies more distant weights and now you will
trust me if I tell you how the energy method works you are you try to prove something like this mystery here kind of comes from that event in terms that I mentioned Of course you need to control of the the energy by this dissipative term here you can't do that because of the spectral gap of property that the it's a technical thing when you can do it and this in principle gives you the exponential decay and complaints could prove what I what perhaps I want to say going back 1 slide this is a mixed space-time norm In reality into the estimates only the good times that because of commutes to the question entirely and the use elliptical estimates to 2 to control the higher space derivatives of the time 1
so this is what I just said he was the time derivatives to build an energy elliptic estimates now 1 of the technical tool that go into closing the estimates remember there is not only in their own own right inside rejection looks quite directly when you write it down to the tools are hardly a new inequalities which exactly allow you to deal with those rates the degenerated 0 1 and in embeddings between Ray Kinsella spaces should take enough derivatives you can control lower by the at infinity norms of lower overturned by the so I'm not going to any of this this this and this is technical I will have 8 minutes left I don't OK so let me then see just a few words about the 2nd problem here and maybe try to hint least that life is more difficult than the first one at the difficulty appears not to be conceptual more technical but it is there and it took us some time to understand so
the reason why this problem is harder conceptually is because you don't there is not you you cannot win it doesn't honor the cell stimulant structural problem doesn't honor the variants the solutions are there but how do you prove that there's their state soap and it will necessitate the use of over so 1st of all the fact that it is not honoring the variance Bill this allow us from using time vector fields as you can see they will not commute well the problem because now when I write down the re-creation for my perturbation the it's with have time dependent coefficients so we have to come up with a correct spatial derivatives that will captured the door the genesis of the vacuum and this is precisely the vector field here we just happens to be a 5 dimensional clashing with his quick
OK so what do we scale times so that's touted new time becomes lofty again so this means dividing by this clearly expanding profile the them In here the land that killed to think of it as something that behaves like a constant and if you right down to the question for the perturbation now you discover that you had this time dependent factors in front and landed deal that is growing exponentially In this new things you that focusing have into the real accede the dollar policy to the town was the spectacles articles right inside this is still causing a 2nd damping effect because it is well known that the expansion produces a sort of a stabilizing effect on it's reminiscent of all who works for instance in a special relativity in general relativity when you have cosmological constant but this is a statement there is a high honor or the energy still there we'll allow you that for small that will give you global existence for small data however is that you cannot classify disability attractive so so all I can say is that the time derivative the powder irritable dissolution decays but you cannot say that in the limits you belong to a nearby member of the the plenary expanding found this energies constructed by using the powers of the Semitic operator just wrote 2nd goal and so so as
to be honest To be honest I can't I can't talk about it appears they have a few more slides in the end but let me let me not going to the technical aspects of this proposed just say that this operator is designed to capture the very precisely so you go in other words you cannot just blindly applied the IRA derivative or DX did you do that you create singularities at 0 and they're hard to handle so you have to do do do it in a particular combination and this operator s is a reflection of the structure of that of that awaited Janet operator LDL and let's leave it at that so I'm not going to the technical details of the yeah that's a good question so and so it's just something that sits nearby but it's not it is not 1 of the the yeah it thinks it they following all the all give you control the the time derivative of that is the case so that gives you convergence to some there and its nearby section of innocent of the result they heard the recently in a talk by by default for crucial and so it is expanding space this year and would that let me spend the remaining 3 minutes discussing perhaps questions that are there I find very interesting the way they're all so here's kind of silly lemon mechanical Lemont perhaps to emissions equivalent but do there exists collapsing or expanding Compaq has supported stars so this is the key but for other values again they should exist in the supercritical range which is between 6 cent for the it's this is an open question which you can show however is that self-similarity and a spherical symmetry cannot coexist for this type of collapsed gonna being 4 thirds is absolutely critical for that to to be able to to be able to combine the 2 again this is a very serious thing to show it it's just some of it's just some of them conservation law political culture which so the question is can you construct existing metric so similar collapsed in the in this mass supercritical range I think this this and this is a very interesting problem and now they come to the question that I believe surgery asked the land and stars that that I mentioned before in the subcritical range where it shown to be condition was stable are effects so a possible mechanism and you will see this discussed something in the physics literature it is some that shot formation this is this is not known and this is really really important so let me remind you of this picture the bigger than the pink elephant in the room is of course this region here the collapse in the region so as I said it is
