 First of all, I want to say thanks a lot for the invitation and I'm really happy to be here I've been here for almost a month now, and it's It's been a lot of fun and very relaxing and it's a really great place to be That'd be sad to go in a few days So today and tomorrow and Wednesday, I'm gonna talk about the Vlasa Maxwell system and the whole space So I also want to say thanks to the organizers for inviting me and for putting together this great Workshop and this great semester program I feel like I've benefited a lot from being here and I've learned a lot from talking to a lot of smart people So today so the Vlasa Maxwell system is a complicated system of partial differential equations that I Started studying a few years ago. I've written two papers on the subject so far and Both of them are in collaboration with Jonathan Luke who is at Cambridge right now and So I'll tell you about those later, but today I'm basically going to try to teach you about the equation and teach you about what's been known in the past and how I'm gonna try to give you a picture about how some of the proof work proofs work. I can't Give you too many details, but I'll try to Give you a broad overview. That's why I'm using slides So to start the Vlasa equation now which was derived by Vlasa in 1938 although people In the West as far as I understand didn't really know about it until 1946 when it was translated into English looks like a simple linear Transport equation you have a DTF plus P hat. I'll explain what that is soon Plus the gradient XF plus capital F, which is another function That product with the gradient of P is of lower case F is equal to zero here lower case F is your unknown that you'll solve for and this is Nowadays considered to be one of the fundamental models for a plasma a plasma is a As modeled as a large number of particles say on the order of 10 to the 23rd all of them are very very small And they're occupying a very small amount of Say in a box they're occupying a very small amount of space in that box But there are a lot of them and they're moving very fast so That would happen if the plasma is officially verified or sufficiently hot So that you can think of temperatures Excuse me, you can think of speeds very close to the speed of light That's why we are relativistic and then in this situation you can neglect Collisions between particles and that's why I have a zero on the right hand side of the equation If you want to include collisions, then you get a more complicated model of which there are a lot some of them have worked on myself and so then the only Way that this equation is nonlinear and the only interaction between Particles is through a collectively generated field, which is capital F Which is a function of t time x and a lowercase f which is itself a function of tx and p But the only way the field is a function of p is through a lowercase f. So here T is time x is space and p is a momentum So this is the system There are a lot of examples of where this comes up physically I won't spend too much time on that but a couple of things you can think about That you might know about already is the solar wind or a Powered-up fusion reactor. This would be a reasonable model for particles interaction Interacting a fusion reactor like you might find at CERN okay, so Let's show you a relaxing picture. This is Vlasov and Maxwell these are so I showed you the Vlasov equation that was a derived by Vlasov and then we're going to talk about Vlasov Maxwell today and The Maxwell's equations were derived by James at clerk Maxwell in the 1800s Actually, there are a lot of pictures of Maxwell if you Google but a Long time ago. I tried to Google for pictures of Vlasov and actually spent a good amount of time on it And this is the only one that I could find. I don't know how to find more but Anyway, those are the two guys The heroes of this equation Okay, so for a while, I'm gonna tell you about a collection of Vlasov models and a sample of results There's a lot of models in this area and so I'll tell you about some of them The main one is the relativistic Vlasov Maxwell system in 3D So here you have the Vlasov equation that I already told you about at the top, but then you have this Thing called the Lorenz force, which is E plus P hat cross B Doddard with gradient P that would be the replacement of your capital F and you also have this parameter gamma in here gamma Normally takes two values plus one and minus one When you normalize all the physical constants plus one, it's thought to be stable and minus one is thought to be unstable, but there's a lot of work to be done in that area and Then you will couple these with Maxwell's equations. These are just the standard linear Maxwell's equations partial Tee plus the curl of V minus J And then partial Tee B is equal to minus the curl of E and you have these constraints The divergence of E is equal to rho and the divergence of B is equal to zero Here rho is the charge density and it's a it's the integral of F with respect to P where P is a momentum variable. Now you see that P is in R3 I'm normalizing a lot of the physical constants to Simplify the presentation Then the current density is a vector as it has to be because it's coupled with Maxwell's equations and E and B are all three dimensional vectors And it's so each component of it I equals one two three is of the form P hat I times F of Tx P integrated against Dp and So these are the charge in the current so then Maxwell's equations are linear in E and B but they have this nonlinear forcing in terms of J and P the charge in the current and then that's plugged back into the relativistic the relativistic less of Maxwell's equation for lowercase F and so this Gradient P term at the top ends up being the nasty nonlinear contribution to this theory and that's the reason why the large-data problem for this equation 3d is still Completely open and so this is really the hard big open problem in this area is the large-data 3d existence and uniqueness problem And that contacts up classical solutions Okay, so the momentum is a P as I've said P is a three dimensional vector then the velocity is a P hat which is P over P zero actually You