[Music] okay okay concerning the the the original lecture date so this is an exception today because for some of the reasons i had to be absent from erlang and could not make it this morning so we shifted it to to this date so usually we have our lecture on thursday in the let me phrase it in this way in the first slot and my proposal is and i think you will all agree that we start at age 30. is this okay yeah later is not possible because then we will not make it up till 10 but we have this possibility and i think okay so so from next week on 8 30 in um [Music] um three so okay uh i will do something with club assistant okay i will do something in this lecture which i usually not do uh i i totally deviate from the mode from from the description of the module because uh this description of the module at least for my for my point of view is far too ambitious and also does not hundred percent fit with uh this uh level you have reached meanwhile so there are much more up to the research topics up to the research line so maybe this is something you can do in the third and fourth master semester and did we do something more basic so i have decided to as i said already a little bit to uh identify a few subjects we would like to cover so the first subject where we know already something about it but not about the analysis of it so the first subject will be time dependent time dependent pdes we have seen those models already so we have seen all the uh time-dependent navier-stokes equation of course also time-dependent stokes equation if we go down then to the energy conservation we have the heat equation and more general versions we have the diffusion equation so this is one class so the second order equation second order in space equations we want to deal with and then of course there's these other class there is the first order in space equation the euler equations the wave equations and so on and okay the wave equations it does not come in the form of a second order in space but the second order in time second first order in space but second order in time second order in space but can be rewritten as we have seen as a system of first order equations so the aim of this first section is to look at procedures to show existence of the solutions for those linear problems so first we are still in the in the realm of linear problems which makes the world much simpler then as a second part we will see how far we will come there uh based on that to some extent as i said we will look at the navier stokes not fully of course and the euler equations so here and then of course the the the problems become more pronounced we have not we have systems we have nonlinear equations and then this is so to speak the first block dealing with time dependent equations and the second block going back to stationary equations but now really looking at the non-linearity so the key word would be non-linear elasticity so we have touched this a little bit in the sense that we have derived such models with hyper hyper elastic material for example but we haven't said anything sorry sorry sorry sorry this is number four this is now the application so we start with number three so maybe you have heard already this keyword so what we have to do a little bit is we have to do a little bit calculus of variations so what does that mean we have seen that a stationary problem can be written either in the pde form or in the form minimize and energy and of course we can tackle both forms um asking ourselves something about equations for example the existence of a solution so we can either say okay we have let's go down to the most simple equation the the poisson equation we can either look at the poisson equation directly at the weak form say okay this is a bilinear form what do we know lux milgram can we check the assumptions and so on that would be one way to go with that and the other way what to say would be to look at the energy functional itself we have a function with directly integral to minimize and what do we know of course you know already something about what our conditions for existence of minima or minimizers in the finite dimensional situation i hope and now but now we are in infinite dimensional spaces and we have to have similar uh similar results and considerations and we'll go a little bit in the direction you know if we have in in finite dimensions or in one dimension if we have a convex functional we can be sure that we have a a minimum or a strict convex function we have a unique minimum which we try to to generalize these notions to not only to multi-dimensions but only to also to the infinite dimension situation so that is something here and then finally the application will be non-linearistic and if time permits i'm not 100 sure about that then number five but this is maybe something we can only do very shortly these are then so-called free boundary value problems so these are problems of the type they basically uh appear for example in phase transition problems where you say i i have a boundary value problem or initial boundary value problem in a time dependent case but a part of my boundary is unknown it's it's part of this of the solution to find and i have a second boundary condition to cope with that so typical examples will be models for phase transition between liquid and solid for example where this separation line or hypersurface between the solid and the liquid part is a priori unknown so this uh so for the last part if we ever will come to that i can i will go back to the modeling book by um howard christopher me but these so these more strictly analysis oriented parts i said i will orient myself at least in the notation to the book of ben schweitzer because they that's the only book which i know which really covers everything in one book but uh we will also give you other textbooks where you have to switch the textbook so to speak from chapter to chapter you can do this but it's it's basically very standard stuff so we start with uh what we will do here what we will do now in section one is um the what is called the energy mod the energy method for the heat equation so first of all we start with the heat equation so with this very simple equation so let me write it down so we have an unknown variable u we have the heat equation time derivative of u minus laplacian of u equals and now we do it as simple as we can do it we make it homogeneous we have this in the space time cylinder so omega is a bounded set so typically at least bounded and open but it has a