 Okay, so the title of the talks is the energy critical wave equation and today's lecture will be organized in two parts. The first part will be an overview of the whole sequence of lectures and kind of putting some perspective into that and then the second part of today's lecture will be the beginning of the actual work where I will set up some of the technical tools that will be needed for the rest of the lectures. Okay, so this talks on the energy critical wave equation are set in the framework of trying to understand global properties of the dynamics of large solutions to nonlinear dispersive equations. So basically the background to this is an attempt to describe the long term behavior of large solutions to dispersive equations and I will start out sort of by the end goal and this is coming now. So the end goal of the studies are what we call this become known as the soliton resolution conjecture. So that's not really a conjecture but more of a philosophy. I think it was first noted in the mid-60s explicitly in the work of Zabowski and Kruskal on the correct degrees equation and it was come to through one of the first attempts at doing computational experiments and this was one of the big successes and closely connected to this was another big success of this story which is the paper of Ferri Pasta Ulam which was in the same direction a bit earlier. So what is this soliton resolution conjecture and it's the belief that for nonlinear dispersive or hyperbolic equations the long term asymptotics of solutions are described by what are known as coherent structures and this is the belief that's been come to be known as the soliton resolution conjecture. So I'll be more specific about this and I think that to systematically establish this is one of the big challenges in MPDE and what does this conjecture say? Loosely speaking it says that in order to understand the long time asymptotics of most hyperbolic and dispersive equations what you need to know is that asymptotically in time so there will be some intermediate regime of times where you cannot say anything and then eventually the asymptotics results into a sum of modulated solitons so these are traveling waves that are scaled and translated plus a free radiation term and that's nothing but the solution of the associated linear problem plus something that goes to zero and so somehow it gives a complete description of the long term asymptotics so the solutions may behave very strangely and in ways which cannot be described for intermediate times but eventually there's a simplification and that's what this is. So this as I was saying this is a conjecture that postulates a simplification and you have a very complex dynamics but eventually it simplifies to this superposition of nonlinear objects and linear objects. So until very recently the only cases in which such asymptotics could be proved was for integrable equations. So integrable equations are nonlinear dispersive equations that can be reduced to a collection of linear equations okay and examples of this are the corrective risk equation, the modified corrective risk equation, the cubic nonlinear Schrodinger equation in one space dimension and there are not that many more examples okay and the other case when such a result had been proved that was also in perturbative regimes where say you start from small solutions or from solutions that are already close to a solitary wave and then you describe what happens. Now in 2012 and 13 with Decaher and Merrill we were able to establish this asymptotic behavior in the specific case of radial solutions of the energy critical wave equation in three space dimensions and we did this in two stages. The first one we proved is the composition for a well-chosen sequence of times and in the second stage we proved it for all sequences of times and I'll have occasion to discuss the differences as we go. So let's now get more specific and describe the nonlinear energy critical wave equation and here it is we have the usual linear wave equation and we subtract this nonlinear term with the specific nonlinearity and we ask that the initial data be in the space H.1 of functions with a gradient in L2 and the time derivative is in L2 okay and the dimension is 3, 4, 5, 6, I is a time integral and the origin belongs to this time integral. So one thing I want to point out right the way is that this equation the rate the way we have here is a focusing equation we have the minus the Laplacian which is a positive operator here and we have minus the nonlinearity. So there's a competition between the linear part and the nonlinear part of the equation and that results in a focusing effect. Now because we are in the energy critical setting the strength of the Laplacian and of the nonlinearity is the same and so there's a real competition between the linear part and the nonlinear part and we will see this more explicitly with formulas in a few minutes. So the first thing is the small data theory and in these problems small data yield global solutions which scatter so what the scattering means we will say that a nonlinear solution scatters if it exists for all large time and as time tends to let's say plus infinity the solution behaves like a solution of the corresponding linear equation and we will have a formal definition for that and I think a so the small data theory goes back to to pioneering works of Cato, Ginevravello, Pescher, Kapitansky and several others. Okay now there's also an auxiliary space in which we put the solution it's that norm this plays the role of a solve of embedding in this nonlinear wave equation theory and I will be more specific of that in the second part of this lecture and it may help you I mean certainly helps me to keep track of these numbers by focusing on the case of 3D. In three dimensions this is L8 okay and in the first lecture I will treat all dimensions and then I will be specific to the three-dimensional case so as to avoid this fractions. So why do we call this equation energy critical and it's because it's critical with respect to the scaling in the energy space so if we have a solution u and we scale it by u of x over lambda t over lambda and then we multiply by the correct power of lambda that depends on the nonlinearity in this case lambda to the minus n minus 2 over 2 then that's also a solution and this depends on the specific form of the nonlinearity and the norm of the initial data is independent of this number lambda so we cannot make it either small or large by moving the scaling parameter and that's what we call this energy critical okay cool the equation is focusing and it has a there are two important conservation laws for this equation in the energy space the first one is the energy and the the second one is the momentum and we'll describe we'll discuss that later but the energy is given by this expression and you see that there's a competition between the the so-called linear part of the energy and the nonlinear part this one comes with a plus this one comes with a minus okay so before going on I should talk a little bit about the de-focusing equation so in the de-focusing critical wave equation what we do is we change the sign in the nonlinearity instead of minus u to the fifth you put plus u to the fifth the power 5 is the three-dimensional one okay in that case then there's no longer a competition between the linear part of the operator and the nonlinear part they in fact cooperate and the resulting energy has the plus sign okay so that that de-focusing equation was studied extensively through the 80s 90s and early 2000s and there's been a tremendous amount of work in that very important works some names I can mention in this connection are Struve, Grillaques, Chattin Struve, Capitanski, Ginevere Vellon Sofer and then Bahourien Girard and Bahourien Chattin and if you look at the combined results of these people which took a long time to come through and there's a big body of work the end result can be summarized very simply it says that any large data gives rise to a solution that exists globally in time and which scatters so the asymptotics is only the asymptotics of the linear equation so in particular there's no solitary waves no traveling waves and everything just scatters to a linear solution okay so that's the summary of course it took a long time to get to this summary but today we can just say this is the summary okay so what we're gonna do now in these lectures is understand that the focusing case in which the dynamics is considerably more complicated okay so let's go back to the focusing case the first thing that you notice is that if you forget about the spatial