 It came about the regularity of the Boltzmann equation. And thank you to Francesco for this kind invitation. It's a pleasure for me coming back in Trieste after my PhD. This is actually the first time that I'm coming back, so I'm very pleased and I feel like at home. So thank you. So what I would like to talk to you today is about regularity of Boltzmann equation that we saw introduced by Laurent this morning. And especially something about the regularity of Boltzmann equation when we consider this equation on bounded domains. So this is a joint work that we did with the Young-Wolch, Young-Wolch team and Alianz Rezcata. So the results that I will present here are mainly taken from the paper that we published. So let me thank you for this introduction and thank you Francesco again. It's a real pleasure for me being here. So I would like to present you some results on regularity theory of Boltzmann equation in bounded domains. This is a joint work with Young-Wolch, Young-Wolch team and Alianz Rezcata. So we will consider the Boltzmann equation that was introduced this morning by Laurent in a form that depends even on the space variable. So with respect to the one you saw this morning, we are considering the inhomogeneous Boltzmann equation where we have the dependence on time, space and velocity. So our function of the density f will satisfy this equation where you see the presence of this term that is precisely due to the fact f depends even on x. So we have a free transport operator, the first one, and the collision operator. The collision operator is more or less the one that Laurent introduced this morning. So this Boltzmann equation was introduced to study the dynamics or ratified gases and it places itself in between the microscopic description of a gas that means like considering the gas as an infinite number of particles and the macroscopic description of a gas that means considering the gas as a continuum. So the collision operator as you see here is made up of a collision part that is due to the gain term that is the positive part that was introduced this morning and the loss part given like this. Now I will just enter a bit more into the details. Let me firstly have a commentary on the free transport term. So we suppose that we have a monatomic gas made up of identical particles that are not interacting between the self but they are just moving without any external forces. So in the absence of external forces the particles are just moving at constant speed along a long straight line. So we have a straight line of this type. So a particle that starts at the point xv in the phase space, so in let's say omega times r3, will move at a velocity v to the point x plus dv. So whenever we require that the function is constant along this line, so fdx plus dvv is equal to f0, for example, xv, then f satisfies the transport equation, which means that dtf plus v gradient xf is equal to 0. So this is the transport equation. We are not considering collision between two particles move along straight line. We will see that this equation is very important for us to understand the behavior even of Boltzmann equation. I will explain you why. So however we have to introduce because we are considering particles that collide between the self, the collision operator. So here you can see more or less the same hypothesis that Laurent was stating this morning. I just underlined here the collision between two particles. So whenever we have a particle v that interacts with a particle that has a velocity v that interacts with a particle with a velocity v star, the corresponding velocity after the collision will be this v prime here, where we have this w that is in s2 in the bowl, so it's a unitary vector, defined like this, and then for the other particles the velocity will be v prime. So I don't want to enter much into the details of all this stuff. I will just present to you our collision operator as Laurent introduced this morning with a kernel that is done in this way. So you have the last term of the collision operator, that is the number of particles that changes their velocity, and the gain term that is the number of particles that will have the velocity v defined in this way, and in our case we will impose some condition on this kernel, so you can see here a particular version of the collision operator. So this is the kernel that we will imply, and the kernel will be dependent on the modulus of v minus v star to the power k, where we require that k is in between 0 and 1, that means that we are considering our potential, and we have the presence of this q0, it is an angular cutoff that we require in this way, so theta is the angle between the two velocity v and v star defined in this way, so these hypotheses are made in order to let things simpler and in order to be able to integrate the kernel and do estimates on the collision operator. Actually we won't really see the operator in all these two hours lessons, not really, so let me just leave it like this, so we are supposing our potential and the presence of an angular cutoff in order to make all the computations simpler. Okay, so as I told you, when studying Boltzmann equation, usually we would like to find the existence and uniqueness of solutions and to see the time decaying of solutions towards an equilibrium, an equilibrium that is a global Maxwellian as the one introduced this morning. This is what we do in general for every domain, what we will consider today is a domain that is bounded, so