little bit we need from time to time a little bit more property and connected we need a little bit more properties and then we have this time to interval so all our variables are dependent and let's see in which direction we rewrite that we will see that later on we have always an x and in t as the independent variables x gathers all the spatial coordinates so x is two co has two components three components and so on and then in addition we have this simple time interval so and as i said i will use exactly the same notation as a numbering as we have it in the book of schweitzer and of course we need a little bit of boundary and initial conditions and we do it here again very simple we take directly boundary condition homogeneous so on the time mantle of this cylinder and then of course that at least something is happening we need an inhomogeneous initial condition so we need something like u equals u 0 for t equal to 0 x from omega so that would be an informal notation of the first equation we are starting with of course you can now say why do we do that we have delt already with this problem we know what the solution is we can give the solution at least even in a let's say explicit form as we know there's the fundamental solution this in some sense the solution for the heat equation with a direct distribution at the point x equals zero t equals zero and by uh convolution of this solution with the initial condition we can get a solution of this problem this is what we did already why we are not we are not satisfied with that because this approach or this knowledge as nice as it is cannot be extended it's really we are stuck to the heat equation so what we would like to do is we want to have here a more general equation we want to have here a general elliptic operator with say at least space dependent coefficients and then we cannot write down a function then we can maybe infer that there is a fundamental solution but we cannot write down it anymore in the way we could do it up till now so what else can we do what else have we done so so up to now [Music] representation with fundamental solution the question is the question is can this be generalized we have done something different which is in some sense not so bad also it's a classical procedure we have done separation of variables remember what this was separation of variables it was it was an approach where we said the solution depends if we have a special form of a solution it is a product of a function only depending on time and only depending on space we plug it in and see what comes out what will come out and what came out is that those space space functions have to be eigen functions for the underlying elliptic problem that would be this here that would be the underlying elliptic problem and then of course the question again is can we write those down and that depends so this is now the elliptic would be the elliptic problem and if the domain is very simple in one space dimension on a rectangle and so on we can even write down those eigenfunctions and compute explicitly the eigenvalues in a general case we can infer which what nature the eigenvalues have they always are countably many they are positive and they have an accumulation point at infinity only so we have a minimal value from which minimal possible value from which we start so this we can infer in general and then what are our approximate solutions so oops and then what we actually do is we set up approximate solutions not solutions of the full problem but only uh functions which approximate the problem and the approximate solution u n let's say could be here something like uh yeah what how would how would it look like it would be something like we look at the sum of the n eigen functions so v i of x this would be the eigen function and then uh we know uh [Music] what uh how the total solution uh of this com with this component in space behaves namely then we reduce the whole thing to an ordinary different to the most simple ordinary differential equation namely time derivative of the function plus the eigenvalue time the function is zero so the solution in time is a decreasing if the lambdas are positive a decreasing exponential so we have here something like e to the minus lambda i t so and now we have still to fit with the initial data so here what would be the initial data uh if we now plug in t equal to zero we would have to adjust the sum of the eigen functions that would in general is not the id initial data so what we still need are some coefficients and these are the fourier coefficients fourier coefficients the first and fourier coefficients of u 0 and only if we in some sense here make an infinity out of this n then we would have the solution but then all the questions comes up in which sense does this converge and so on and so on but what we can maybe keep in mind here that what we are doing here is we are doing we are first finding approximate solutions and then we have not worked this out and we will also not work this out this procedure in total then we have to deal with the questions how do these approximate solutions converge and is this limit which comes up in which sense is this limit then a solution of our problem so this could be a procedure to do this in general so the procedure will be we approximate the problem such that we can we come back in the approximation problem to something which we can solve so what can we solve we can solve linear ordinary differential equations here even explicitly that's the way we did it here um that is what is then called the spectral methods and can even you can even make an a numerical method out of that so what this means we approximate the problem in space and keep it in time as it is and solve these problems which are approximated in spa in in space continuously in time this gives us our approximate solutions and then we have to see that those approximate solutions converge in which sense so but that is a procedure we will do then later on before the for the wave equation um would could do it in the spectral form and could also do it in a more numerical inclined form and say okay what i do is i discretize in space for example with the find it element method and leave the time as it is by this discretization in space i then get an uh here also linear because we start with a linear problem in general nonlinear system of ordinary differential equations i have to know that i can solve those and then i have to see how this limit in the n which then is the number of nodes or