derivatives and just consider the time derivatives and consider functions that just depend on t we have an easily constructed solution here this expression is a solution in the three-dimensional case and at t equals 1 and something catastrophic happens right it becomes identically plus infinity so there's definitely a finite breakdown in finite time breakdown now you could say well but this solution clearly is not in the energy space right certainly not in the energy space but the thing is that for wave equations we have finite speed of propagation and so if you truncate that solution at time zero let's say in a box of radius two at time one in the box of radius one it's exactly equal to this to the solution and so therefore you have finite energy solutions whose norm tends to infinity in finite time okay and this is what one calls a type one blow-up and by the way to anticipate some of the things that we will be thinking about and studying it is still unknown whether type one blow-up at infinity can occur that's an open problem we will see that at least in the radial case that cannot happen now the interesting thing about the energy critical case is that there's not only this type one solutions which are arise by ODE but there's also what we call type two solutions those are solutions that break down in finite time but then whose norm remains bounded and we call those type two blow-up solutions so it's not the very imaginative terminology but also notice that they're not mutually exclusive type one and type two type one means that the limit is infinite type two means that the supremum is bounded but it could happen theoretically that for one sequence of times you go to infinity and for another sequence of times you remain bounded that would be mixed asymptotics and again in the general up to infinity well this kind of thing is not understood in the general case but in the radial case there are no mixed asymptotics in the finite type blow-up and this is something we will also prove okay now the first examples of type two solutions were constructed by Krieger Schlag and Tataru in the three-dimensional case then in the four-dimensional case by Illeria and Raphael and recently by Gendrige in the five-dimensional case I think also in the six-dimensional case okay let's go on now should we expect solid tone resolution for this equation okay so once you think about this in just a few seconds of thought tells you that it can only happen for solutions that remain bounded in the H1 cross L2 norm because if you have this decomposition for linear solutions you clearly have boundedness and for the solid tone part you also have boundedness because is there rescaled translates of a single function and so if the solid and resolution is going to hold you will have to have boundedness so to understand solid and resolution for this equation we restrict ourselves to the solutions that are bounded in the energy norm and so what are some examples of solutions which remain bounded in the energy norm and that exists for all time let's start looking at that I mean there's a certain zoology of what are the possible objects that arise here and you like to understand so the first thing is scattering solutions so what are scattering solutions formally there are solutions for which the final time of existence is infinite and such that there is some data in H1 cross L2 such that the difference of our nonlinear solution and the linear solution corresponding to this data goes to zero in H1 cross L2 okay that's what we call scattering solutions and I'll be more specific about this in the second hour today and clearly for scattering solutions the H1 cross L2 norm remains bounded because it is so for the linear equation okay so for any small initial data as I said from the local well posted mystery we have bounded H1 cross L2 norm so let's now go to other examples of bounded solutions in the H1 cross L2 norm that exists for all time and so like before we forgot about the time about the space derivatives let's forget about the time derivatives now if you forget about the time derivatives what you're left is this nonlinear elliptic equation for n equals 3 is Laplacian of q plus q to the fifth equal to 0 okay of course we consider nonzero solutions otherwise it's not so the zero solution is not so interesting now this nonlinear elliptic equation is an equation with a very long history it arose in the solution of the Yamaver problem in differential geometry so this is the work of Ovan, Talenti, Shen and many others so the Yamaver problem is a problem of whether you have a compact revanian manifold in dimension 3 can you make a conformal change of the metric so that the resulting metric is a has constant scalar curvature and it's in the solution of this problem that the study of this equation was very important okay now we will call this the class of nonzero solutions to this elliptic equation sigma it's just a name so what are examples of objects in sigma the first example well is this guy so this specific function is a solution to the nonlinear elliptic equation so we certainly see some features of what these solutions might look like from this now stationary solutions obviously do not scatter why because if you have a linear solution and you look at the energy restricted to a fixed set let's say to the unit cube and you let that tend to the time tend to infinity that will always go to zero that's not difficult to show and therefore for static solutions that's constant so it will not be going to zero so they don't scatter so that's a quick way of seeing that static solutions don't scatter now so we see already this example of a global solution that doesn't scatter so this this stationary solution has some interesting properties the first is that up to sine and scaling this is the only nonzero radial solution to this elliptic problem and this is a theorem that goes back to the work of Bohorzai in the mid-60s and then Gieders-Nehenberg in the late 70s okay so for radial solutions we know them there they are the other thing and this is the fundamental work of Gieders-Nehenberg is that up to translation and scaling these are the only non-negative solutions and this is the so-called moving plane method that proves now nevertheless and this is important to us I think at the at the time when when Ding found this it was regarded more of a security but we will see that this is important to us there are variable sine non-radial solutions and they're not just a few there are many there's a whole continuum of them and Ding's construction was functional analytic and you could see almost nothing about what these solutions looked like from his construction then more recently Dalpino, Mousseau, Pacare and Pistoia have constructed other examples that are much more hands-on from which one can read properties now our W I remind you from the previous slide it's W it had all these characterizations but it also has variational characterization if you look at the sobolev embedding let's say L6 is contained in gradient in L2 in 3D with a best constant the W's realize the best constant okay and that is a result of oban and talenti although in the radial case it was already proved by bliss in the 1920s okay now this variational characterization results in the in the fact that this is the elliptic solution non-zero elliptic solution with the least amount of energy and because of this it's called the ground state so in in 2008 with Merle we established what we call the ground state conjecture for the nonlinear wave is equation and this says that a solution of a nonlinear wave equation whose energy is strictly below the energy of W satisfies the following dichotomy if the gradient is smaller than the gradient of W then it exists forever and scatters in time in both time directions and scatters in both time directions now if the gradient is bigger then it breaks down in finite time in both time directions and finally the case of equality here does not arise there is no such function because of variational considerations okay so the part of the energy space where the energy is smaller than the energy of W one can understand completely the dynamics and then there's a threshold case when the energy equals the energy of W and this was completely described in a work of Decair and Merle that followed shortly after this one so that in our proof of this a ground state conjecture we obtained this as a result of a method that we developed to study the longtime behavior of solutions to critical dispersive equations and this method worked in defocus in cases and in focus in cases below the threshold of the ground state okay and this method