we are in presence of a bounded domain because maybe the gas is inside a box or something like this, so we need to understand the interaction of the gas with the boundary of the domain where it is confined, so we will require some boundary condition and of course we will have to consider the interaction, so the collision not only between particles, but even with the boundary of the domain. So suppose that we have a domain omega that is sufficiently regular so that we can write this border as a function with a situ function, we will decompose the boundary of the domain in three parts. So this is our domain, we will have this gamma minus that are the particles that are on the boundary, so they are more or less here, with a velocity that is incoming, so it has a scalar product between the normal, this will be the normal, maybe not like this, but like this, that is negative, this is the incoming boundary. Then we will have the upcoming boundary, gamma plus, that is made up by particles that have velocity that goes outside, so the scalar product is positive, and then we will consider the grazing set that is made up of particles that have a velocity that is tangent, so the scalar product between the outer normal is equal to zero, so this will be gamma zero. So we will see that actually this set is the most important one because it's the set that creates the singularity for the solution. I will explain to you what. So this is the composition of the boundary. Okay, so once we have a boundary, of course, we would like to impose some condition. So the condition will be imposed for incoming particles, so on gamma minus, so on this set, for particles that are incoming. So the easiest boundary condition that we can impose is just prescribing the flux of the incoming particles through a function g as written there, is the inflow boundary condition. We will require that incoming particles will have, like a velocity given by the function g there. So the distribution of the incoming particles is given by this function. Then we can, for example, require that we have a specular reflection boundary. That means that whenever this is our, in this direction, omega is here, a particle comes from here, this is the normal, and it's reflected in this way. So this is a reflecting boundary, and you can find the definition there where we explicitly write how we can find the reflected velocity starting from the incoming one. Then I can have another boundary condition that is the bounce back reflection, that is the particle, it's the boundary with a velocity and come back with the opposite velocity. So it's just reflected in this way. And then there is another boundary condition that is like more a stochastic probability condition in the sense that when the particle is the boundary with a velocity v, then it's reflected with a velocity that depends on the average of the other velocity. So it's not really well determined, we just know that it's going outside and can be whatever. The point is that the average of the density, the average of the function is given by this condition where we have here the global Maxwellian with a temperature T that will be actually the temperature that the gas will have after the collision with the boundary and will be reflected like in a diffuse way in the space. So actually we don't really have the control on the velocity that the particle has after the collision with this boundary. So just a slide on known results about the existence of uniqueness of solution. We have some results that are for weak solution and that exists like the Bernoulli answer solution. So we have a list of people that work in this direction. We will work more on the perturbative framework. So we will be able to find existence and uniqueness results for stronger solution, but solutions that are closer to the global Maxwellian, the equilibrium that we saw this morning. So the first results were obtained with domain with particular geometry by Ukaia, Zano and Girot. And then there were results on general domain by Google. Then as far as the time decay towards an absolutely Maxwellian of this form written here, we have results by Laurent and Villani that state some polynomial rate of decay in HKE spaces. And then some results by Google with an exponential rate of decaying. So just to give a picture of the previous results on this part. And then what about regularity? So this will be like a summary of what you will see in the detail afterwards. These are the some existing results on continuity and formation of discontinuity for solution of Boltzmann equation on bounded domain. So in a paper in 2010, Google was able to prove the existence and uniqueness of solution of a strong global solution in this perturbative framework. I will enter into the details sooner. It was able also to prove no matter about the for any boundary condition that I showed you before, actually. It was able to prove the time decay towards the absolute Maxwellian with an exponential rate. And actually the most important thing that you have to remember in this case is the fact that whenever the domain is strictly convex, then it is possible to prove that the solution is continuous away from the grazing set. I will show you immediately why this grazing set is the worst set that we are considering. Just remember that away from this grazing set you can prove that the solution is continuous. We will understand why. And actually one year later, Kim proved that it is possible in the case of a