won't see this notation in the Physics literature very much as far as I'm aware this P hat notation be the physics literature may use P half for something else actually But this P hat notation as far as I'm aware was introduced by a glassy and Strauss in their early works on this equation in the 80s And so P hat is P over P zero P and P zero is the relativistic energy is the square root of one plus P squared. I'm hiding The speed of light it should be present in the whole equation, but it's not it's normalized equal to one here And so in the relativistic case velocities are bounded by the speed of light which in this case is one And So this is the full model As I said before gamma equal plus one is a plasma physical case in that case a stability is expected and Gamma equal minus one is the stellar dynamics case and In that case instability might be expected for certain initial data you have examples of that in the four relativistic of last up was on that I'll mention later and Okay, so that you might ask well there are these constraints here How do you satisfy those well it turns out that there's a There's another equation that you can derive for row and the j that is a consequence of the system and that equation itself predicts that The constraints are propagated by the equation if they're satisfied initially, so you don't have to worry about that Yeah, so that's a relativistic glass of Maxwell in 3d. Oh I should say especially for students in the audience. I'm really happy to Answer questions and I often find it easier to answer questions on the spot rather than at the end of the talk so Please feel free to Raise your hand. I'm happy to try to answer any questions that you may have Okay, so this is relativistic loss of Maxwell So it that's What that actually was was so-called one species relativistic loss of Maxwell for one particle Okay, if you again if you look in the physics literature most of the time You don't really see one species less of max So what you see is in species less of Maxwell or as I'm calling here many species less of Maxwell so you actually have In equations so alpha represents any integer between 1 and n then DTF alpha plus b sub alpha hat Yeah, but dot gradient X F alpha is equal to an M plus gamma e plus p hat cross B That gradient P of alpha equals zero and that represents n different equations and then you have the same Maxwell's equations and the same Divergence conditions and here the velocity P hat alpha is p over the square root of M alpha plus P absolute value p squared and M alpha is represents the mass of Each individual different particle that you are modeling here and then the only Difference here is you have the charge and the current and they're now the charge is a sum of the F alphas and e alpha where e alpha is a Physical constant and the current is j which is a similar sum of the alpha and p hat alpha times F alpha and so all of the Unknowns are summed here in this way and Normally and the rest of the talks I'm only considering the one species case normally for many Types of results that you prove for these systems. It's sufficient because of the structure of the equations To just write your proof out carefully in the one species case and then things continue to work in the Many species case so that does need to be checked on a case-by-case basis Okay, so that's the full system and Now I'll give you a very incomplete and very short list of previous results for this equation So it turns out that not many different Families that people have worked on this equation. So there's a school There's actually a couple of schools coming out of a couple chains of schools coming out of France a One are really School coming out of the US. There's and then there's another school coming out of Germany and That really win you and then there's some new papers by some people in Canada but That really encompasses most of the mathematical work on Lots of Maxwell's is not the field where you have thousands of papers And so you have a different leon's a glassy Strauss and then glassy Schaefer Strauss Was the advisor of glassy and then glassy was the advisor of Schaefer and Gerhard Brine comes out of This is a German school And actually I'll tell you about a lot of these papers later on so let me not mention them in too much detail You also have results on glass up for some I Won't really read the names because in a upcoming slide. I'll tell you more about these people and what they did Okay, so This is lots of Poisson and Studying scaling so scaling is a huge tool in a linear partial differential equations For relativistic lots of Maxwell. It's hard to really Effectively Find a way to quantify. How do you scale in? In particular because if you legitimately rescale of lasso Maxwell you get and you you get less of Poisson So you get a completely different equation and the equation is not In some in certain senses equation is not really you in scaling variant although you can that's not There are things you can do like consider a massless Relativistic lots of Maxwell. We're not doing that, but I'm just I just wanted to make the point that It's not completely easy just to derive heuristics from scaling for relatively simple as a Maxwell because When you rescale you get lots of Poisson and here is lots of Poisson So you have DTF plus now you we change P to V So actually most people who study Lasso Maxwell mathematically also use V I happen to prefer the P notation because it comes from physics and then you can differentiate between the variable in V and The variable in P. So V is a three vector and you get V dot grad X F plus gamma E dot Grad V F equals zero and so this so they're sending the speed of light to infinity kills the magnetic B term and You're only left with the electric e-term and here you have gradient X dot e is equal to row and e is equal to minus the gradient X that phi where phi is a potential That I'll tell you more about and then you will you continue to have the same Charge except we change the notation. It's the integral of a DV and then you continue to have the integral of FTV Okay, so now in comparison to Lasso Maxwell where the large data problem, which is the big open problem