has become a standard to to understand the global in time behavior of solutions below the ground state now from the from the beginning when we applied this to to the nonlinear wave equation we realized that the nonlinear wave equation was very suited to this method it fit with it very perfectly and in the back of our minds it was always the idea that we wanted to eventually see if we could find a model for which to prove soliton resolution and the fact that this nonlinear wave equation fitted so well with this method made us decide to study it in this case okay so that's how we came to the nonlinear wave equation I have to say that I have no expertise in wave equations prior to this but it was kind of forced by what was happening coming to this I think also Frank didn't particularly have expertise in this direction we were just exploring and yes right well same with me and so we particularly it's not that we had decided we're now going to study nonlinear wave equations it kind of came to us that this was what we needed to do okay so let me now this explain what other non scattering solutions are you know these solutions of the nonlinear elliptic equation can be viewed as traveling waves that don't travel the speed of traveling is zero now we are going to go to true to true traveling wave solutions okay so what are the traveling wave solutions they're also non scattering and they're obtained by doing a Lorentz transformation of the elliptic solutions so you you observe that the nonlinear wave equation is invariant and the Lorentz transformations and I'll define them in a few minutes and then you view this elliptic solution as a solution of the evolution equation of the nonlinear wave equation and then you do the Lorentz transformation to it and what you end up with is a traveling wave solution and it has to travel at speed strictly less than the speed of light so the directions have to have length strictly less than one okay so in our in our formulation the speed of light is one okay that's how we're normalizing things so so these are the traveling wave solutions and here we explicitly view them as a traveling wave solution in the direction given by L and this is the formula of what you do to them from the elliptic solution to get the traveling wave solution okay and I'll describe I'll discuss this a little bit later today in more detail okay so these are these are examples of traveling wave solutions and we will see that it can be proven that these are the only traveling wave solutions but that is already a result okay so the ultimate goal as I mentioned was to establish solid solid term resolution and so let's explain what what should solid term resolution be for nonlinear wave equation for the energy critical nonlinear wave equation so we have a solution who whose energy norm remains bounded until the final time of existence and you want to show that there's a number j and elliptic solutions qj directions lj which are sub speed of light and that have the property that if you take any sequence of times converging to the final time of existence which may be finite or in or infinite then you can find scaling factors and centers xj and this have to be arranged so that the different solitary waves don't see each other and that's expressed in terms of this orthogonality of parameters conditions so we'll get back to that a little bit later on and a linear solution which is the radiation term such that our solution equals the sum of modulated solitary waves plus the radiation term plus an error that goes to zero in the energy norm as the as the tn's go to t plus so this is a very concrete description of what you would like to to prove and the subject of this lectures is to see how far we get into that okay so as I mentioned before in the radial case and three dimensions this has been proved in the work with duke and Merrill in in dimensions three and five this had been proved for the finite time case you close to w in the paper with also with duke and Merrill and so now I'm just what I'm going to do in the remaining part of this hour is give you a broad sketch of some of these results and the results that came for a later okay which are the large data results in the non-radial case okay and then the rest of these lectures will be trying to prove these results okay and in this part of the series the aim is to reach up to the radial case and then in the next part in and the June-July is to go to the non-radial case so I'm leaving the non-radial case for the end as an incentive for you guys to return okay all right so the so as I was saying that the this decomposition was first proven for a well-chosen sequence of times and then for any sequence of times and I'll try to explain both methods now so let me prove the result for all sequence of times well not prove a sketch an argument of how the proof goes okay so this was a using what we called the channel of energy argument which is a very powerful argument in this in this problem so what one the heuristics of what happens as you approach the final time of existence that leads to this a soliton resolution is of course energy cannot be lost because it's conserved but what and the solitons the sum of solitons has a fixed amount of energy depending on this none and on this elliptic solutions so there has to be some mechanism for which energy disappears and how can energy disappear cannot disappear in size what happens is that it moves out to infinity so this soliton resolution phenomenon only holds when you have infinite spatial domains and what happens is that energy just gets displaced towards infinity and then in the finite part you just get the soliton parts and then the radiation part takes care of the energy moving out to infinity okay but of course this is only heuristics there's no proof in there so you have to find a way to quantify this and to measure it mathematically and that's what this channel of energy method does okay so there's a main in the radial case there's a main fact and the main fact please don't read this too carefully let me summarize it in the following way suppose I have a globally defined for both times radial solution of the nonlinear wave equation in dimension 3 and suppose I know that this isn't w or minus w or its rescale versions okay then there always has to be either for positive time or for negative time energy left outside a light gone for all times that's what this says there's always this lower bound there's always a block of energy outside these light cones and this is this energy that is being moved away okay and this is what captures it this fact so this is a dynamical characterization of this w which doesn't move for w it is obvious that this cannot hold okay it's a simple calculation because w behaves like one or absolute value x at infinity in in space and then you see that this just fails so if you're not w you have this property okay so how do you use this okay oh first I'm going to tell you how I prove how one proves this thing and here again don't pay that much attention to the slide there's a property of solutions radial solutions of the linear wave equation in three space dimensions that's important here and this is an elementary property in the sense that I think D'Alembert could have proven it okay so it's a very basic property and what it says is that if you have a radial solution in the energy space then either for all positive times or for all negative times and any number are not that you choose there's always some energy left over so there's a lower bound now there is an exception there's one one kind of solution for which this doesn't hold and that's the what corresponds to the Newtonian potential which is a 1 over r which of course outside the light cone is a proper solution of the energy critical wave equation okay and for that it fails but what the real mathematical statement is is that once you do the orthogonal projection to the complement of the Newtonian potential then this is true okay so that's the meaning of this statement now how do we use this in the nonlinear problem the way you use it in the nonlinear problem so this is a property of linear solutions but it holds for all are not suppose we have our solution our global solution that's defined for all times both positive and negative I choose an R naught that's very big so that the energy at time zero outside the ball of radius R naught is very small okay I can always do that is it's so small that the small data theory applies and then the nonlinear solution is close to the corresponding linear solution and then by a finite speed of propagation the truncation if I look at it outside that light