the degrees of freedom in the discretization works so that would be the approach discretize in space leave the time to get the uh approximate solution what we will do here is the is the other way so he so here what we the the what we did here is as i said approximate in space continuous in time and this leaves us to this this leads us to odes systems of odes so here the alternative which we now proceed for the heat equation the alternative is just the other way around discretize its time discretize in time continuous in space so what we always are doing i are using here that these are not just n plus 1 or k or d plus one variables that there's a strict distinction between the spatial variables and time they have totally different physical meanings so what does that mean what kind of problems do we get here so maybe we do this first so to get a glance let me first do this to get a glance and then we will do a little bit of technicalities before we come back to that um let me first write down the discretization that we see it as it is so what we will do what what means what does that mean discretize discretizing in time so what we do is we have we we have a set of discrete values in time of course we start with t0 and then we have t1 t2 they may be equidistant but it's not really necessary and then finally we have the final one that is t capital n so the number so n plus 1 is the number of our discretization points in time and that is the capital t so uh okay we make it a we we we could leave it in this general form but we make it equidistant uh for the for the for the analysis consideration there's no restriction if you really do this as a numerical method and actually it is a numerical method then it would be would make quite some sense to do different time steps so equidistant is means that the tks are just k times delta t and the delta d then of course is the capital t divided by n so we have this time these time steps in in in the time interval and what we then do is we substitute we substitute the time derivative which we have so we have approximate solutions we're looking for approximate solutions which only live at these time levels so we look for uk these are functions of x i leave it first informal not specifying any spaces and what do we want to have we want to have a time derivative okay we have something discrete we cannot write down a time derivative but we can write down a difference quotient so the third derivative is substituted by the difference quotient and now it depends from which side you look at that if you look from uk you are so at the interval going from tk to dk plus one that would be the forward difference quotient here and now we have to decide uh where to take our spatial part and a possibility or the numerically not the too bad possibility is to say we take it at the end of the interval okay why is this now what is this now of course this is now in omega and we have we have our uh boundary value boundary uh prescriptions as we have it so here we inherit the dairy clay conditions and what do we have here now for the for the uk so now we can argue in the following way so at t equals to zero so for k equals to zero we know the solution given by the initial data so u 0 is just the given initial data so if we are now at some time level going from tk to dk plus one where we already know the uk so this is now known in every step then we have to have here a problem for the uk plus one but what is this this is just an elliptic boundary value problem and it consists of the laplacian and a derivative three part a zero order part with a positive factor and you might remember that this situation so to speak makes the cohesivity even better so if we only have the laplacian we are we depend on the boundary condition really to make a coercive problem out of that so we need pancreas lemma but with this part here in addition we don't need that so you see this is just a example there's much more generality here behind that hey sorry sorry sorry sorry it's a miswriting that is so this is the unknown this is the unknown of course and for this unknown we needed we could ask our so what is this is you might have seen this or if you do now the second part of normatives of pde you will see this very quickly this is what is called the implicit euler method so it's actually a numerical yeah euler of course had no computer when he invented it i'm not sure for which reason he used it whether he really used it to compute solutions or whatever but nowadays this is one of the basic methods one could do different things here which are numerically better as for example the so-called crank nicholson method where we do here uh the convex combination of both values we could as we now only want to do analysis the efficiency of our method is not the point we only at the end of the day we need convergence we don't need an a specific rate of convergence but there are also of course a lot of other other arguments why the implicit euler method is nevertheless a good method also we take here a one-sided difference quotient which means that the order of this approximation is not very good but that is not the point here okay so let's keep this in mind what does that mean if looking at this specific example it means that what we are doing is we are substituting by this time discretization we substitute we substitute our system by a system of elliptic boundary value problems for which we know there is a solution and which properties that this solution has so we'll rely on lux milcram and those things so the procedure and of course this is a procedure which we can always so to speak uh iterate let's assume we can we can do this kind of problem at the end of the day then we will see then this is not the end of the story then we can do all kinds of uh of inhomogeneous data we can do other boundary and of course then we can also ask ourselves what about nonlinear problems then we can do a similar approximation for nonlinear problems for example if you have a nonlinear problem where for example here the diffusion coefficient depends on the u that would be one possibility or the the heat conduction coefficient depends on the temperature and we could say we we make an iteration i'm not saying that this is now good to do but it's a principal approach make an iteration and we the iteration which just start with the initial iterate then we put this initial iterate in this non-linearity so to speak we freeze the nonlinearity and then we get a linear elliptic pro and then a