in the field all of there are others That's really the most visible one For that problem, it's open for Lasso Maxwell for plus a Poisson the problem has been closed since the late 80s in the early 90s and you have this paper by Lyons per Sam and they can prove a global classical solutions via the propagation of high Moment moments in the whole space And this is really non relativistic And then at the same time roughly you have this Fafel Moser result from 92 and Schaefer its simplification of Fafel Moser's proof from 91 You have to explain why the simplification was published before the actual paper, but it was They had he has global classical solutions via bounding the momentum support So Fafel Moser is this a very interesting guy who came out of the German school that I told you about before He's this this paper was his a PhD thesis and it took a long time to publish and He wrote very few other papers and he's not in math anymore and Schaefer Simplified his proof. This is it's really much shorter and he acknowledges that he was simplifying Fafel Moser's proof Just because of the speed of the way things get published his paper was published first So the Fafel Moser Schaefer theory is a compact support compactly supported initial data and the Lyons per Sam theory is non compactly supported initial data the So that's a real advantage of non compactly supported initial data on the other hand As far as I know it uses their proof uses dispersion in a fundamental way and It's still unknown Whether or not their proof can be generalized to other domains in the whole space Whereas the Fafel Moser Schaefer's theory works in say the Taurus and the whole space Okay, so another thing you can say about so these These are large-shaded theories work In both the stable and unstable case so both for gamma equal plus one and minus one On the other hand, you have blow-up when gamma equals minus one and then you can do that roughly Okay, so you can have blow-up and global distance for the same model in the same spaces This blow-up when gamma equals minus one is actually for the relativistic of Lasson Poisson is not for the Newtonian plus or Poisson that you see here So you replace this with V with P hat P over P zero and then you have Old very rough results of Horst and these are explained very nicely in glasses 1996 book and then you this is These are these are more of existence proofs of blow-up so the point is that one gamma equals minus one the the energy of this system is negative and You can calculate the energy that's negative and you can use that to prove that the lifespan of solutions is finite and then in 2008 Limoume has Rafael proved a stable self-similar blow-up for Relativistic Lasson Poisson and for a massless scale invariant version of relativistic Lasson Poisson, so they have these very nice profile decompositions like you saw in the earlier talks of from the first few days of several people and Then for this model another thing you can talk about land out damping You have the big initial proof of mohovalani from 2009 and Then there was a quiet period for a couple of years and now there's an explosion of works on both Land or damping for the Lasson Poisson equation and for an ore damping for dirty fluids you saw Professor Masmoudi series of talks on that so I won't really yeah, and there is a like a huge number of people who have worked on that both before and after Mohovalani so I won't instead of just mentioning a few names. I will not mention any Okay, so this is a Lasson Poisson Actually, if there's time I'll come back to this later. I don't know if I will But okay, so now I'm going to return to relativistic Lasson Maxwell with in the stable case with Gamma equal plus 1s. I'm just gonna show you the equation again so that You don't forget it Okay, so how do you study this? Well one thing you can do is derive a conservation laws and it turns out that like many wave equations There are a lot of conservation laws for Lasson Maxwell. Some of them may be more useful than others Impartee or some of them may be not useful at all, but it turns out that There's this old paper from I think it's 1984 of glassy and Strauss where they derive a huge number of conservation laws for Lasson Maxwell and Many of them have they never been used Or people don't know how to use them but so So normally you think that all of the conservation laws are known by the physicists long ago, but as I looked actually I looked in a lot of the physics literature and I Didn't see many of these conservation laws I don't know if they just knew them or they didn't bother to derive them or what but Many of the conservation laws that you find were that the first place that I found them was in this 1984 paper Of course the big ones were known forever like the energy but so you have this This equation for the energy density you have E you can define E to be 1f e squared plus P squared plus 4 pi The integral of P0f and then you take a time derivative of this and you can see directly from the equation that this Is equal to a divergence of E cross B Plus the integral of pkf dp as you see up there. So this is an exact equation that can be derived and okay, so if you have sufficient decay conditions, then you automatically get the L2 conservation of E and B and R3 and the P0f conservation of the BDX Integrated against the BDX this is equal to constant the constant is the Initial data of the similar terms This is just easy integration And then this term as long as you can get this term to disappear by integration by parts Then you're in good shape Okay, so you have an you also have another much less well-known conservation law that's It's actually not a conservation law. I'd say a quantity that remains finite by the Propagation of the equation, so I'll tell you About this you have I call this the good conservation law and This and this is it so Wow Here CTX is the backward like cone emulating emanating from TX This is what you get This is the domain that you integrate over when you integrate the when you invert the 3d wave equation with zero initial data it's just It's the cone not including the interior so you integrate from zero to T and you integrate on the