cone doesn't affect things and then I can use the linear result okay and I will give this proof in detail I mean this is just I I'm just a kind of giving you a taste for what we do all right but yeah you have a solution initially which are outside of R zero and then we fill it in and so it has to remain small then the influence what they know they yeah I I need some chalk so in here they're the same so you control what which is going on outside outside influence what which is going on no no because of right it just happens there and it gives me the lower bounds that I want are outside the call right and so it's a cheap trick but it's it's a very powerful tool but never never is that truncated data influences what what is outside of the closing call yes but we only care what's outside of this no you see somehow what goes inside the light cone is unknowable so we decide that we don't care and we somehow manage with the information outside okay this is the point of this strategy all right so then that was only in the radial case in 3d there's a corresponding inequality that is what we used in this results near the ground state in dimensions 3 and 5 and that's we take our 0 equals 0 then we can only take our 0 equals 0 and then you have the corresponding inequality and but this notice that it's claimed only for odd dimensions so let me just explain the proof of this result let's say in the finite time blow up the other case is very similar so the first thing is that as the solution is bounded so it has a weak limit s t tends to 1 and one can show that this weak limit is independent of the sequence of times you need to give an argument but that's the case and then you look at the linear solution with this data time 1 and that's what the radiation term is okay so that's an easy description in this case of the radiation term then of course you look at the nonlinear solution with the same data time equals 1 and I'll be near the linear solution because everything is very nice at time 1 for that solution so you can use a small data theory at time equals to 1 and now we look at this nonlinear solution and this nonlinear solution is what we call the the regular part of our solution and what you can see easily from finite speed of propagation is that there's a clean comparison outside this truncated in inverted cone the two things are equal and all the bad action is happening inside the light cone so I don't know if I can draw another picture so outside your solution is very nice and the whole blow up is happening from what's happening inside okay that's what you need to understand anyway so so we now take this is the singular part of the solution and we break it up into blocks now these blocks are nonlinear solutions and but they are localized in space time and they have certain orthogonality properties with respect to each other so that you can think of them as independent pieces and these are what you know formally then nonlinear profiles associated to a Bahurijer are profile decomposition okay just think of them as blocks and then there's an error and this error is small but only in a weak sense for instance you can think of it as tending to zero weakly in the energy space but not necessarily strongly okay so now what we want to see is that each one of these blocks has to be a w that's that's our goal that's what the soliton resolution says okay so if one of the blocks is not w we use our dynamical characterization of w and then there has to be energy here but there is no energy there because u-w is concentrated in here so all of these blocks have to be w and now all that remains is to prove that the errors term has to go to zero in the energy norm and for that we use the second of this outer energy or well we can use either one of them and that's the end of the proof so you have to of course to take this with a grain of salt because the proof is rather long so as we as we will see towards the end of the week okay or early next week so now let me explain the other argument the argument that worked for only one sequence of times so this argument proceeds in a different way and the first the first part of this argument is to show that there can't be any energy even near the boundary of the cone in this region as you approach the blow up time there can be no energy so in fact that the energy has to be in something like that so that's the first part of the argument and our original proof of this fact used again this outer energy inequality then you combine this with virial identities what are virial identities in this context there's nothing more than the Pozhev identity that was used by Pozhev to show that the only radial solution is w and there's wave equation analogs and this this are the two analogs okay so these are the virial identities now we have this formulae and you add them and you get this very nice fact that the t derivative plus an error is a time derivative and if you combine that with the fact that the energy is in a region like this you can conclude oh I'm sorry seem to have lost myself oh here it was so what you conclude from this is that this a Cesaro means of the time derivative inside the light cone go to zero okay so some average of the and this is where you need to now pass to us to a convenient sequence of times because of course because Cesaro means go to zero doesn't mean that ordinary means go to zero that the ordinary thing goes to zero but it means it for by a Tiberian argument for a subsequence and that's our subsequence so this is a very classical argument okay so from Cesaro means to ordinary some ability you go to a subsequence so for a subsequence the things have t derivative that goes to zero if they have t derivatives that goes to zero it means that each block has to be time independent but what are the time independent solutions the solutions of the elliptic equation and the elliptic equation solutions in the radial case are w so now you have a different proof that these blocks are w but only for a sequence of times and then you conclude using the weaker of the two outer energy inequality so I'm gonna now go to the discussion of the other dimensions and the and the non radial case I will do it rather quickly so that we can then start with the actual detailed proofs and study okay so what happened after that well as I mentioned there's other applications of these techniques and right here let me first start with a fact that was proved by caught myself and slug and it's and this sort of paralyzed us for a while and it's the fact that this outer energy inequalities even what the ones with our zero equals zero are false in even dimensions okay they're just false you can write down counter examples so okay it's life I mean what can you do now the second ones the ones were are zero are equal is equal to zero which are the weaker ones have the the following peculiar feature is a if the dimension is even but of the form for 812 and so on they are true for this kind of data and for the other even dimensions it's for the opposite kind of date it is nothing to do but to accept things because this is how they are okay now for odd dimensions with Lori Lou and slug we showed that this outer energy inequalities even with they are not have an analog with this true which is true in all odd dimensions but the exceptional functions get to be more and more as your dimension gets larger and larger so the projection the orthogonal projection that you have to do is to a space of increasing dimension as and the dimension goes to infinity as the ambient dimension goes to infinity but the constant is always one half so that there's a uniformity in the constant so using this fact my student Casey Rodriguez was able to prove for all odd dimensions the radial case along a well-chosen sequence of times now for the even dimensional case there's a there was a problem because it's outer energy inequalities do not hold and there what we did is we we first looked at the case of a related problem which is the wave map problem the wave maps are the hyperbolic analogs of harmonic maps and in this problem it had been known for a long time that under a certain symmetry assumption called equivariance there was no self-similar blow-up in the sense described here and this was proven by integration by parts by Christodulu and David Darzadeh but an important fundamental ingredient in this proof was that the energy density is non-negative for wave maps even though they could be focusing the energy density doesn't change sign and now for the nonlinear wave equation in the focusing case this is the thing that fails completely and how did you use that the energy density was bigger