 So, before I start, I do want to thank you, well, to Frank, for organizing the program to Yvonne and to Fabrice, for putting together the summer school, to all other organizers, to putting all this trimester program. It's been really great, really enjoying my time here. And also, I guess I want to say thank you to the staff, because it was a support and during, I guess, Robert's train and saying that I will be set in living in a few days. This is a great place. So, I'll start my talk, and I will talk about focusing and less equation and going beyond the threshold, but I'll explain what it means. So, I'll be talking about the focusing, unlike Benoit, who's talking the defocusing, but I will be considering the entire RN, nonlinearities, conserved quantities, scaling, scaling invariant. It's pretty familiar for this audience, so I'll just flash it quickly. And I'll consider the case when the scaling invariant index S will be positive, typically less or equal to one. Okay, and the ground state, which is also probably familiar to most people in this audience. So, for example, from Steven Gustafsson's talk, although, again, I'm considering all of they are. So, there's a number 1 minus S just to show that we are in the regime between 0 and 1, but it's not really important. When S is less than 1, then Q is in L2, and in particular, in one dimension, it's written as the SESH. To certain power, if S equals to 1, then it's explicit, and we know this from Carlos Koenig's lectures, and it's in L2 when N is greater or equal to 5. Okay, so just I'm pulling this. Since I said going beyond the threshold, let me define the threshold. And before that, let's look at this quantity. So, there's a mass, which is L2 norm, scaled to the 1 minus S in the energy to the power S. It's a scaling invariant, and it's interesting to look at the relative size to the same of the ground state. And, for example, in the mass critical, when S equals to 0, so it's S equals to 0, right? So, there's only mass here, so when the mass less than mass of the ground state, we know the global existence due to Weinstein. So, when M of U equals to M of Q, then there is minimal mass blow-up solutions, and it was characterized by Frank Merle. If we look at the energy critical, S equals 1, then when energy is less than E of the... So, here's the S equals to 1 of energy of the ground state. We know the dichotomy, which goes to the work of Koenig Merle in 06. If it's equal, then there is also characterization, WW plus minus by Tomein Frank. And there's similar situation, and I'll explain a little bit in more details when S is between 0 and 1. So, the threshold, which I will be calling, is exactly when the initial mass, scaled invariantly with the energy, is equal to that of Q. So, in the notation, I will normalize so I could compare to 1. So, if S equals to 1, then it's this caligraphic E. And in general, it's this threshold divided by the Q. I shifted the S, so I'm considering positive S right now, just so when S equals to 1, for example, then we exactly have just the energy quotient. Okay, so by now, behavior is well understood when this mass energy quantity is below or equal at the threshold. Okay, so I'll give you an example of 3D cubic NLS where... So, if we look at the normalized gradient, so this is the part which is the change in this time, and then the mass energy. And if we look at the energy and bounded by the Gagler and Nirenberg from the left side, so the potential and drop the same term on the right, so then there's corresponding bounds. And it's better to see it visually as one of my favorite graphs where you can really see what's happening. So, too fast. Okay, so first of all, so here's the 1, right? So, that's the line which is the threshold. Okay, and if we... and this is the gradient squared. So, if our solution leaves on the... and it's constant, right? This quantity Y is constant. So, it can leave only on this horizontal line, so it's obviously bounded and then there's scattering. Or it can be on the right-hand side and then obviously the gradient can go to infinity and blow up, right? So, I'm coming from Washington DC and there's like always this political presence and right now you can think of... so this is like the boundary of the... what is it? First Tuesday in November, right? And there's like these two candidates and who knows which one will blow up. Anyway, so let's go to dichotomy below the threshold. So, this is the result which is in the energy critical case. And the point I want to make here is that usually it's written with the gradient, but it actually doesn't matter, you can state it was the L... with the potential part, Lp plus 1. And the other way is... so it's well known, so that's why I'm going a little bit faster in the slides. So, there's a dichotomy, right? There is a similar result for S between 0 and 1. And I guess originally we did it with Justin for 3D cubic and LS for the radial setting and together with Stomar for the non-radial. Then my student, Christy Goivara, she extended to all the range using this best of Stryhurt's estimates. But there's a... using just the specific LPLQ norm by Kazanavi, Fang C, and there's a similar work by Akahori Nava. Okay, so again, we are below. And