non-convex domain to prove that there is a discontinuity actually that can be formed exactly starting in point of non-convexity of the domain as you see here and entering inside the domain. So this was the result of Kim in 2011. So this is just a picture to let you understand what I'm going to explain you in the following slide. Okay, some regularity estimates were obtained previously in the case of Vlasov equation for BVs and estimates, Sobolev estimates and older regularity by Yanguo, but for the Vlasov equation, not for the Boltzmann one, and in the case for the equation in a half space, so a very particular case, and then they were generalized for Vlasov equation in convex domain with specular bounded condition to all the regularity results. But in the case of Boltzmann equation very rare results existed up to the results that I'm going to present you. Okay, what's the perturbative framework? So we take the global Maxwellian, defining this way, so it's the exponential minus V squared over 2, and we look for solution of the form F equal to the square root of mu F. So we have that now we are imposing that our F is given like this. So it means that if the equation for F was this one, now the equation becomes, I just substitute the square root of mu F. So now the Maxwellian does not depend on time, so this will be the square root of the Maxwellian, dt F plus, and the Maxwellian does not depend on X, so I can turn it outside. It's got a product, X F, equal to Q, and you remember that Q was the Q gain minus the Q loss evaluated in square root mu F twice and here as well. So I just have to divide by the square root of mu here, and I will call this term is the gamma gain, and the other term is mu, how did I call it, is like mu square root mu F F. So this is to explain why now the equation for the little F is this one. Okay, so from now on we will consider this as the Boltzmann equation in which we will work. I just write here the form of the kernel, but it's precisely the computation that we just did. Okay, so since we changed the equation in a way, now we have to change the boundary condition accordingly. So we just have to divide by the square root of mu in this way for inflow boundary condition, for diffuse boundary condition will be like this. For specular boundary condition and boundary condition we won't see because we divide in both sides and we can remove the square root of mu, so no problem. Okay, once you consider Boltzmann equation with an initial datum actually, so you fix the value of the solution at time t equals zero, you need a compatibility condition of the initial datum with the boundary data. Data, of course. Because you are imposing that at time t equals zero the initial datum will be over here, but since it will be defined up to the boundary it has to satisfy the boundary condition. So we will need to define this compatibility condition for the initial datum that precisely consists in requiring that the initial datum satisfies the boundary condition. So the inflow boundary compatibility condition, the diffuse is there, the specular reflection and the bound spec reflection compatibility condition. So in the assumption of our theorem we will always require that our initial datum will be positive and will be satisfied, will satisfy this compatibility condition with the boundary condition that we have here. Okay, so let now enter a bit more into the details of our, of the geometry and of the behavior of characteristics for our problem. So we will require that omega is a bounded set that is a smooth, so we will require the existence of a c function so that omega can be seen as the negative part of this function, so the set of point that are negative here and for this reason the boundary will be exactly the set of point that will be equal to zero. So whenever we have this we will say that the set will be strictly convex if this quantity, this inequality is satisfied, so in this case it will be strictly convex and will be easy to compute the normal because the normal, the outward normal will be the gradient of c with respect to x over the modulus in order to have modulus one. Okay, now characteristic, so the characteristic of the Boltzmann equation actually is a straight line as soon as we don't touch the boundary and we don't have any interaction is a straight line with constant velocity so the derivative of the characteristic with respect to s will be the velocity and the derivative of the velocity will be equal to zero, so we are precisely moving along straight line of the type that I write there. Now we have to see the interaction with the boundary, so if we are on our set, for example we take a point here, x, v so let's say that v is in this direction it is important to see which is the time that this straight line needs to touch the boundary backward, so this will be our actually the time needed will be t, b, x, v and the point that we have here will be x, b, x, v so we are going backward coming back to see when we touch the boundary of course when you take a point here you can touch the boundary but if the time that we are considering here I'm not considering time but suppose that I take a time t if t is less than tb I will never touch the boundary it means that the condition, the initial condition will be reached before the boundary t means that I start from zero and the time t arrived here but if t is less than tb it will be reached before by the initial condition okay so this is the backward exit time and now we will