Exterior of the cone or the absolute value y minus x is equal to t minus s and then you have the outward normal to the two sphere which is the unifector y minus x and then you can So what you do is you take the previous Equation that I wrote down for e and you integrate Against the whole cone not just the exterior and then You use the divergence theorem and you integrate my parts and you get this and you do a complicated two-page calculation and You get that these Quantities are positive and bounded above by a constant which depends on the initial data minus Another concept which depends on the solution, but it's also positive, so you can throw it away and It's not bounded by anything So you don't want to keep it here But you get e dot omega square plus b dot omega square plus e minus Omega cross b square plus b plus omega cross e squared Then you integrate over this domain And then you get p hat p zero plus one plus p hat dot omega squared f dp d sigma and This is a beautiful conservation law. That's a very useful in several contexts First I know it's only been used in four papers in the field Maybe five if you count two of them as one or one of them as two excuse me But it's not perfect because two independent components of e and b are missing So it does not control all of e and b on The backward light cone But on the other hand many terms as we'll see coming up many terms to estimate Involving integrals over the backward light cone which make will make this conservation law very useful Okay, so I can't explain that yet, but you'll see eventually So also there are characteristics for the Vasa Maxwell system and they're written down as follows I use a capital X and capital V and capital V hat where V capital V hat is just V over the square root of one plus V squared and the characteristics are the X of t equals dv equals v hat and dv of t is equal to e of evaluated x of t v hat cross b evaluated x of t and We abbreviate x of t and v of t for simplicity as x of t as xp and v of t as xp and the characteristics really depend on all of these things And you have what are not exactly initial conditions, but like initial conditions. Let's say that When you have x of t t xp and v of t t xp Then that is exactly equal to lowercase x and lowercase p which are just the variables This is basic characteristics 101 stuff But we will also use these forward characteristics That's when you start with s equals zero and you go forward in t and the backward characteristics That's when you are at t and you go backwards to zero Then with this notation of forward and back backward characteristics for a solution to Vasa Maxwell It turns out that you can express it as the initial data F of t xp is the initial data Evaluated at capital X and capital V. So the you can you also do things like this for Like 2d Euler in workicity formulation and things like that So you can simply express the solution as the initial data, but it's evaluated at the backward characteristics So all of the information about the dynamics is hiding here in the backward characteristics Okay, so it turns out that the Jacobian of this x v transformation particularly at tt is the identity matrix and Then when you try to calculate the dynamics of the Jacobian You see that it doesn't change I won't actually do that calculation, but it can be done. So at least formally presuming you have a nice enough solution then the This change of variable here the these equations here when you further differentiate them with respect to x and p and do some clever calculations you get that the The Jacobian determinant is the same as it was initially and so it's just an identity so this these characteristics are formally assuming you have a nice enough solution so-called measure preserving and when you have it when you have this expression here for f and when you have the these sort of lq spaces you just plug in the You can you take the f of lq norm you plug in the expression here you apply the change of variables That I just mentioned and the Jacobian is 1 and so the every single lq norm is a conserved This is another collection of conservation loss Yeah, well because well because what's divergence free are you mean? Yeah, yeah, this is true for a lot of right Yeah, you can say it that way I mean somehow you don't need to come when you don't need to do any calculation You just say it's divergence free so it's conserved Sure sure Yeah, there's yeah, absolutely Okay so either because the vector field is divergence free or because you can calculate the Jacobian exactly and See that it's again. It's propagated to be the same as the initial data then you get that these These are measure preserving And the most popular conservation law in this business is often the L infinity conservation law, which says that The L infinity norms are just bounded Okay, so you have some more conservation laws that are used less That I'll tell you about quickly So since the characteristics about Lassa Maxwell are measure preserving For a almost arbitrary F, but it's just a function of all of F then the DX and the DP norm of a Of F is a conserved just the same exact way that the LP norms are conserved So this is Another conservation law for all of these of Lassa systems, not just for Lassa Maxwell or Lassa Busson and lastly there is a Relatively less known and less used Morowitz type Inequality so you there's a lot of morose types inequalities for this. There's this is just one so if you integrate from Zero to infinity of time and you integrate over X DX over X cubed and you get E dot X square plus B dot X squared and then you write p hat dot X squared F integrate against DP just for this term and that's bounded I think people were really happy to have this conservation law when they when it was found but As far as I know it hasn't been For in this business it hasn't been used. Yes You would like to have things like that I mean you so there are a lot of Bad things that can happen in plasmas. You have a lot of steady states You have these a BGK type equilibria, and you have other things and you have So you can't So you have to be more careful when you try to talk about a statement like do you have scattering or not? But