if we... we can state it was the gradient or it was the Lp plus 1, the potential term. So a dichotomy, if it's less or greater, then we get either global existence and scattering or they blow up in the positive and similarly negative time. There's this... we can also do this sequence if we don't want to restrict to some radial. Okay, so now we come to the threshold itself. So this picture shows just the... used from the work with Stomar where we classified, but we got the idea from... they got some moral work on the energy critical and LS. Where there's also... so there's a solution QQ plus Q minus or WW plus WW minus. I think we've heard from Carlos Koenig and Jochen Krieger talk about those solutions. So there's a classification. Okay, so that's the question. I want to go beyond or above the threshold. What can we say about the solutions? Can we get some kind of dichotomy? Can we classify? Can we get dynamics? Okay, so let's start with some known results. NL stick was 3D cubic NLS because basically that's all what's known. So there's a work by Nakanishi and Schlag in 2012 described the global dynamics of H1 solutions slightly above in this regime where they classify and give different manifolds depending on whether it scatters to zero, to Q, blows up, etc. And the Bicciano constructs a co-dimension, one manifold invariant by the flow of H1 half. Solution is also close to Q in a similar regime. And then we got some criteria for a specific 3D cubic with Justin and Rodrigo Platt and also then Jelai Svartama. And that was the question where physicists would ask, for example, if you take the Gaussian data, then can you find exactly the amplitude for which you can guarantee that it will collapse or it will blow up? So I'll show you a picture here just to what is known. So we're looking at the Gaussian and for the purpose you can just think about alpha one. And this is the picture for 3D cubic NLS where S is one half but this can be generalized to other ones as well. So the blue line, this is the negative energy, the well-known glassy, Zaharov, Petrishivtolanov, lots of negative energy criteria. So if we take the amplitude more than 2.91, then it will guarantee that this initial data will blow up. So then the purple lines, this is where this threshold occurs. And it says that so below the amplitude it's guaranteed global existence and scattering or above it will blow up. And then we have then two criteria here, green or yellow line. So depending on what the initial variance is, it will tell also that there will be a blow up. So you can see that theoretically we have a gap, we can't tell everything but then the red line that's numerical simulation gives the number and that number actually showed up in the work of Kuznetsov and his collaborators originally when they were trying to also look at the criteria for the blow up and use it on the Gaussian. But we can also do it even for the case where it's energy supercritical because we don't know really anything on the global existence but the blow up we can guarantee. Okay, so variance turns out to be an important tool and we've seen in like lectures of Carlos Keneguer, varial identities show up. So this is just the basic one for the NLS. So we're going to try to use to see if we can get something like a dichotomy above the threshold. So the first theorem which I'm going to talk about is that we look at the finite variance and we look at the mass energy threshold being above one. And we get the following restriction. So we multiply so we could control so that's the first derivative on the variance squared. There's the energy and the variance itself. So we assume that and then now you can start recognizing sort of dichotomy statements. So if the mass times the potential term is less, right, so then that's the area where we expect that there should be global existence and it is but we need to require that vt prime is positive. Then we can show that this will be holding for all t in some maximal time of existence and then we can prove this scattering correspondingly in h1 or h.1 depending on which we have and those are the typical restrictions coming from the previous theory. And the two if the gradient is more than that of the q normalized with the mass then we expect and also we need to have the negative derivative of the variance. Then there will be a blow-up. So there's some sort of dichotomy we can get above and let me talk about the proof comments on the proof and then I'll do some remarks. So again if we look at 3d cubic and the less just for the numerology although we can do it for any intercritical. Then here's the improved Cauchy-Schwarz. So if you forget about this f4 term, what do we have here? Is this the standard Cauchy-Schwarz? Nothing else. So if it's in h1 then we can actually improve it and this is first I guess due to Valeria Banica but it's pretty simple to see if you take a function and multiply by the quadratic phase and I'll be mentioning quadratic phase again later on. But if you look at this, I mean this is the variance that's the first derivative so it's exactly this, nothing. And now if we use the expression for the gradient or for the force power in terms of the energy or the variance second derivative then we can