than or equal to zero it was to control the so-called flux of the solutions and it's this control of the flux that allowed for an integration by parts prove of the lack of self-similar behavior and what we did eventually here is in 4d with Caught, Lorentz and Schlung we reverse the analogy between wave maps and nonlinear wave equation usually that you think of the nonlinear wave equation as a simplified model for wave maps and the wave maps is more complicated but what we did is instead use the wave maps to prove something for the nonlinear wave equation okay so there's a very simple change of variables under which your solution to the nonlinear wave equation solves something that looks like a wave map equation and if you look at the corresponding energy it will be non-negative the energy density provided the C is small enough and there then we use the radiation term that shows that on the boundary of the light cone you have the smallest condition for the u because you have it for the v and then an iteration argument to go inside the light cone and you can prove it inside and then this works and the final step was to use this outer energy inequality with r not equal to zero to show that the dispersive term goes to zero and here in the four-dimensional case the data for which it is true is the one for which the time derivative is zero but the this a virial identity argument gives you that the in the error the time derivative goes to zero and so that you can plug this in but then this doesn't work in dimension six because it flips the the pairs for which it is true and then with a ha or g a who's my postdoc at Chicago we were able to do this also in all even dimensions by combining two virial identities and not using this channel of energy and I'll have occasion to prove to go back to this later anyway so the the the next the conclusion of all this is of course now for a sequence of times we have it for all dimensions in the radial case how about the non-radial case and the the summary of that which we will see in full in the last lectures is that for a well-chosen sequence of times one can also prove this in the non-radial case so one has for a sequence of times the full soliton resolution for the energy critical wave equation in all dimensions okay so I'll stop with this overview now and with shift gears and instead of speaking in broad terms we're gonna go to specifics so now we continue but instead of continuing we start okay so this is a restart we start from the very very beginning so the first part is the local theory of the Cauchy problem for the nonlinear wave and I will discuss this in at some length so I start out with a linear wave equation and I will need to draw a picture very soon so we start with a classical linear wave equation so we have a linear equation evolution equation with the right-hand side initial data and initial derivative data so there's a general method to solve all such problems and that's the Fourier method right so we take the Fourier transforming space and then we get ordinary differential equations and we can solve okay by superposition so the Fourier method gives you a formula for the solution so there's the cosine term on w0 the sine term on w1 and what we call the Duhamel term which is this integral and we summarize saying that w is the s which is a linear solution and the Duhamel term and the term the notation I'm using is the hat is the spatial Fourier transform okay so cosine of square root of minus Laplacian t its meaning is this on the Fourier side I just multiply by that the same for the sine and this term this thing means this and so on okay so this is standard notation so one of the main properties of the linear wave equation is this finite speed of propagation so I'm gonna draw the picture now so the finite speed of propagation tells me that if the right-hand side h is supported away from this cone and the data w is also supported away from this cone and in here the w has to be zero that's the finite speed of propagation for the wave equation in odd dimensions there's something better which is the so-called strong Huygens principle which says the following let me draw another picture so so now the support is contained in here then in all dimensions if this were parallel in all dimensions the support is contained in here but in odd dimensions this is where the support is okay and what's the difference between even and odd dimensions okay so if we're in the three-dimensional space and we see a jet go by we hear it for a minute and then it disappears and we don't hear it anymore and that's the three-dimensional case if we're in two dimensions and we throw a pebble in a pond we see the ripples all the way out okay so this is nature so this things will play an important role as we saw so now we start with a local theory of the Cauchy problem for the nonlinear wave equation and oh as I'm saying from now on I will stick to 3d no more fractions to remember well there will be fractions but there'll be specific fractions there will be a four-thirds and a few others coming in but okay so this is our nonlinear wave energy critical focusing wave equation now for for the study of these equations there are some important estimates which in the trade are called strickers estimates okay so the strickers estimates play the role of the sobola estimates so in the study of wave equations or generally dispersive equations the strickers estimates they play the role that the sobola estimates play in the study of the Laplacian okay and this many strickers estimates I'll give you some examples of the ones that we're going to use okay so if you solve oh so here I'm talking about the linear wave equation that we had here so the h refers to this h and the w refers to this pair okay so the first estimate tells me that the soup plus the l8 norm plus the l5l10 norm and then the l4 norm of one half derivative and then the other version of one half derivatives is controlled in terms of the initial data and the right-hand side and there's the four-thirds and this fractional derivative one half derivative is imposed to us by the equation this is the maximum amount of derivatives that you can gain okay so I'm sorry fractional derivatives appear even though our problem has no fractional derivatives in it okay but that's life and it's the control of these two space-time norms that we think of as like a sobola estimate and there's another strickers estimate which I call s sub 2 where on the left you have the same quantity but on the right instead of having fractional derivatives you just have the l1l2 norm of the h and this one is obviously easier for some things because there are no fractional derivatives but sometimes you need to also use the other one okay and because we have a fractional derivatives an important technical tool is there's some mystery here this four should have been four-thirds okay is this estimate when you apply half a half a derivative for to u to the fifth it behaves as if it were true derivatives in this chain rule setting and then there's more of the similar estimates here okay and these are used in the proofs to do fixed point arguments okay so the small data theory is carried out through a fixed point argument and to do this fixed point argument it's convenient to use an annotation if i is a time interval the l8 norm or i l8 norm in x will be called s of i norm and l4 in i l4 in x norm will be the w norm okay you don't like the notation I'm sorry sometimes we will call the s norm the l5 l10 norm okay that depends on what we're doing at the time so let me define what I will mean by a solution a solution on an interval i with zero in i will be a function which is continuous with values in h1 and its t derivative is continuous with values in l2 in the interior of the of the interval i and for all compact sub intervals we will have control of this space time norms and the solution will satisfy the integral equation that's what we will mean by solution okay so it's a standard standard terminology by the way at the end of this first series of lectures I'll make available to people these notes okay and then but I won't give you the second installment until after the end for obvious reasons okay the main result on the local Cauchy problem is that if there's an arbitrary initial condition you have a unique solution which is defined on a maximal interval of definition which we will call i max and t minus and t plus will be the end points of that maximal interval of definition and to prove this one uses the contraction principle which will be and combine it with a chain rule to show that if there is a delta not that there is a