yeah, I mean formally you expect Once you know the Global regularity then you expect to be able to say a lot hope to be able to say a lot more things and I mean you do have Several small data theories that I can tell you about and you have decay and you can work on problems like scattering for those and things like that but No, I mean as far as I'm aware they don't really use this Conservation law yet. I mean it's something you can try to do in the future Okay, so now I'm going to tell you about a few different theories Solutions actually I'll tell you about three the first one that I'll tell you about is the 1989 the apparently on theory of weak solutions For this one, I'll be brief. So if you So for weak solutions you satisfy the equation in the sense of distribution So you multiply The equation by a test function and you integrate by parts and you put the derivatives on the test functions All the test functions are compactly supported So you don't really worry so much about what happens at infinity and then you have the following theorem so if you assume that initially the L infinity norm is bounded and you assume that the L2 norms of EMB are Bounded and the first moment of F is bounded in the P and the X integral then You can have a weak solution for arbitrary initial data And so given So for these solutions, you know, it's the equation that's satisfied in the weak sense and the equation is known to be a positive and it's known to And it's known to be satisfied for almost every time and you have For almost every time you have that the conservation laws Continue to be finite So that's good The big open question for weak solutions is of course the uniqueness and the long-time point-wise behavior Of solutions and that's it's not just a big open problem here. It's a big open problem for many theories of weak solutions Okay, so that's the weak solutions The Bernard Lyons did this in 1989 for the non-relativistic plus a Maxwell system as I said a few minutes ago For weak solutions the difference between a relativistic and non-relativistic is not a big deal Because you have a compact you're multiplying by a compactly supported test function and so you're basically working in compactly in The compact support setting but at the same time Ryan wrote a expository paper in 2004 with basically the Bernard Lyons proof and he wanted to sort of Revitalize the Subject and bring new students into it and he explained the proof in the relativistic case Okay, so I Can quickly Give you a one-slide explanation of how this works So you regularize the relativistic plus a Maxwell system you write down a different system of equations That happens to have the classical solutions That are smooth in some sense and you call that system F and E and B and then it's Satisfied for every end and you prove that for the regular a system you have a uniform balance on The conservation laws this gives you control to take limits and Then you have this averaging lemma This was really a key contribution at the time There were a lot of different averaging lemmas and but this was the one that they used so if you have a Solution h to the following linear system dth plus p hat that grad xh is equal to g0 plus gradient g1 where This if this set of equations is satisfied in the sense of distributions Where you only assume that hg0 and g1 are L2 functions where you will they only need to be L2 in the V variable in a compact ball Say center at the origin, but you don't care, but there are L2 in the whole time and the whole space then when averaged against a test function You go from L2 to h one-quarter roughly in T and x so this is a This is this is basically proved by using the Fourier transform it's sort of in some sense a Related very loosely to the end of damping or war damping type of phenomenon where you have this transfer of Fourier modes and Gives you a little bit of regularity and Then once you have this h one-quarter regularity, then you have Enough compactness to take Limits strongly in some sense and the the impossible term for Plus a Maxwell is this one so this is a nonlinear you have Ian the Lorenz fourth term times fn and normally These conservation laws give you a weak convergence, but we converges and not closed under products But this averaging llama gives you strong convergence and so Now you can take a you can take a limit here because you have some strong convergence That is the three-minute. I don't know how many minutes it was about that is the short Outline of the proof-of-weak solutions. That's all I want to say but since I want to give you a picture of the whole field Decided to explain it So okay, so another type of theory that you have is generic global solutions This was actually an old paper of Gerhard Rhein. I believe it was his PhD thesis in recent years you've had Papers like this for Navier Stokes, so I thought it'd be interesting to tell you about this old paper for plus a Maxwell So Yeah So you're so Consider initial data f0 with compact support and easier B0 RC2 These are the standard like all the assumptions that people used to make in this theory and then You assume that you have a global Solution that satisfies these Bound conditions so r0 is really it's not on the slide, but it's the constant that gives you the support of This is the bound on the support and then you assume that initially the derivatives of the fields have some decay and For example, there are other assumptions you can make but you assume that the k and the gradient of k where k are the fields You assume that they decay a little bit in time and they that they decay in The space-time cone in this way then Under if you have a global solution then satisfies these bounds then you have an open set of global solutions so Ryan So for such a global solution there's a small ball of initial data that leads to global solutions satisfying the same condition So given any global decaying solution you have a ball of global solutions So you have generic global solutions in this sense Yeah, so there's similar maybe from results for a 3d numbery stokes Was it a Gallagher and plancheon Yeah, who did this to do some Completely different result for a completely different