parabolic problem but this level we have already solutions of linear parabolic problems and then we can we know there's a solutions we get a sequence of approximate solutions and maybe we can conclude that there is convergence and so on so this is always an ongoing story we can always go level by level by approximating in each step more complicated problems okay so before starting with that we will have a look a little bit uh yeah what gives us so this is no problem to write down this uh this procedure and we know uh for each step this these functions uk exist then we can make functions what we will then do in time out of that for example we can connect those values linearly in time or we can just take it piecewise constant on this time intervals that would be two simple possibilities then we have sequences of functions in space and time but how do we now conclude that we have convergence and in which sense convergence of at least a subsequence and how can we see then what this limit is and here we have to go a little bit in this all this problem of compactness in infinite dimensional spaces if we would be in a finite dimension space everything would be easy if in a finite dimensional space we know every every bounded closed set so basically every ball in a finite dimensional space is compact meaning if i have a sequence which is bounded i know there is a convergence subsequence and i don't have to deal about i don't have to think about different notions of convergence because i know all all norms for example are equivalent on finite dimension spaces in infinite dimension spaces this is different and in general we cannot we will never get something like a compactness in in a in in a norm space sometimes one can do this for you will maybe remember the theorem of azalea ascoli which basically gives a characterization of compact sets in the space of continuous functions with a maximum norm so that would be a possibility we show that the conditions of the acelascoli theorem hold true and then we know the set is compact there then we can extract a sequence which is strongly convergence as one also says this will play a strong a big role if non-linearities will come in play as long as we don't have non-linearities as here we can be satisfied with v convergence so how is this for example in in l2 if we can show that a function is a sequence of functions is bounded in l2 we know there is a subsequence which converges weekly in l2 and the same thing with h1 or in general with every hilbert space so we have to have a notion of solution for which it's not it's it's it is sufficient that we have weak convergence such that we can identify the limit then as a solution of the problem but this we will do we'll look at weak solutions so uh we have to know a little bit about uh for example compactness properties of those spaces but what are the spaces let's have a little bit an interlude i will not make this in all details but only to um to to to indicate a little bit uh before doing this interlude uh again let me conclude this this line of thought so we need we need boundedness boundedness results a priori bounds as they are called and from those we can then with weak convergence properties we can conclude uh we can go on yeah what kind of boundedness can we get here from the heat equation so what is the energy method for the heat equation so the idea is and later on this will be very very even easier you you know all this already so what we do is we take the heat equation so again energy method formally formally means that we said we say we have something like a classical solution everything what we are doing is is okay and let's see what kind of formulas we get out and later on we will see how to justify those formulas in a more general setting so what do we what would we do we take our equation put everything to one side so this is just 0 what i've written down here now we integrate in space and we integrate in time over the interval from sum from 0 to some instance of time small t and now we might if we do this the same thing is true if when we multiply here with any function and we don't take any function we just take the solution so this is the answers and later on it's the same thing as if we would have a weak formulation and taking as a test function the solution itself what we already did so now what can we do the second part is clear this is just here partial integration no boundary terms because of the boundary initial boundary so what we get here in space and time is so to speak the directly integral the gradient of u times great interview so this term here this chords for the weak formulation because that's what we did here uh doing partial integration to come to this form what do we do with the first part and now uh if this all is is classically okay then we have a time derivative time derivative of a function times u but the chain rule tells us that's the same thing as one half the time derivative of u squared so this is the one half the time derivative of u square and then we can space integral is still there but the time integral we can interchange these we can integrate out so we get here one half i'll write it maybe first explicitly yeah u square minus this is a plus sorry minus and now this i bring to the other side minus the same thing as t equals zero so we have here this term so what do we have here we have our data the only data in the problem which is just the l2 norm of the initial data squared and what do we do we can estimate is even an identity so-called energy identity we can estimate the same term for arbitrary instance of time by this term here so here we have now u evaluated at t so the spatial function if we fix the t also in l2 squared and here we have this additional time space term this at least is greater or equal to zero that means uh this term is always less or equal than this term so in terms of the data we have here in a priori bound and in which space is this a priori bound it is spatially in l2 but in time we have every instance of time estimated this is pointwise so we have l infinity in time what we have here is an is an a priori bound what we see here is an a priori bound and we expect that means we expect solutions he expects solutions let me write it down in in those in those norms which we can bound which we can control by the data was is this this is l infinity in space uh sorry in time um what did we have l2 in space and in additionally what we saw here this is l2 l2 so to speak this is l2 in time and now we have