rewrite it exactly like this. So this extra term which comes from this improved is essential. Let me redefine v and write it in this way. So take a square root. So then what we have is the following. z prime squared is some function phi which is written like that and again I guess it's maybe after lunch so to keep you a little bit more awake let me show the picture because then it's easier to understand. So this function phi behaves like this and the important part is that it's only defined up to a certain value but there's a minimum and it's always greater than that minimum and what's important is this phi of x0 is positive. So then what we get is that this z prime once it starts at 0 for example we can always say that it will be greater and then we can use the continuity that it keeps away from the 0 so it can be if it starts positive then it'll stay positive if it starts negative it will stay negative and it exactly will split the dichotomy so it's kind of a trapping argument. So if it's positive then of course it will be greater than this for all time of the existence that will guarantee that the second derivative of the variance is positive and through this inequality it shows that the either gradient or l4 norm is bounded and so then we get global existence and the other part is similar the other way so if we start on less than 0 we'll stay like this for all the times but then it means that the second variance is negative and the standard argument gives a finite time so only in part one we need to show now the scattering and that's the where we use the well dimensioned I guess 3 so we use the techniques like in dimension 3, 4 of Keenick-Merle and then for high dimension of Kilobyv-Vizhan here so I'll switch the numerology and I'll talk about the 3D quintic and the less but either s equals 1 or s less than 1 it will be handled it's just easier to see than I don't have to write the mass for example okay so we assume if so there's a little statement here that if the limb soup has T goes to T plus with the u6 less than that of the ground state in dimension 3 and 4 we have to assume that solution is radial then the time is infinite and use scatters so now for the scattering it's sufficient to show that it's the corresponding Strychard's norm in this case each one admissible LPLQ so is bounded and for that the proposition that we prove so it is the following the proposition 2 if A is less than the 6L6 norm of W then the s of so this is defined as the supremum of all the Strychard's norms on the interval I such that the energy is bounded by this E0 and the L6 norm is bounded by A so in principle this is if we were to look at the gradient then this is known that's what the but we have to work just a little more to get it for the 6 exactly because the sobole it's only one side embedded so it doesn't follow so we need to generalize and there's like a little bit more in the characterization in the beginning for the word but anyway so it follows the variational characterization in the beginning but otherwise we'll follow the strategy known for the gradient okay so a few remarks if we look at this set which is sigma of functions in H1 with finite variance and that's where we have this condition which allows us to go above the threshold then if we define the set sigma sub B so this is where we have for the LP plus one norm positive initially or is not positive but greater than the corresponding of the Q or less than this two sets so one is sigma sub B or blow-up set than the scattering set they are stable by the forward flow of NLS so the same variance sets moreover this solutions can have arbitrary large mass and energy so that's why it will be above the threshold and that can be seen by let's take initial data v0 for example which is the at threshold and we'll give the quadratic phase to that so if we just write out what it is here's the expression then we can see that if we choose gamma large enough then the energy can be very large so we can definitely get beyond the threshold and in fact this quadratic phase originally motivated us to look at this problem because again physicists were asking the following questions if you look at the Q and you give a quadratic phase can you actually say what will happen with the Q it doesn't necessarily stay under the threshold and you couldn't tell so originally when we looked there's a word by Kaznavi which says that if gamma is very large then you have scattering but that's all you can get but as a consequence or corollary of this result we have that so if you start with this initial data and the gamma is negative then in positive time it will blow up in negative time it will exist globally and it will scatter so here's the visualization and vice versa if the gamma is negative then we get the picture reversed and this is true not only for the Q but you can take any v0 which is at under the threshold for example the quadratic phase and it will characterize exactly the behavior of the solution and the same for w so I didn't include it here just because with the w you have to be careful it's for high dimensions because if you ask for the finite variance then it's from dimension 7 but ok now what can you say about so this was just the characterization