equation, but they proved generic global solutions for now for 3d numbery stokes So, you know that you have an open set the set of Gallagher if to me plancheon prove this for numbery stokes Okay, so that's another type of result that you can have I won't Basically the 30-second summary of that result is you use the so-called ST decomposition Originally given by glassy Strauss and you prove estimates So that's all I'm going to say since I'm going to tell you about the STD composition later I'll just won't mention that and I don't think I'll have time anyway Actually, I plan to use slides and then Talk on the chalkboard, but I don't think that's going to happen My I'm making an effort to go through it slowly through these slides for students And the chalkboard stuff who have to come tomorrow. Yes, but so this is Relativistic glass of Maxwell for large velocities. This is the precursor slide to glassy Strasse in 1986 of continuation criteria So you assume that the energy is fine initially You assume the L infinity norm is fine and initially and you assume compact support So there's soup in P such that there exists x such that the initial data is non-zero of its finite That says there's this support of the distribution function f is finite And so you have a C1 with compact support initial data and you have C2 fields and the fields satisfy the constraints and Then you have an unique solution to relativistic glass of Maxwell in on a time interval Those are the assumptions then if you assume that the support that the distribution function is zero For large momentum For all times so you assume there exists a band of continuous function cap of t such that this The distribution function is zero for all times are up to time t actually the way I'm Stating it is up to time t then the solution can be extended beyond t and if this kind of assumption holds for all time then the Solution can be extended globally actually is people don't really talk about it that much today, but there's a it's on the slide So I'll just explain it. There's a Technical condition in glassy stresses proof that says that you're so to do this you approximate glass of Maxwell and because they are using these Ck function techniques and things like shouter estimates and these Techniques that people don't use that much anymore, but they are very popular at the time then they have to assume that the every approximation So also satisfied the same compact support assumption But if you use modern techniques like a so belief spaces and energy methods Then you can remove the assumption eight on the approximations by for instance working in h5 Or h4 or some others HK Okay, so this result was Originally given by glassy Strauss in 1986 and they introduced the s teak the composition of the fields Which allows you to decompose e into an initial data term e zero and e s and e t And then the the game is to estimate e s and e t successfully and then you also In 2002 you have the Clinton and Staffelani approach where they use the Fourier analysis They took the Fourier transform of the whole equation and they wrote down their own Fourier space representation of the fields and they use Fourier based methods to prove The estimates, but then at the end of the day they get a similar result and you also have the Boucher-Gauss-Pallard So these two papers are basically at the same time. I don't actually know the sequence But these are the publication dates, I believe And they have a third representation of the fields They express the fields EMB in terms of distributions of linear a wide chart presents Potentials and they use what they call a division llama I'm not really going to talk about Either of the last two representations at all in my course I'm only really going to talk about the STD composition So I can explain why right now the the idea is that We want to use the conservation laws as much as possible and Is the Fourier based approach sort of kills these complicated conservation laws like the good conservation law and it may be that someone Can come along and figure out how to use them using Fourier methods, but number one we don't We use harmonic analysis for sure we use it quite a lot, but we don't use the Fourier transform because it hurts the conservation laws and We don't number two. We don't know how to do better. So you can certainly use harmonic analysis You can use the Fourier transform and try to Split up into terms where you don't use the conservation law in terms of what you do But we don't see an advantage to doing that We haven't been able to get better estimates that way And so here when you say conservation laws you mean conservation laws in cones. That's that's the point Yes, the conservation laws in the cones, right? Right Right, that's that's right. It's hard to think how to adapt the Fourier transform to that We don't know how to do it and When you do it you get a big mess, so and on top of that For all the things that we're doing the harmonic analysis in the physical space has progressed to the point where I Don't know what the advantage of doing that is so for the four of the types of results that we're proving There's certainly great advantage to use it before you transform for other Things that people do but for this equation for the types of theorems that we prove it seems like I Don't know what the advantage would be Okay, so let's finish by talking about wave structure in Vlasov Maxwell. So again, you have the equation so this is a wave conference and the It's a wave semester and the reason why I decided to talk about Vlasov Maxwell as opposed to something else was because it has a wave equation hiding in it and here it is so you have the Maxwell's equations I you you take a time derivative of E and you throw it onto B and J and then you plug in the Equation for B and so you get minus the double curl of E minus J and then you just have simple vector identities which gives you that the Laplace N of E minus the gradient of the divergence of E from the double curl and then you have this partial T J hanging around and you know that the Divergence of E is equal to rho so you end up with the gradient the Laplace N of E minus the divergence of rho minus D T J