it is not difficult to show that it's not only nearly unstable at the moment you have correct will pose this theory and you have a growing mode it's it's not me it's very easy but to forget this should be a lemon but it's really unsatisfactory because it doesn't tell you anything about the the problem real instability the particular it could it be that of the nearby guys of decay of collapse at perhaps a corrected rate the collapse at all so this is just as vital and you don't have to go outside spherical symmetry asked this question and finally I would like to I would like to sort of point of this viewpoint that I believe studying this this family problems is really it is really the right thing because you really want to understand how the understanding of Holly creations of state affect the collapse this is partly motivated by General relativity if you think of the most famous example collapsing star in general relativity it's the Oppenheimer spiders solutions which by rich which corresponds to the case you don't have any the pressure so by the result of crystal doll from the 3 believes you can show that there's a desire generically unstable solutions that generated in the Hugo outside the home a Jamaican class you generically go perform naked singularity so you want to see what will you can you can devise various conclusions out of the 1 conclusion that I would like to draw out of that is that studying a nontrivial accretions of state is important problem and this is just a Newtonian version of the problem but realistically commissions estate there and this is just the final comment in his principle quite a hard problem even for refuses so this is
my favorite of relativistic astrophysics physics book of Byesville recalled we're able to learn this stuff a lot and they discussed 1 of the most high-density regions for for the for the the creation of state and here's what Eldridge says to Jemison leather trim costs and apparently so it is in isolated so if you look at the physically attitude there's a there's a huge disparity between what people think is the right thing Christians that anyway are and just so this is a teaser single of the the trailer for this wonderful textbooks because because it's a very serious book but as I browsed through it every every now and then I find the gem of this of this but I think it the results of the so it seems that things but and all of a sudden I say that if we do this so called can think what you obviously you if you generically perturbed a cell similar expanding great if you generically perturbed that perturbations that have positive energy then they will the 6 generically unstable and these guys are stable really only coordination Oncins it's going there is the state Illinois takes over another self-centered yeah yeah yeah exactly the case reason I'm not I I'm not going well and if you leave that the exact securing the gold and the labor which was stuck at 1 of the starting point for this project they are for some reason they really say they will not be interested in this in this plenary big drop a constant of integration in the 2nd order the which allows them To go down 1 level and get an explicit solutions which is this In this individually set out the catalog so but this is this is a finding part of the Mass was also to understand the exact structure of this of this parameters this if he considers it resulted in the loss of course hold the government all actually it actually goes up to here I did not tell me what happens it's 650 at 650 together just to get a steady state but it is supported all the way up to it's infinitely it has intervened support and it turns out to be so in this case the steady is exactly the steady state of the critical acclaim wave equations it's 1 of the items the some constant here but then I believe 1 something like that so this is that this is the rate of decay of of respect for the whole country will In this work well it's it's quite different because here here you are working on a complex and supported the main so you know you really have to rely on the vocals in this theory for vacuum interface problems here it's different here the waste finish so it has the flavor of it but it's quite different I'm not sure how hot so this result is also due due to change so this equality case all of this is due to jet in search in this strange have to be this is the is what would this result shows you know in a way that the 4 thirds case which is billionaire was stable is in fact a widely unstable In the movements of the loans to the review it's a it's this is entirely radius of the papers for shoppers is entirely radio but on that it's it's told also from perturbations it as isn't so I sort of alluded earlier today that there is there's a reason why the real case so simpler not just for obvious reasons but also because it reduces the whole problem to just a quality when they're really question when you have when you when you work there in Europe honoring the case to the is not irritation of animal engine so you have to account for the for the richest which does not service has have decided they like structure but it satisfies a certain transport equation that allows you to 2 comptroller so what they said recently that it is a question of what you want is a willing to local time resulted in the field of the development of the universal 1 saying I'm not sure if understand your question but always saying I want a lot of the long long time for although there lot of of haven't produced in Vietnam said the additional case is proving to be straightforward and the participating in the in the term result of the make claims I don't think so should than in the regular tennis 1 of the things that you have this damping effect so it will be better it will be the 1st of all it's not what you have for instance for of for Friedman Willamette Robertson Walker space-times you have to have a stability .period irritations which so we have to have a cosmological constant for instance rate from the procedures use you can control the who who will of course this is not a free boundary problem because this rebounded from G hours more of you kill them the the