here the a2 norm of the gradient but we are in the dirichlet situation with a gradient we can also control the l2 norm of the function itself in space so we have the full h1 norm here so we have here h1 or more precisely h10 of omega so now i've used already a new notation of first bases and i hope you have some feeling what they they mean they're supposed to mean but they are in in fact they are new function spaces what we have here what is new here with these spaces these are spaces on the one hand side they are quite simple spaces because they are only on an interval in time and here there are l2 lp l infinity spaces or later on w1 spaces so spaces we know but what is new with these spaces is their values are not in the reals they have values in a banach space so we interpret this x t or t x function now as a function in time with values in a banach space to cope with this x dependence so how do these spaces relate to each other in principle one can avoid this notation because uh there is a sort of so these are these spaces are called bochner spaces bochner space means you have some lp or sobolev space on a time interval with values in an banach space so we will i will not twelve too much on the botner space we could now do all this theory and this would keep us busy next four weeks or so it's i'm not saying that's uninteresting but that should not be the focus here uh let's frame it in this way we can avoid bochner spaces in the sense that we write them down in a way that we don't formulate it as a in this form here but we re formulate function spaces more classically on the space times cylinder omega t so for example the the the most simple bonus base is of course this one here a2 a2 of course this is supposed to be the same thing we would expect as just being the l2 space on space time and what else do we have seen we have seen for example l2 h1 and here we would say this should be something like those l2 functions so those l2 l2 functions those l2 functions on the space time cylinder maybe i now abbreviate this also with uh with q or qt such that what do we want to have we want to have that the gradient and now we have to be as we have the x and the t derivative we always have to make sure that it is clear what we mean that therefore an x here is not bad to write that the spatial gradient is again is again an l2 function now let's use the abbreviation so and okay uh another way to write that so what we later on we'll also need is this space here so what again what is h minus 1 h minus 1 that was in shorthand notation for the dual space the topological dual of the h10 so the set of all bounded linear functionals on this space and uh so we can use any banach space here as the image space we're not using this space so what we have here is we have here a dual space we know the lql2 space as a hilbert space is with a with a with a natural isomorphism why are the scalar product isomorphic to its dual space so we always identify the l2 with this fuel what we will also do later on so what we expect here is that we have a dual and the dual so what you expect here is something and that will also turn out that the whole thing is just the dual space of this space here which we can write down in a non-bochner fashion similar to this space here there so in principle we could avoid this bochner stuff nevertheless it's quite helpful and let me just summarize a little bit of results which we have for it first of all first of all if we start we have so x is a bonus base let's make it general p is a number between 1 and infinity and we would we ask ourselves what about the space lp 0tx so the first question is we would like to speak about integrable functions so what we like want to have here we would like to have functions whose piece power is integrable so first of all we have to speak about integrability and for doing that we have to speak about measurability of functions with values in barnard spaces but this is very analogous to the lebec theory which you hopefully know how does one do it in the lebec theory opposite to the riemann integration theory when approximates the functions by what is called a simple function a function which only takes up a finite number of values so we have quite a simple image space and the the pre-image is then subdivided in those subsets which which are mapped to these finitely many values so for such a simple function we can for example write down what the integral is this is a sort of a a general step function so we just take the value times the measure so at zero t we take the lp the back measure f0t times the measure of the subset and then we sum up over those values said you should know this at least if you know a little bit of probability theory that's a discrete measure or whatever whatever so and now we so this and now we ask and then the next natural step is we take the limits of those functions that would lead to the to the notion of strong measurability and but now as we are here with this infinite dimensional uh spaces we there is a theorem theory of pettis and i only phrase it because we will not use it to to to to to to d more in detail it basically says this notion of uh of lebec measurability as this notion of strong measurability which we have just defined by this limit process is the same as if we do the following we take the c we take a function from this space put a general element of x prime on it a general linear bounded function and then in this composition we again now in the reals and if this function is now in the class in the usual the back sense measurable for all possible elements of x prime then it's exactly the same notion but this that this holds true needs the separability of x so again what means separable so we have hilbert spaces that's clear these are spaces which uh where the norm is induced by a scalar product and which gives us we are there is a representation theorem a natural isomorphism between x and x prime and of course having this between x and x prime we also have one between x and x double prime but on the other hand in general what we always have is a natural embedding of x to x double prime by just so what what is how is this element of x double prime defined uh take take its argument so an element of x prime and evaluate this element at the given element of x so so just so to speak the point function at x prime this usually is an embedding and not an isomorphism if we have it as an isomorphism then we call this space reflexive so the lp space is for p having this we can always