and similarly for B you have the second time derivative of B put that on E you plug in the related Maxwell equation and you get the Laplace N of B minus the gradient of the divergence of B minus this curl J term and the Divergence of B is zero there So you get to so this is the wave structure for max Vlasov Maxwell and for now I just get me equal to plus one So a key idea in all of these studies of Vlasov Maxwell that I mentioned is to write the electromagnetic field as wave equations with complicated Forcing and I wrote out carefully a complicated forcing on the right-hand side So you have the integral of these integrals of Dp And now you want to invert the wave operator and you want to prove estimates for EMB but It appears just naively from looking at this that you lose regularity and that is a nightmare and it's would be fatal if you really lose regularity Fortunately, you don't but there's a price to pay for not losing regularity So to prove existence for all the continuation criteria the goal is often to use this representation to get good estimates for EMB and by the basic control that we have is in terms of the previous conservation laws you have other control but for the sake of presentation, let me just mention that and The goal is the one big question For now and for the future for people want to study this in the future. Maybe students is Do we have is there a more useful understanding of the structure of the right-hand side terms that you can use to get better estimates? Right, so as I've said the right-hand side involves derivatives, but the conservation laws do not and this makes the Forcing look extremely problematic, how do you integrate by parts to remove the derivatives? That's a big question and in 1986 class these trials introduced a decomposition that we call the ST decomposition So you write s as dt plus p hat grad x and then you write t as grad y minus omega partial t and omega is just the normal vector y minus x over the value of y minus x and using this decomposition you can Successfully remove the derivatives on the right-hand side So for one you plug in the blast of equations So as of f is equal to minus e plus p hat plus the Lorenz force dot pf and then you can integrate by parts in p and Then you can use the divergence theorem on the t term after inverting the wave operator Okay, so that was a bit too brief. So let me explain it in more detail. So The main observation is that you can so if you are here and you have something like the gradient x This is this gradient p term in this p term you can directly Take the gradient p term out because it commutes with all this stuff And then you can just integrate by parts and get rid of it but for the gradient x terms Even after inverting the wave operator you can do that So what you do is you decompose The x derivatives here I call them y derivatives, but they're the same as this operator So it's omega i over 1 plus p hat omega of s plus b i j omega comma p t j Where t and where b i j is this operator delta i j is the standard Dirac. That's one when i equals j and zero elsewhere and then you have this omega i p j of a 1 plus p hat omega and You have this decomposition and as I said s when you have s you can plug in the velocity equation And integrate by parts and the p variable and when you have t you can integrate by parts on the wave cone So s is the operators and t are the tangential operators and you can prove that b is bounded by p zero squared and That's what you get. So I just said that before it were on the slide Okay, so this is the philosophy And this is what you get I won't do the calculation. It would be quite horrendous to do so But then just for the e-term you get You can decompose e as e zero plus e s plus e t where e zero is a integral of the initial data that I won't write down But you can just estimate it using whatever assumptions you make on the initial data and e i t and e i s are these formulas up here. So you Have c t x this integral which is what you get after inverting the wave operator And then you have these kernels h e t times f dp sigma over t minus s squared So that's a more singular and then you get this yes term which is h e s I j k tilde i f And k tilde is the Lorentz force and I think it is the normal again and then you can write down exact expressions for h e t and e t s And They are right here. So I want to show them to you exactly because they are important to look at Because this is a big part of the problem and And So this is h e t and it's 1 over pi dot omega squared on the Denominator and then you have omega i plus pi squared in the numerator and one minus p squared in the The other term in the numerator and then you can split h e s Into this complicated monster. I won't read it to you So this is the problem is how do you estimate this is the part of the problem is how do you estimate these things? These are point-wide expressions and you have a similar thing for b except it's more complicated. So e i t is Linear in F and e i s is nonlinear F because you have the Lorentz force and the F also Okay, so it turns out that let me just write one thing On the board. I hope you can see it it turns out that You can prove as to mess like this So the big problem in this business is that or one big problem in this business is there are many This is not the only one is that You get these singularities and these singularities grow like powers of the P and In the worst case that's the worst estimate right there And so if you just go through and count everything and you ignore the numerators then it looks like You have a p zero squared squared term from h e t and so so and then Here it looks like you have a one over p zero and here it looks like you have p zero cubed so I'm just counting this Denominator and ignoring numerator and that's that's This kills everything. So if that if that were actually correct Then you'd just be dead and throw away this equation or not really by you. There's nothing you can do because These this growth is Is just so if you're you're bounding F By you use the characteristics and you're bounding F and derivative of F by E and B and then you're bounding And F itself and then you're bounding E and