see that actually with the definition that I've given of a backward exit time and backward exit point the set of grazing the set of grazing particles that was the gamma zero actually will play an important role why? because if we consider let me see yes I have a point of non-convexity here so let's take this point here x, v and consider a closer point here not so close but kind of it is x' with the same velocity so I want this velocity and I want this velocity here now I try to come back so for this particle that has this x and v when I come back I precisely reach this point so x, v will be this one and actually I reach the boundary on a point x such that x, v x, v belongs to gamma zero is a grazing point because this vector is tangential to the boundary while if I do the same for this particle actually and I come back more or less on a straight line I will arrive at a point that is very far from the other one so actually the time needed to go there is much longer than the time needed to go there and the particle the position that I reach here is very far this is x' v so we see that in the presence of such a boundary with a non-convexity then we have a singularity of course so why we have a singularity and I was saying that we were looking the Boltzmann equation but think of a simpler case is this the case, yes you have a stationary transport equation this one so you don't have time you just have the transport term with a position in this way you don't have the initial condition because you don't have time but you just have the incoming flow condition on the boundary for example there what's the value of f at the point x' v in this case is precisely the value of g on the boundary so it will be gxbxv so it will be gxb minus tb xv v v so if the time is discontinuous even if I take something that is very regular as an inflow boundary condition the solution won't be continuous so the fact that the time tb is not continuous will introduce a singularity in the solution so this is precisely the idea for this transport equation that is stationary it's not at all our case but it's connected you will see why so actually we will define this sigma b the grazing singular set in this way as the set of particle so this sigma b is the set of particle xv that belongs to omega times r3 such that the scalar product between n evaluated in xv times v is equal to 0 so are all the particles coming from the inside and going to the boundary in a point that is a grazing point so it's like the injection of the grazing set inside the domain and we will expect that the solution will be singular precisely on that set of course we are considering boundary condition up to now apart from the little idea of the incoming boundary in flow condition we haven't really followed we haven't really looked at what happens if on the boundary we have other condition so actually we can play a bit with characteristics apart from the case of diffuse condition we have to assume that we know which is the velocity with which the solution is reflected we can really follow so if we have diffuse boundary condition and we start from a point x with a velocity v for example this what will happen we will go on the boundary here we will be reflected with a velocity it's called the rv prime x0 v0 and we take a time t0 so we will be reflected with a velocity v1 for which we only know that actually this color between nx1 and v1 is greater than 0 this will be x1 v in this direction v0 so we are reflecting we are touching the boundary again again and so on we can go on up to time t equals 0 so at time t equals 0 either we are on the interior either we can be on the boundary but we are starting from the initial condition this is the formal way to define this but this is precisely what I just picture you here now in the presence of a specular cycle what happens so you touch the boundary again and now you are reflected with a velocity that is precisely specular so you do like this you see that it is like this and then again and then again and so on but now you know which is the velocity actually because it is obtained through the formula that I showed before that is this one written somewhere here so you can really construct the specular cycles while when you have bounce back cycle actually the nicest one because it's very easy you start from a point maybe in the interior let's think it's here you come with this velocity you come back to the boundary then you are reflected with the opposite velocity and then again and then again and then again so you won't move you stay there okay so it's actually very easier to very easy to even it doesn't seem so but this is the definition of the point you just have two point that is x1 and that's all so you have a way to define the backward trajectory that can be seen as generalized characteristic and actually you can follow everything till time t equal to 0 and the initial condition this for sure will be the case if we are considering just particles that are moving with the transport equation it's not the case with Boltzmann equation because you have interaction but we will write that to there so let's take this standard transport equation that we have here so we are considering this dtg plus v scalar equal to 0 and let's see what happened with the different boundary condition that we have so I already explain a bit to you what happens in the presence of the inflow condition so you have your domain you take a particle here at a time tx with a velocity v so if