go back and forth because that means it doesn't make it doesn't make a difference whether we view a space as a so to speak original space or as a dual space if we look at our p with a p strictly between strictly between one and infinity then only for p equal to two we have this identification with its which is dual space in the other case we have an lq such that 1 over p plus 1 over q is 1. but now again taking the tool we are back to the lp so in this sense we always can go back and forth between dual and non two world space in the reflexive situation the non-reflexive situation is much more complicated so the dual of the l1 is uh l infinity but the dual of the l infinity is something more complicated i hope i didn't i haven't looked at functional analysis for quite some years so hopefully i still able to do it okay so uh what we usually will assume is reflexibility so that's not this we can work with all uh p spaces so uh what i've said now in words and we're not going to write down gives us now a definition of this space that just means that it's this space where the piece so uh where's where how is the norm defined here of this space so what we do is we take the norm function so the substitute so to speak the modulus in the scalar situation or in the vectorial situation we take the b the idea the piece power and then this integral this lebec integral is supposed which exists as an element of either r or infinity is supposed to be finite that selects those functions which we have here so that is this is the definition to define it is the definition of being an element from the space and that is then the norm we have on this space and this is down in a banach space okay there's a basic theorem of bochner i'm not sure whether i'm not going to write it at the blackboard maybe maybe only a few ingredients from it a few ingredients from it not everything that we have some working knowledge of these functions so um one important thing is the following estimate we would like to have and we have it so uh on the one hand side we have this integral this integral is an element of x of the banach space here and on the other hand what we can also write down is we can look at this function here this is this we should type a t of course we look at this function here this is now a scalar function we could also ask ourselves about uh the integral and what we would like what we what we expect from our knowledge uh if this is a it is a scalar or a a vector in r and we would expect something like this here and this holds true so we can always work with this estimate and the same thing is if lambda is here it's called lambda for some reason why lambda yeah if we take in lambda okay i i said i i stick to this notation lambda from x prime a dual element and then i can do both things i can take look at the integral which is an element of x and i can put this into the lambda so i can take the dual pairing between the lambda and this element of x so this just means apply the lambda to this element of x and that is again the same as if we would we can so to speak put this through we uh put the lambda at all the uts now we have scalas and now we do the integral shouldn't write the x here always so these are maybe the most important things how how can work to this and there are further further uh things which basically say the theorem of the labex theorem of majorized uh convergence so i have a converging situation and have a dominating uh integrable function and then the uh as a consequence the conversion is not only point wise but it's also in the integral in the l1 in the lp sense this also holds true here so um one two will do we try to avoid this one to uh to do deal with those fun in general what is how does one work with such general functions which are not so regular we also did this to some extent for example did we do this look at the at the um oh god when english at the trace theorem did we look at the proof of the trace theorem i'm not sure okay one one way to do is that when first week works with smooth functions proofs these estimates which one has in mind for smooth functions of course avoiding all these uh in the in in the final result avoiding all these higher order derivatives which one would have uh due to the smooth functions and then do some some density arguments saying okay these move functions are dense in these norms which appear here and then i can go to a limit and have it for all the functions and a more explicit way to do that is to approximate a special a function from these general spaces by a sequence of smooth functions by doing a convolution with a a special function and that is a that is a do a tool to deal with that that is a tool we would need if we would like to prove all these things but i said we will restrict ourselves a little bit only to have a look at the results so the next question is what about the dual space and the assumption again is now from now on as i said f x is reflexive we have a p between 0 and infinity 1 and infinity so 1 and infinity is excluded and we have the conjugate of it so we have a q such that 1 over 1 over p plus 1 over q is 1. and now we can say something about the dual space here about this thing here and this is when i say it is i always mean it's isomorphic too it's isomorphic too so maybe it's better to write it like that so uh if we would have his colors then we would know what the dual space is the dual space would that be lq as i said and now okay as we saw it already in the example here at the in the last line what you would expect and what actually is true is that here in the image space we have the tool of the x so it's quite formal we just put the prime forming the jewel here and here okay so what does that mean what does this identity means it means that if we have an element u from lp and we have an element let's say v from v here from lq then the okay i would write it the other way around okay sometimes i'm in conflict with the book but i said i stick to the book so i keep the formulation here of the book so the formulation is and that's it's uv so this can now mean several things it can first mean the following uh it's something like a generalized color product so if this would be two and two then we would have here the scalar product and uh as similar as in the in the in the real valued function now we have here in the interior the scalar product of the let me first write it down okay let's do it in several steps okay let's assume the x is a bonus it is a hilbert space then we would