B by F and E and B again And you do that through These expressions and so then in addition to all that you're losing powers of P Then you're dead So it turns out that We're still dead because we don't know what was this into the system, but We're not as dead as this because there's a lot of cancellation in the numerators That you have to study carefully. I was going to show it to you, but I think I'm out of time but so these all of these numerators Decay have cancer have Can't have a certain amount of cancellation in terms of the denominators and that really Gives you a little bit of hope and in addition to that this H s term can be split into a piece that Is better and And the piece that is not as good but can be bounded in terms So you have a you can split ES in terms of a good tier a good term that can be bounded in terms of the good conservation law and Another term that's not a singular that can't be found about a good conservation law. So that's what you do basically you bound ET directly point wise and then you do more advanced things and then you split ES into two terms One is nicer and the other Is Not as nice but boundable by the good conservation law Okay, so let me have more stuff to talk about but I have I can talk about it later. Oh Okay, so then I have 12 minutes, um, where should I think where should I go? Okay, so Time to decide what to do in 12 minutes Okay, let me explain Can I stop using the board the slides? Okay, so I want to tell you about How I want to give you an idea how to prove the classy Strauss continuation criteria because it's instructive to a lot of what's done in this field and There are at least in terms of existence, so let me just write down the equation Okay, so since it's so basically What I'm trying to do is give you a flavor for how you prove the glassy stress continuation criterion without all the headache so Then we have less of possum and I'm going to do this backwards because I don't know How far I'll get but I will Get far enough so that you have a picture So local Okay, so, um We're gonna start here. So you have last so this is What I claim here You can't go look it up anywhere except for in a set of lecture notes that I gave to my students in the class I taught last year but you have less of possum and but it's it's not like original at all either it's Anyone any expert who understands how to prove local existence in silver of spaces and understand lots of possum can do this So this is lots of possum that I told you about you can prove local existence in HK and you can prove that the solutions are global if the integral from sorry This is not really precise. Yeah, you can prove that this I mean even this way So I have to write less You can prove that the solutions are global if this Integral is finite you can prove that Solutions can be continued beyond capital T if the integral from zero to T remains finite and You so you need bounds on the elementary norms of e the gradient of e and the gradient in x and v of f times a weight Let me not tell you what the weight is actually I can it's not so bad The weight is p zero natural log p zero to three over two Natural log of p zero. What is the next board that I should use? The first one. Okay. I remember in someone else's talk. There was a preferred order so Okay So this is what I want to prove No No, you have locally. This is large data large data for blasts of Poisson is well known and this this is how to Continue a solution If the support is compact so if you're Yes Yes, oh, I Didn't write that Yes, the integral is in time Yes, if the integral remains finer then you have global existence And then Q of s is one plus The soup so that Q of s Yes, also in what was the question Oh Technically, no this local existence theorem Does not does not require a comeback support but so Okay, so then this is Okay, so the theorem says that if Q of capital T is finite then a of s is in L infinity So in particular it's in L1 and if T is infinity then you Then you're in oh Yeah, there's a little ambiguity because if you're now infinity You don't know that you're in so if that's finite then this is No Okay, okay, so then My understanding that there is one direction, which is like let's say trivial because if that one is finite Propagation, you know that this will also be I mean, I think the points really yeah, this this direction of the last the second one This implies this You think this direction is the important one or the other one? Yes. Yes. Oh, okay So if you're compactly supported and you know that These things are finite Then it's automatic that you remain compactly supported by the conservation laws Yeah That's right absolutely so the If initially you're compactly supported and you have this direction then you have both directions Okay, oh That took me a long time so Okay, so I'll say one more thing and then I'll finish This just gives you a flavor of the kind of thing is that you do okay, so take a differential operator either being gradient x or gradient v and then Differentiate the equation then you get a then you integrate along the conservation laws So you get no there Yeah Yes That's right. Okay, so if you so you you can differentiate the equation like that and then you integrate along conservation laws and you get an expression like that and then You bound it and now you integrate in time along s and you bound everything and what you what you end up with is Okay, so this is roughly speaking what you get and now You prove estimates and you use the ground walls inequality So there's another term Okay, so you get expressions like this and now you use these are the L infinity norms and Now you have to prove a hard estimate for e and that depends upon the support and Then you use grommels inequality and conclude that you're fine. Okay Thank you. Yes, you mentioned this continuation Result by glasses throws. Yes, if I won't remember it implies some global existence in for small data, is that correct? Yes. Yes. Oh, there's a idea actually right. I only mentioned the Generic results of Rhine just because I only want to mention one, but there is a huge collection of small data results for Lessa Maxwell and 3d in particular. Yeah, okay