delta not small enough such that if the s norm of the linear part on that interval is small enough there is a unique solution on that interval with initial condition u0 u1 moreover if we call a this norm on the initial data we see that the difference between the non-linear solution and the linear solution is has this kind of control okay so that's what I meant earlier when I said that if you are small then the linear solution is close to the non-linear solution and this is the specific estimate okay now this gives you a solution in some time interval and then you want to expand it to get to the maximal interval of existence and to do that you have to have a little uniqueness argument that says that if you have an overlap in the time intervals and they agree in the middle they agree in the union okay so you can do that and of course this is for intervals which contain the origin but we can translate to any time t0 for instance I was talking about time one earlier okay now what's very important for us is something that we call the finite time blow-up scattering criteria we have to to find a way to describe the fact that t plus is the final time of existence okay suppose that okay one thing that can happen is that the h1 cross l2 norm goes to infinity we have time type one blow-up but how do we describe type two blow-up where the norm remains bounded first so let me just say in a few words what happens at that time if the norm remains bounded but you can't continue the solution what happens is that the gradient squared of your solution is concentrating like a delta mass and that's why you can't continue continuously okay now how do you measure that and it's this s norm that measures it if the time of existence is finite that means that something does blow up and what blows up is this dispersive norm the s norm so the l8 norm becomes infinite so if the if this norm is finite then t plus is infinite but more than that this tells you more than that it tells you that the solution needs to scatter so scattering is also described by the s norm is finite so there's a solution of the linear wave such that this difference goes to zero okay so that's what scattering that's and it's all combined in terms of this assumption so if this holds that's equivalent to saying that t plus is plus infinity and the solution scatters okay so it's a good way of codifying that and okay so I'm saying all these things for forward time you can go backwards and get the same then there's one remark that I want to make the fact that we know that the solutions are constructed by the picar iteration scheme gives us the finite speed of propagation for the non-linear problem for free because remember that in here we included the right hand side in this finite speed of propagation and so if you define un plus one in terms of un and you know that un has the right support property you can conclude that un plus one has it and then by the picar contraction mapping prims the un plus one converges to the solution so the solution will have the right support property okay and so if support of u zero u one intersection with a certain ball is empty then you are outside there and you can say the same thing that if two solutions agree in the ball b x zero a then they'll agree outside the inverted light core but of course the for the stronger horeons principle you cannot do it because things get smeared out by the non-linearity so okay so I just briefly I want to mention that this notion of solution is one possible notion there's other notions of solutions that you could have and they are important instead of having the l8 l8 norm you could ask that the l5 l10 norm we find it okay and so your the immediate question is is this equivalent are the two notions equivalent and the answer is yes so if if you have this property and your equation you don't need it to be true in the in the du Hamel sense but just in the distributional sense and then you can conclude that you have a solution in in the du Hamel sense okay so but notice that you always have to have this auxiliary spaces now it's a very natural question to know if you know that u is just continuous with values in h1 cross l2 and solves the equation in the distributional sense is it a solution in this sense okay now this is true in dimensions 4 and higher which was proved by Fabrice Plancheon and it's open in the three dimension okay it's actually an interesting open point in the theory yes yes yes yes it should be true but I don't know how to prove it nor does anybody else as far as I know okay okay why would we want to use this space sometimes because when you use this space you don't need to use fractional derivatives the disadvantage it has is that the exponents are different in in t and x and I will show you in a few minutes why that's a problem or why that may be a problem and which is why you may want to have both notions of solutions so the next thing I want to give you is something that is a kind of the furthest you can go with this local theory of the Cauchy problem and it's we call this the long time perturbation lemma okay and we call it the long time perturbation lemma because the size of the time interval is not important in the proof of this lemma okay and this is a kind of this is one version of it it's purely a technical thing but it's an important technical thing and that's why I'm mentioning it here suppose you have an approximate solution it has finite l5 l10 norm is continuous with values in h1 cross l2 and it solves this equation with an f and the f is sufficiently small and we have u0 u1 sufficiently is close to the data for capital U or by u arrow I will in general just mean the vector u d dt of u okay it's just a shorthand so under these conditions if the epsilon is small enough this allows you to have a solution to the actual non-linear problem which we will call u which is close to this capital U and how close it is is by an epsilon that's controlled by these things and this is a very useful tool in this theory so let me show you what are some results that one can prove using this thing it's a perturbation argument to you there's no quick way of describing it but it's the one where where you use the fact that for the non-homogeneous for the du Hamel term you have a wider range of Stryker's exponents yes this I'll go to this using this proposition this proposition I think goes back to the work of a coriander keel Stafilani tako kantao on the energy critical shreddinger equation that's where such a result was for specifically formulated and it was important in their work and it's important in all of these other works since okay so the n is independent of the interval and therefore the epsilon star also is independent of the interval yes there's a exactly and that's why it's called long-term perturbation but what is it is it a two page proof no no no it's a maybe 10 page proof and I don't want to give it because I mean this is a I would say harmonic analysis result okay it's a oscillatory integral result but of course I'm not going to give such a it's a very technical thing so there are other versions of this proposition where instead of the l5 l10 norm we use the l8 and the w norm what are some consequences of this theorem first suppose we start out with a compact set of data okay so we have not just one data but we take a compact set then we can give uniform bounds of the maximal time of existence that just depend instead of on each individual data on the compact set that's the first thing the second thing is the thing that Gustavo mentioned for continuous dependence it tells you that if you have some data and a sequence converging to that data then this sequence all have time of existence as at least as big as the one of this data and in that interval they converge to the solution and I think in the early versions of local well-postedness this was absent this hadn't this was a an open problem until this kind of version of local well post anyway all of these results are needed in this theory so now I'm going to discuss Lorenz transformations remember we discussed the Lorenz transformations of these elliptic solutions to get traveling wave solutions right that was the previous hour okay so and remember that this depended on this vector l now let's take the vector l to be one zero zero okay yeah no no there's a factor of little l so our vector will be little l times e1 okay thank you then if you look at what the Lorenz transformation does it boils down to this so the x1 variable gets changed to x1 minus lt over square root of 1 minus l squared the x2 and x3 variables are left untouched and the time variable gets changed to this so