say okay we can write this down and this is just the scalar product of the hilbert space and that would be so to speak an obvious color product then of l20tx x being inhibit space now if x is not in hilbert space but uh only a bonus base your reflexive space then we have here the in the same notation in similar notation the dual pairing then we have the dual element applied to the element of the space and here i would write it the other way around but here the dual element is written here to the right any anyhow okay so and in this sense what we here define is now uh so this makes sense what is written here and this defines just a bilinear form here again on our spaces and it's this bilinear form which so to speak is that you will pairing which gives the relationship between [Music] an element of the dual space and an element here so the the surgeon is if i have any element of the dual space for this element there is a unique we from lq and so on such that applying this dual element to an element u is given by this and in this sense this is now the dual pairing between these two two uh norms so tool bearing just means application of the dual element to the element of the space and basically it's it's quite often is an extension of an underlying scalar product so now uh up till now we had no time derivatives of course we also need time derivatives so the next space which we need with time derivatives is so we need a time derivative and a time derivative so if u is from l p 0 t x then we need a distributional time derivative why so why it's so so important always to refer to the distributional derivatives we will uh encounter the situation that we will have derivatives in in different senses for example classical derivatives speak derivatives whatsoever and we want to identify them and we need so to speak a third common relay relational point which is always more general than the other ones and the most general version is the distributional derivative and on the other hand side the distribution derivatives fulfill it's unique so it's something if a is the distributional and b is the distributed derivative then they're both the same and it fulfills all the all the rules which we know about derivatives because distribution derivative only means put everything to the test function and so this we do here too so we just say what do we have so the time derivative of u and uh to indicate that this is a distribution is this with these uh brackets so if only one argument is so this is different from this here with two arguments if only one argument appears it means that's not really a function but it's a distribution and we need our very smooth test functions for the distribution so the c infinity functions with a compact support on the interval 0 t and now the things become simpler because now here we can rely only on reals we don't need here something with x prime also and so what do we do how is this functional uh defined let's let's say for an element phi from this yeah what do we do we integrate in time between the derivative which we don't do not have and the test function phi and put the derivative let's say by in parenthesis by partial integration to the other side so it's defined by this function minus ut time derivative of the phi so it's basically the same notion of of distribution distributional derivative what we know already so and then quite similar to the classical subolef spaces we then say we say okay we are not satisfied that this is just a distribution just a function we want to have a function representing this distribute this distributional derivative and in the if we look at the space for example at the space w one p zero t up till now that would be so to speak the classical superleft space but now we have have it x valued what do we want to have here we want to have two norms what do we want to have we want to have that the function is in lp of course zero t x and it should be uh and the time derivative the time derivative in this sense here should also be a function should be represented by a function and this function now is supposed also to be in lp 0tx and then of course we have the corresponding norm and the norm now in the space the norm in this space now is and then this is a norm it is a banana space the norm is we have to take uh just the norms of these these two ingredients and here is defined and therefore i will stick also this definition it's just that we add up these two norms [Music] maybe you find this now at the first glance a little bit irritating because up till now what we did for example for p equal to 2 what we did we took the square of this and the square of this and took the square root out of the sum but you can see it in the following way what we have is we have a tuple here a two tuple of this number and this number and to make a norm out of that we need a norm at the r2 but which norm we take is absolutely unimportant of course it changes the number but all those norms are equivalent of course we could also here take a power p and here power p add up and take the p root would be an absolutely equivalent norm so why make it complicated if it if you can make it easy but of course this also applies already to sobolev spaces but there's more traditional really to take this other for other other equivalent form of the of the norm okay um that's again now a bonus base all these spaces so we never will lose this property from time to time first if we start with the product space and from time to time to have the properties we would like to have we need the reflexibility in addition so for reflexible x spaces everything is okay okay so trace the trace is here much more simpler than in the spatial situation because we just have an interval and the trace means we want to evaluate a function at one point of this interval if we would have a continuous function then it would be obvious then a continuous function we can evaluate um so maybe let me come to that later on we can first clean the blackboard so why do we need for which which ingredients of the problem we need the factors we need something like a trace or we need something like continuity in time which would not make sense or which would be unclear which part of the so we have three parts in our problem here you can have a last look at it with these three parts for which of these three parts we need this trace property you see this last chance before i erase it yeah but that's a trace in space and that is so to speak already integrated in the space x if we say