and the theorem that you can have the global existence versus the finite and the global existence you can show that there is a scattering to 0 that's the question and what can you say can you tell something about the blow up or dynamics of the blow up solutions ok so I'll now retract back to s equals 0 setting or to the mass critical setting and give a little review here and we'll go from there so ok as mentioned before the minimal mass solutions are known in the L2 critical setting when m of u is equal to mass of Q and all of them were characterized by Frank that they are pseudo-conformal transformation of the ground state so then if we go just slightly above of this mass of Q then originally they were numerical and heuristical work of this log-log-blow-up solution Landau-Lenman, Pimpiniklau, Solemn, Solemn 88 there's a work by Freiman in 85 which observed it and then first analytical construction was done for the 1D quintic NLS that's L2 critical by Galena Perlman and then the fantastic body of work the systematic study was done by Frank Merlin Pierre Raphael and so these are the main features in this L2 critical of course so I should probably mention that there's a Bourguin 1 solutions and then if you start going into the inhomogeneous there's Pierre and Jeremy and also by Nika Karls and if you start putting several log-log solutions then there's a work by a change of fund and not interacting, if they're interacting then the recent work by Yvonne and Pierre if you do go, so there are all kinds of perturbations but on this work if you go anywhere and that's all which is known in this regime when S equals to zero so now I want to go into the mass supercritical and energy subcritical because if you go, if we are in energy critical then there's a type 2 and in the energy supercritical there's also some, well there's recent work by Frank Pierre and Igor on the type 2 so let me just stay in this regime and see what we can say there so basically analytical description is known only in a few specific cases and mainly it's an adaptation of the stable log-log when you take a symmetry of the solution and you kind of lift it up to an equation which has the symmetry so for example you know that the L2 critical is 1D quintic and so if you can take the 2D quintic and show that this extra term 2R over dr can be dominated so if you can show the stable dynamics away from the zero then that's another generation of the blow-up solution so in the first such solution was done by Pierre for the 2D quintic and the less and then the generalization for all the quintica was Raphael Zeftel and the only thing that they required to have initial data of higher regularity and then the equation itself and then later with Justin we actually for example in the 3D quintic they required H3 data and with Justin we could improve it and require just the H1 and then for example if you take the 2D cubic and the less and for example use the axial symmetry and lift it to the 3D cubic then we have a construction with the axial symmetry so it's a blow-up on the ring and there's a similar construction but using the approach of Raphael Zeftel by Ian's wires on it but that's basically all there is to this regime let me show you the picture which is by Gadi Fibic so if we look at the so this is the 2D quintic and the less and if we think about this radial so let's say a radius equal to maybe 3 or something so it's if we just look at the radial coordinate then the blow-up happens exactly at the radius equals to 3 and that's a stationary standing ring and with time it evolves exactly like that so excuse me in physics there is this notion of the strong and the weak blow-up solution so strong means that you accumulate the L2 mass at the core or at the center of the blow-up and weak you don't however there is still the concentration of some norm for example the scaling invariant norm and if we look at the for example coming back to the 3D cubic NLS so the scaling invariant norm will be the L3 norm so if we look at how the L3 norm accumulates or concentrates then it turns out to there are two regimes so one regime which would be typical and it would be the strong blow-up and there's another regime which is a little bit wider and it will be corresponding to the weak regime because it won't be accumulating the L2 norm so we were surprised to see this and then we try to think of the scenarios which can actually provide this kind of solutions and the preliminary analysis which was just in Homer which when it was based on conservation laws was the following idea so the profile would have to be still similar like this and to keep the second regime we would have to have that R0 the center they have to be traveling and it would be T-T to the one-third and also the lambda which is the rate of the blow-up or the contraction towards the sphere would be twice as fast so two-thirds and the profile would be the although we are in 3D it would be this one-dimensional so we asked Catherine Soulem and she confirmed numerically at that time and then there is also work by Fibich in 2008 who does the numerical confirmations and the specifications of this possible so that was when I was at Arizona State and the University of Arizona is just 