somehow it mixes spacetime but in this very specific way and so this is for the linear wave equation and it's true that if you have a solution of the linear wave equation and you perform this Lorenz transformation then you still have a solution of the linear wave equation the question is is it still a solution in the energy space and the answer to that is yes and it's given by this lemma and you see that what you have to do is estimate h1 cross l2 norms instead of on parallel hyper planes on tilted ones provided the tilting is strictly less than one okay but c depends on l yes and blows up as l goes to one yes it has to okay but for each for which l there's a c and okay and this is a very important compactness thing of this family of Lorenz transformation now the proof uh if you if you start to sit down and write the proof it boils down to the following if i define v hat by multiplying by the wave group e to the itc this is true and to prove this you only have to do plancher l theorem and then do a change of variables in frequency space and then this comes out okay so it's elementary if you see it the right way and also this is continuous in time in in this norm so for each fixed l i'll define y and s this way which is the inverse of the Lorenz transform that we had before so to invert this Lorenz transform you change l into minus l that's all that that you do okay and i write this down here so now suppose we start with it globally defined solution of a non-linear wave equation we can certainly define is Lorenz transform by this formula by the way for this is the same formula that i had before that looks so horrible but when you really write it in coordinates it's not so horrible it's very simple now since u is global in time this is well defined as an element of l8 log and this is why we use the l8 norm before because since the x and the t have the same power you can do change of variables in space time norms and see that that gets preserved by the Lorenz transform and that was the reason why we didn't just work with the l5 and 10 norm initially but we needed these l8 norms because we needed to have the Lorenz transform okay so it's an unfortunate technical aspect of the theory but anyway you can overcome it overcome it so at least we have it as an element of l8 log but now you have to prove that it's actually in the energy space and that's a proposition that we prove with duke and merle that if u is a global finite energy solution and you do the Lorenz transform this is also a global finite energy solution and the the strategy for this proof is somehow uh truncated on outside cones again so at the beginning of the theory we didn't know how to prove this so we didn't and then eventually when we realized that the important thing is what happened is happening on outside of the cones then we realized how to prove this okay okay so the next part of this introduction goes to the profile decomposition which is the Bajorijerar profile decomposition so this is a you can think of it as a completely functional analytic thing uh machine on the other hand it is not really that because in concrete cases you're taking some solution and breaking it up into its blocks of energy and space time i mean so it's both it's both things so what is this a profile decomposition and what does it measure uh so this profile decompositions were first studied in elliptic settings going back to interpretations of pierlillian's work on concentration compactness uh they were works by struve patrick gerard and brazil's corone and it was a brazilian corone who first noticed that there's a certain orthogonality in the parameters and certain Pythagorean identities that are turned out to be extremely important here okay so uh and you can think of this in terms of concentration compactness in these evolution problems to see these profiling compositions as characterizing the defect of compactness in the embedding given by the strickers estimate so you can think of the l8 norm of the wave equation being controlled by the h1 cross l2 that's an embedding of some kind and one side has derivatives the other one doesn't so uses ask is this compact the answer is of course not because there's a group of invariances which is non-compact how do you describe the this defect of compactness as it's in the terms of this profile decomposition of course that's not the only way to think about it because what you really think want to use this for is to pull solutions apart okay okay so we start out with any bounded sequence in h1 cross l2 and we start with a bunch of solutions of the linear wave equation and a sequence of parameters for each j a sequence in n and the lambdas are from 0 to infinity the xj's are in r3 and the tj's are in r so the xj's and tj's are the spacetime centers of this solutions the lambda j's are the scales for them okay and we say that this sequence is orthogonal if the parameters are such that different solutions don't see each other because of the location or their scaling okay and that's what the meaning of this condition is and now we have this solutions ujl and we say that this is a profile decomposition for this sequence if the parameters are orthogonal and if we denote ujln the scaled and translated ujl and w jn the difference between the solution for the u0n's and this sum of modulated profiles then this is limit it this is small and in what sense it's small well first of all the h1 cross l2 norm is bounded where these are the time zero expressions and second the space the striccarts norm of this difference is small as n goes to infinity and then j goes to infinity okay and that's what we mean by saying we have a profile decomposition this is precisely what i want to say i was saying that we have a profile decomposition okay and it takes a while to to get used to working with this thing i have to say for me it took years but anyway eventually it becomes second nature you do it enough okay so bahouri and girard for n equals three and then bulute for n bigger than three proved that for any bounded sequence you can always extract a subsequence which has a profile decomposition and the error not only the l8 norm goes to zero but also the l5 l10 norm goes to zero and the l infinity l6 norm goes to zero okay so all of this kind of the striccarts norms go to zero but not the h1 cross l2 norm this need not go to zero and it's not that you can't prove it you cannot do it okay and this is one of the reasons why these channels of energy are useful because this is what we used to show that our errors are actually zero in the stronger sense okay i'm going back to the first part of it that's for any bounded sequence of initial data yeah it is you can think of it as a sequence in h1 cross l2 you don't can you don't have to think of it as an initial date of course you should think of it as an initial data but it's so that's why i meant to say that this is a purely functional analytic statement okay you are you are writing s of t that's so that they are really taken as an initial data right right but of course if you have a different equation you could do similar things okay angle one how are these profiles ujl constructed they are weak limits of the original sequence so what do i mean by that for each j if i do the reverse scaling and translations on my original sequence and i take the linear solution with that as data and i take the weak limit that gives me the profile at times zero so what these profiles are are all the weak limits of scaled and translated versions of my data to which i apply the solution operator the fact that this holds is a consequence of the orthogonality of the parameters and the fact that for each little j less than capital j this expression that remember this is the error goes to zero weekly in h1 cross altar i'm sorry what is this little n arrow zero zero what is your name goes weekly to the function zero zero n is the parameter that tends to infinity okay so that's the notation yes there's nothing conceptual now so so you're rescale with uh you're rescale with some parameter you translate with some parameters and then that hill or the other one yeah and then yes so and if you remember i said a profile decomposition is a sequence with these properties the parameters of these properties and the limit goes to zero in this sense and then for any such thing you can prove that this limit goes to zero and that this goes to zero weekly okay so these are kind of functional analysis exercises okay but i'm sorry there are still these these for solutions of the linear problem yeah yeah i haven't said anything yet i'm working out there okay now there's also the important Pythagorean