x is h10 then this is already integrated it's for the initial data because in time but what we we have to have a notion that in time it makes sense to say at t equal to 0 there is some condition of course if this only is an a 2 function in time that doesn't make sense so we need trace notions and we need finally we need continuity in time in our notion of solution but nevertheless i think it was good that you mentioned this even it was not correct but because it tells everything what is what that is this is the advantage of this approach everything what is concerned in space is already encoded in the x it is strictly separated here okay so uh what's the math so let's just put down this this notion of of the of trace so we have an element we cannot do this for an ll2 or lp function of course why should it be easier having it an x as an image space a part of r so we need more we need some weak derivatives to have this trace and this is enough to have this w one p so uh in this p i have not specified that but i think that p can include the one here so this p is between one and infinity [Applause] so what about now the trace let's let's do a computation which would be obvious if we could evaluate the function so let's say we start again with with a term like that so so to speak with a with the up to the sign with a distributional derivative so we will later on specify what the what the what the phi is and on the now we do partial integration so we put the whole thing on to the other side and now we don't want to have it in general at t at zero but at t zero so we go from t zero to t partial integration now we have the derivative here and this we can do because this is now an x-valued lp function it makes sense and we have here so so what is this this depends of course on on on on what we have on the phi if the phi has compact support in the full interval that's just zero that's just a distributional derivative uh now we are a little bit more uh if if we have no restriction on the support then we would get the boundary terms so what we only want to get we want to get one boundary term so what we do here we take those test functions here very smooth compact support but now there comes the difference here the t0 is included here so the compact support can go up to the t0 but it is beyond the capital t so we have here no contribution from the t0 but we have here a contribution from no contribution from the capital t but the contribution from the t 0. so in if this now we turn everything around and say if this identity holds then we call u 0 so u 0 would be something like in classical terms u of t 0 then we call u 0 the trace of the function at this point so and the notation is do we have a notation for that is something like okay so from this now we can conclude the fundamental theorems do you know the phantom you know the fundamental theorem of calculus of course and now we have also the fundamental so to speak the fundamental theorem of calculus for botner functions so what does that mean what we would like to do is we would like to integrate the derivative and we would like to get so what we would like to have is we have two two numbers in the interval uh t1 and the t2 we select two numbers we would like to integrate from t1 to t2 over the derivative this is the formula 1017 like integrate over the derivative and in classical sense scalar valued and for continuously differentiable functions of course this is then of course ut2 minus ut1 and this also holds true here with a very slight exception namely the exception is for almost all so up to a set of measures 0 of points t1 and t2 this identity holds true in this context of the w w one p functions so uh to turn the thing here around here to take to turn this identity around we can now define the trace we can define the trace in the book it's the german word for trace the trace at t 0 is now a mapping from 1 p 0 t x 2 yeah x this is still here an element of x here of course and what do we do we just take this identity here and we have to restrict ourselves a little bit with the phi so this value has to go away so we have to take here the test functions with phi at t 0 equals 1. so the trace is just so we we map u to uh now we have the minus we bring the minus to the other side to minus um g zero t uh u t time derivative here and yeah keep it in the integral and here then the time derivative on the other side so these all these terms make sense and now the test function is the same as above the t 0 can be touched has to be possible to touch the t capital t not and uh now we need to have phi of t 0 equals to 1. we scale them to 1 there or otherwise we could also take a general run and subdivide here by phi of t 0 of course okay and what holds true is here in this case maybe i put it here with this kind of functions we have this has to do with the fundamental theorem we have ut equals trace of how to write this with a t yeah i have to write it like that trace of u at t for almost all t so up to a a set of measures 0 this is really this is this is the the value which we have but it's at a set of measure zero we might get the wrong value and only in the situation when we have a continuous function then in in the case of a continuous function we can so to speak select continuous representative and then this almost everywhere goes away so that would be which we can if we think we cannot do it completely today that would be the next step which is an uh first a simple step namely to do the same thing not for lp functions but for continuous functions so you have here something like like c uh zero t x continuous functions on the interval with values in x and then the next step is an embedding theorem do we can we is it possible to get if we know something about the function in lp if we know something about the derivative in lp do we then know more uh about the function with respect to continuity and this this will hold true and this will be then maybe the final things which we then will need and having that uh then we can go back to uh to the energy estimate and what we will then do next time largely is to not then deal with this what is now not on the blackboard anymore to deal with these time discrete approximate solutions given by the implicit euler method so what we have to do is we have to derive a priori estimates see which kind of conversions we can conclude and then we have to see what we can conclude for the limits and in this way come to an existence proof okay this we will then do next week [Music] you