100 miles so that's 160 kilometers south and Zaharov is there so at one of the conferences I was excited and I talked to him about that and of course Zaharov he is like I know all this long time ago he knows and he predicted well he did, I looked up so there is work Zaharov Rudakov and he does the resumention and even at that time there is some numerical simulation so but finally we were able to do the rigorous construction with a journal with Galina Perilmon and there is also a similar result obtained by Frank Raphael and I will talk about this now so we call it contracting sphere blow-up and let me talk through the formulation so if we fix a little E okay and that will be I should step oh okay right okay I should keep my hand down okay so there is a radial solution with a prescribed energy so energy can be any real number positive, negative, large, small so we start at the time t zero and we will backward in time so we will backward to the time t equals zero so then the solution will behave as follows so there is a the blow-up core which is described by Q so it's in the radial coordinates Q and lambda there are two more parameters there is the space and the speed v of t and the leftover part, the remainder okay so the Q is the sesh the Q of t so that's the sphere is exactly t to the one third times the constant which is that number lambda t was which the blow-up happens towards the sphere is t to the two thirds exactly with the Q zero squared okay and then also there is a description for the theta of t and the v of t happening like that and then of course we need to control the h and give the control for so in L2 we have bound on the h in h1 and then the variance localization so note that this says that for example the h1 in h is bounded by t to the minus one one over t okay so also note that so gradient that's like one over lambda so it's the contraction towards the sphere is t to the minus two thirds and the localization is t to the one third okay so that's the statement of the theorem okay so just to visualize how this happens this is I had a colleague back in Arizona state, Rodrigo Plattich who did this visualizations and so the intensity of the color is proportional to the rate to the gradient of the L2 norm so if it gets really red then it becomes large and this is the first time and then as time goes on then you can see that the so the sphere moves towards the origin and then it basically so contracting is happening towards the sphere but the sphere itself is moving towards the origin that's how the okay so I'll make some comments before I go to the remarks on the proof so this NLS is the 3D cubic right so it's a third dimensional but we do use the one dimensional profile sesh ground state now via scaling as we can scale this blob dynamics can have an arbitrary mass so that's why it's an above threshold or even when we can think about M of q plus epsilon in this log log it's not the log log gem or in fact the third comment is that the rates of blow up in contractions are very different from previously known so numerics show that this dynamics is stable under radial perturbations however if they're not radial then I don't think they're stable that's I think the work by Fibich there right it splits so physically I mean I probably can't see this dynamics however still it's an interesting and so since there's a similar work by Frank Pierre and Jeremy let me make a comparison so I'll show that we start with four parameters as I stated and then we derive equations imposing two conditions and we get so the two parameter family now since we know there's this t to the one-third we expand those parameters in the t to the one-third powers and as I can understand in MRS use the ansatz with five parameters so there's an extra parameter b r squared and three a priori conditions used to impose these five parameters to reduce to the two-parameter family to obtain the profile equation which has a priori two unknown polynomials which depend but these are independent of time if I understand that correctly and then eventually we prove so we get localized variance and h2 bounds which gives us convergence in h1 and it also allows implies the control on energy so we can construct the blow up with any prescribed energy and so in MRS there's a localized bounds and there's a convergence in L2 we give precise values we only do the 3D cubic and the less but I guess this result it's the general setting so let me give some ingredients of the proof okay so we start with the following ansatz so there's a phase there's a speed and there's lambda so now lambda is multiplied so there's always this option you can either divide them also here it's a r minus q t so although it's written like this it doesn't look like it but all these four parameters they are time dependent okay so we write rescale coordinate so we write the equation for the u right so when we substitute into the equation we get the following it's very long but the point is let's see where the red shows up here in two places so we want to get rid of this term and so we impose the following condition so that makes a zero and then the second condition is we want this to be scaling as lambda squared so that's our second condition which we impose