expansions okay and these are extremely useful if you want to look at the energy of your sequence there's a bar missing there you can use the energies of the profiles plus the energies of the remainder and you take the limit as n goes to infinity this is true for each capital j and the same thing is true for the l6 norm remember h1 embeds into l6 that is also true and from the two things if you go back to the non-linear energy the same is true for the non-linear energy because it's true for for each one so there you're already seeing the non-linear problem as we will see that by extracting subsequences and changing the profiles possibly we can only we can always assume that the tj n's are identically zero or they tend to plus infinity or tend to minus infinity so this is always the case and i'll i'll say a little bit more in a minute okay now i just want to make a comment here it is tempting to think that this Pythagorean expansions also hold not just for h1 plus l2 but for each one individually it turns out that to my dismay this was false okay so there's something where the energy conservation plays a role and this you can find a counter example but there's alternate versions of those and i'll just say this very quickly we separate the profiles into those oh maybe they have to go back here i said that we can always assume that either they're identically zero or they tend to plus infinity or to minus infinity so the first case we call the index j core and the second case scattering is just terminology okay and it turns out that you do have Pythagorean expansions for each individual piece provided you keep the core terms separate so you have to keep all the the scattering ones together and all the scattering ones together you cannot break them up but you can break up the scattering ones i mean the the core ones i'm sorry yep where did the scattering go it's in this part the scattering ones are good enough that you can split them all individually i mean no i'm sorry the core ones are good enough that you can split them the scattering ones are in here but they all get added together they can't be split up they're there it's just that i can't separate them i'm still confused i have the impression that the the scattering ones are on the left hand side no no no no on the left hand side i have my original sequence on the right hand side i have the core and the scattering ones but i'm piling them i'm splitting them up differently the the core ones i just i can put the sum inside okay no no i don't get rid of anything it's just how i sum them okay yeah so what i would say is that at this point it becomes interesting to know whether there's some uniqueness in this profile decomposition because it could be that if that the fact was that you didn't choose it correctly and that's why you didn't get the right Pythagorean expansions okay so we said that a profile decomposition was a sequence and so on but why is this the only profile decomposition right you could have two different profile decompositions in principle for the same sequence of times or for the same sequence of initial data now it turns out so the first part is that if you have a profile decomposition and you change the parameters in this way to lambda tildes which are just the limits exist and are are non zero and so on and you change the translations and you change the times and you get new profiles in this way then this is also a profile decomposition and that's how we can reduce always to the case when the tn is identically zero or tn over lambda j goes to plus infinity or to minus infinity so there is this ambiguity and this is inevitable okay but this is an ambiguity that we can understand and then there's a theorem that says that this is the only ambiguity so it's a lemma if we have two profile decompositions of the same bounded sequence and there's a and they are either both finite or both infinite then the only way which this could have happened is what was described by the previous lemma so there really is uniqueness in the profile decomposition except for the ones that you can describe by those transformations and this is something that we prove with educare and merrill so this is some precision on the profile decomposition okay and then the next statement says that if you have a weak limit of the kind that create the profiles this has to appear as a profile in the profile decomposition so all weak limits appear as profiles modulo this transformation okay so it takes a little bit of time to digest this this is kind of the background and now in the last few minutes we will pass to the non-linear versions of the profiles which is what we use in non-linear problems okay and these are the non-linear profiles so a non-linear profile associated to this linear profile is a solution of the non-linear wave equation which asymptotically looks like this linear solution and of course this limits after extraction in n we can always assume exist because we're admitting negative infinity and plus infinity as part of the limit so for any sequence after extraction the limit exists it's either plus infinity minus infinity or finite so it's not that hard to see that you always have a unique non-linear profile associated to a linear profile decomposition so these guys always exist and now we rescale them and one can understand what the time of existence of these non-linear profiles is because of simple formula and these non-linear profiles are the building blocks for solutions in an approximation here but before let me just say that this limit when the lambda j tjn tends to minus infinity where then t plus is plus infinity and when it goes to minus infinity it's the opposite so in the cases when you have scattering lambda jn over tjn that gives you scattering properties for the associated non-linear profile and that's why those are called scattering j's okay and then let me just finish today with this approximation theorem that gives you a wave we have a bounded sequence which admits the profile decomposition let me take the solutions with this initial data suppose that all the associated non-linear profiles scatter so it's one situation all the associated non-linear profiles scatter then we look at this remainder term which is the non-linear solution minus the sum of the modulated non-linear profiles minus the error the linear solution corresponding to the dispersive error then this difference goes to zero and the solution un exists for all time and scatter so we have a way of writing our solution to the non-linear equation as a superposition of solutions to the non-linear equation with these non-linear profiles plus errors so this is the easy case when all of these uj's scatter forward if they don't scatter forward you can do the same provided you stay away from the final time of existence of the uj's and provided that uniformly the spacetime norm the dispersive norm remains bounded so this is the way in which these non-linear profiles are building blocks for the linear for the non-linear problem okay and I think I'll stop here continue next time thank you yes yeah yeah it's okay Frank it's okay you can ask so that I clarify yes okay so you write everything in dimension three but okay dimension rate is typically this kind of lemma in large dimension are a little more quickly yeah a little bit more complicated and some the the surprising thing that we found that very recently is that some some of the uh uh stickers norms don't work for the errors to go to zero and if you look at the kill tau endpoint and you use that spacetime norm that will not go to zero necessarily so you know you have to you can't use these things like a black box you have to understand what you're doing okay yeah so yes I'm sorry if you compare this kind of profiling composition from the elliptic case and say the specific case which are the main differences or the main well the the main difference is that you have a time right so the if you want to do this with just the l infinity of l6 norm on the error then it's basically the elliptic case okay but if you want to include these other spacetime norms then you have to include this tricker's estimates in the proofs okay when you have a perturbation you don't need the perturbation yeah so the perturbation lemma is the fundamental tool to prove this I mean the one you were messing yes that's the fundamental tool to prove this approximation thing if you ask how to prove this you have to go back to the perturbation but then is there an analogous to this perturbation lemma in the elliptic case or it doesn't make a question no it's not needed okay okay yeah it's just no there's no no need for that