t is a smaller than tb sorry xv it means that to reach the boundary I should have had a longer time to go there so 0 will be somewhere here so it means that I'm connected with the initial condition so if the initial condition is g0 I will say that g at the point txv will be equal to g at every point as inferior so if I took here I have to take x minus t minus s vv and if I take the initial value this will be the value at time t equals 0 of x minus I think yes so I'm connected to the initial condition the point for which t is less than tb our point coming directly and propagated by the initial condition while if t is greater than tb or equal actually that's why you see why it's necessary to impose the compatibility condition because in the case it's equal to tb you have the choice between the initial condition and the boundary condition so they have to be compatible to have something that makes sense so if t is greater or equal to tb actually this can be here it means that you have the time to reach the boundary so actually the value of g at time txv will be equal to the value of g at time s this will be always the same x minus t minus sv for all tb st but then you have to stop with the boundary condition that is little g at time t minus tb that is the time needed to reach the boundary xb v so with the inflow boundary condition you can always know that either you come from the initial condition either you come from the boundary condition and everything is fixed so actually this condition is the nicest because we can cope with very easily and actually we will be able even to write the shape of the solution in this case while if we are considering the specular or the bounce back reflection we are obliged to use this kind of generalize yes yes not at all not yet I will explain to you why I'm just trying to give you an idea of what happens for the transport equation in the case of the several different boundary conditions and then I will explain to you how you can use this to cope with Boltzmann equation no no no actually the diffusion in a way is a kind of thermal condition but not the Maxwellian one for example just this one now actually in the case of the diffusion here you cannot say anything because you don't know where the velocity is talking about specular and bounce back reflection in this case you have to follow these cycles actually even if you are imposing something at the boundary you are imposing either specular reflection or bouncing back reflection that won't change the velocity of the particle so the only data that we have is the initial datum you just have to know at what time you stop to the initial one because you are just changing the velocity either through the specular reflection either bouncing back but the information that you are propagating is always the information of the initial condition so coming back through the cycles that I just defined before in a way that was a bit technical you can link the solution to the initial datum in this way following the generalized cycle that I just defined and of course for diffuse reflection is a bit more difficult to explain what happens because you don't really know that the velocity is the average velocity so you don't have one specific velocity so the link between the initial condition and the other one is a bit more difficult let me just go a bit further into the details of our grazing set according to the different geometry of our set I will try to do the same because it was nice I like it we can have different grazing sets so let me start with this point remember the grazing set the point on the boundary such that their velocity was tangential with the boundary tangential like this here this is a grazing point or here for example and then we can have another one here I like it let's say like this ok so you see that they try to find three different situations the first one is the situation in which when you are on the boundary here and you go a bit on the left and a bit on the right you always stay inside the domain this is a non convexity point is the first one there so it means if you want to say vigorously that the time to come backward to the boundary is always positive either if you consider V or if you consider minus V so let me just underline the fact that even if I'm already on the boundary I'm not considering that the time is zero because if I move a bit I can go actually if I'm here and I move with this velocity backward I will go there if I'm here and I'm moving I will go there so the time V will be different from zero in both cases and actually this is the worst part of the boundary now you have this case where you have two possibilities either if you are here with this velocity you go backward and you arrive here so this is actually the outward inflection grazing boundary and either you are here and you go backward and you have zero in this way while if you go with the other velocity it's positive so this is the incoming because actually you have to look at this velocity when you consider this velocity you are entering when you are considering this velocity you are going outside so you have this two set and actually what you can imagine is that if a singularity is created here for the upcoming boundary nothing happens the upcoming grazing set outside while for the incoming could be that the singularity propagates inside as it was the case for this one but actually this is not possible you will see later and then we are left with this other possibility