and then the equation looks like this so now if we sort of look at this equation this capital u looks exactly like a 1D so that's where we can think that it's a sesh plus a left over okay so then the question is like what can how do we get for the dynamics for the chi okay so for the chi so now instead of writing in the complex I'll write it as vector notation so here's the equation that we get so that's I so we switch from lambda to the w and there are terms organized in the following way so there's a Hamiltonian which is corresponding to so because it's a one-dimensional so the linearization we're doing around the sesh which is corresponding to the 1D cubic analysis so that's the Hamiltonian corresponding to that and then there are linear terms because so we substitute plus sesh right so this term which will be containing the linear terms with the sesh and this is the term which will be containing the linear terms with the chi and then chi squared and chi cubed we're organizing this higher order terms okay so okay and then so what do we have here we have w and q right and the chi itself so we write w, q and chi in terms of the power series okay so power series t to the it turns out that this will be just the odd even then this is the odd and here's the chi okay so then the claim is that we can substitute and solve this with the following parameter so we can fix we can choose w to be 1 and then the sequence for the w's q sub k will leave q to arbitrary this will help us manage the energy later on it will be adjusted to what the original is and then chi we can manage to construct as exponential decay provided that we can there's this solvability or the orthogonality conditions such that this right hand side we can correspondingly control on some time interval of course this t0 depends on n and there's a I need to put the restriction on r so that's okay so then we do the approximate solution so we construct an approximate solution okay so it's exactly like what I wrote the ansatz before and u is the sesh plus our chi is the sum in the powers of tk of chi sub k there's a r slightly cheating because it has to be a truncated version but that's not worrying about this right now and this u sub n satisfies this equation there is a right hand side but the right hand side eventually we can manage to have the control in terms of the corresponding the powers t again it's on this interval okay so there's also the energy estimates and that's where the q2 comes in okay and then we go to the exact solution so if u sub n is the approximate solution on this interval then for any epsilon between 0 and this time t0 of n we set the u to be solution such that okay so the initial data t equals epsilon is un of epsilon right and then we're well with this evolution so then for h of t which is the difference between this exact and the approximate we'll set up so now we need to make all the estimates and we set up the lopna functional so it has so I'm trying to be not very technical this moment but so w there's a combination of the mass there's an angular momentum and the energy and this is the difference between of u and un in the first derivative of this so then okay so as a lopna functional we need to get the bound on the the first derivative of this we need to check the cursitivity but that actually comes not the cursitivity comes not difficult because we know linearized operator in this case so there are extra terms we call them capos which we can control but that's and then we have to get the so this is all on this interval t between epsilon t0 and and the comparison estimates and localized bound so we this is the where we actually need to get the h2 bound so eventually in the compactness argument but all of this we need to do so that constants would be independent of epsilon right so then finally we can fix the t0 and for the sequence of epsilon j is going to 0 right because this will be the epsilon going to 0 we set u sub j to be an exact solution as I described over here with epsilon equals epsilon and u sub j of t0 converging to u0 so now we just need to run the compactness argument up to the subsequence we can do that then run the compactness arguments to complete the proof okay perfect thank you thank you for your attention comments questions just a technical question so capital N you have you choose it big with respect to universal constant that appears right yeah so you need the approximation to write in your results of the threshold in the first case if you had infinite variance and non-radial things you could have glow up in infinite time if I have infinite variance infinite variance and non-radiality there was possibly infinite glow up wait but okay okay so maybe get quickly to that but in all of those I assume the finite variance ah where I guess where I is it over here over here no yeah so that's why you get finite positive time either you have finite variance or radiality or if you have the infinite variance then we can show it for a sequence exactly and in your new results this also happens we don't okay so that's a separate though yeah we don't I guess you can probably redo the analysis but we don't do it yeah I mean yeah so right I probably can redo this yeah yeah yeah speaker