that is the case in which we are always staying outside so this is the convex grazing boundary because actually this is the point of convexity of our function of our boundary so if you move from one side on the other you always stay outside the domain so the Tb is always equal to 0 so just to note if you have a point xv that belongs to gamma 0 v the convexity since Tb xv is equal to 0 you have that xb is equal to x and this is the same actually for points here gamma 0 e- for a point gamma 0 e- Tb xv is equal to 0 so it means the same did I mistake? ok so now that we have this definition we can have a further look at our grazing set that I defined before so actually what you can see is that this set can be further divided into several other sets using the definition that I just wrote so let me just underline this is gamma 0 e- plus gamma 0 e- gamma 0 v and this one was gamma 0 singular so what happens actually our sigma b is defined as the set of point xv I was already written there so it's defined like this so you have that nxbxv v is equal to 0 so for example you see that if you take a point that is here as I that is here sorry on a convexity point this point will be precisely inside this set because whenever you compute as I told you Tbxv will be equal to 0 xv will be equal to 0 to x and so we know that this point are on the boundary are the grazing one that x times v is equal to 0 sorry nx times v is equal to 0 so it's ok this set is included down there this is the case even for this set because of the same reasoning and you can see because I put this set here in this part but whenever you consider points here or here they are not inside because the time to reach the boundary is positive for both of them this way this or this one and when you are here at the boundary you are not at all you are not obliged to be like tangential even in this way so they are not inside but so this is just the part let's say sigma b contains gamma 0v gamma 0e- so this is the part of the grazing boundary that is inside this set when you have points that are inside the domain which are the points that can propagate the singularity so which are the points that either are propagating from here either they are propagating from this one so this is what I tried to write with this gamma sb the point that propagates from gamma 0s and gamma e-b the point that propagates from gamma e- gamma 0e- so actually I just put the point on the boundary inside the same set so that's why you can see this so actually it's called like singular set in an improper way because it's not really containing all the singular points but what it contains and what is important is that it contains the point that propagates inside and actually so we are not considering the point that propagates from gamma 0b and gamma 0e- because they are going outside the domain so we don't care about them so what it turns out is that Google proved some regularity about the time backward exit time that you can see here so actually this tb is always lower semi-continues then we can say that tb and xb are a smooth function whenever the scalar product with the normal is less than 0 so actually whenever you are coming to the boundary with a velocity that is not tangential if you have a set for example that is not convex this is the case and then you can prove that when the tb when you are considering x0 v0 that is in sigma b e-b that means that it is propagating from here tb is continuous while if it is propagating from here so on this side and on this side it is discontinuous so the discontinuity is precisely propagated by the singular grazing set that set there so in order to cope with regularity we need to do something around this point so in order to realize that let's remove them either if we have a set that is not strictly convex we have to remove this part in order to prove more regularity either we try to consider sets that are strictly convex so that we avoid them in the case of a strictly convex set what happens is that no it's not strictly convex if you have a convex set strictly convex set they have singularity singularity exists but they are on the boundary like here for example all over actually but they don't propagate inside the domain so everything is easier while as soon as you have an unconvexity point the singularity can propagate inside so this is what happens so of course I'm trying to cheat a bit because I'm not really considering the set that then the particles are acting even with the boundary again in the case of transport equation but as soon as we have inflow condition for example this is the case so we can define the discontinuity set properly in this way so we have that the discontinuity set D is made up of two parts the first part is the discontinuity that we have on the boundary we have the presence of grazing particles so it's made up of the three terms gamma 0s, gamma 0v and gamma 0e plus because we know that gamma 0e minus is not producing any singularity so this is like the particle that stays on the boundary for all time they are not moving and then we have the particles that propagates but the only one that propagates inside the domain are the one coming from this so in the discontinuity set we include the 0 so the point on the boundary for all time and then the one propagating inside from the singularity set while we can even describe the continuity set in this way as the union of three parts the first part is the initial condition so the condition that we have at time t equal to 0 all over the set because if we are taking an initial condition that is regular of course then you have a continuity part that is propagating from the incoming set due to the condition boundary condition that we are taking regular so it's propagating inside and it's propagating as well from the part coming from gamma 0e minus because we know that tb is continuous at that point and then we have all the sorry that one we're not propagating we're just on the boundary for all times and then the one that propagates from this the continuity set are points that are propagating either from the initial condition because the initial condition propagates inside and propagates regularity either from the boundary condition so gamma minus and gamma 0e minus in time all over the space so this is the case whenever we have inflow boundary condition if we have a specular boundary condition or bounce back condition actually we need to cope even that once you reach the boundary with a singularity then the singularity is reflected or in this way so actually trajectory in the case of specular reflection or bouncing back reflection are propagating singularity and when they reach the boundary the singularity propagates if you have an inflow boundary condition whenever you reach the boundary with a singularity then the inflow condition prescribes you the incoming particles so erase the singularity and this is the case also for diffuse boundary condition because whenever you are on a trajectory propagating a singularity and you reach the boundary the diffuse boundary condition diffuse the velocity of the incoming trajectory and erase actually the singularity so it's no more propagating inside can be created again from a non convexity point whenever the singularity touches the boundary is erased this is more or less what happens so this is in a way the picture of what happens in the case of the transport equation with the free four boundary condition that I just showed you now we will try to make a link with the transport equation and the Boltzmann equation so we need a reference case that is the linear transport equation with inflow boundary condition so it's not yet the Boltzmann equation but we will arrive by step so we are considering now a step further that means that we are adding something linear here a new that is positive and that term here makes the things not homogeneous but with some regularity that are enough to prove that we can find a solution of a linear transport equation like this with initial condition and inflow boundary condition so actually whenever we require some regularity assumption on age and new that we have here I don't want to enter into the details but I mean you can just a bit understand why everything is very nice in this case you require some compatibility condition that is again to cope with the fact that whenever you are on the point, on the boundary at time t equals 0 the compatibility condition has to exist in order to give the continuity for the solution and then the solution of this linear transport equation is given by the Duamel formula so whenever you have the Duamel formula you have two possibilities so you are considering your function txv so either t is less than tb so it's always the same reasoning it means that I'm coming from the initial boundary condition so through Duamel formula the solution is given by e minus the integral 0 t nu s ds where nu actually was supposed to depend on txv and with the notation nu s I'm considering as written there, nu evaluated at x sx minus t minus s v v so along the trajectory here then you have the initial condition of 0 x minus tv v plus the integral from 0 to t yes I'm here e minus 0 s nu tau d tau h evaluated at time t minus s x minus sv v in ds so this is precisely the value of the function whenever you know that t is less than tb and that you are coming from the initial condition you can even have an equal here okay while if t is greater than tb then you are not coming from the initial condition but from the inflow boundary condition so you can write everything again so f t xv but instead of putting f0 here you have to consider g and actually you have to stop at the boundary because you won't go afterwards so you have minus the integral from 0 to tb now nu s ds g I have to take g at t minus tb that is the time that I need to reach the boundary xb v and then the term coming from the presence of h plus the integral from 0 to tb e minus the integral from 0 s nu s d tau h at t minus s x minus s v v so what I wrote here is just a compact way to put everything together using characteristic function of the set t less than tb and t greater than tb so this is precisely the shape of our function so everything is fine in this case because you know how the function is done so you can do estimates on the function you can see something more about the derivative of the function and something more about the regularity of the function because you have something that actually if you want to find something about the derivative with respect to time of this function you just need to derivative I mean it's a long time there are a lot of terms that has to be derivative because you have the presence of time in nu s of time inside f through x f0 through x then there then there but you can do it actually you have to be patient and you will see I wrote in one of the slide all the computation so actually in this case Google proved in 2010 that whenever you have omega that is as boot bound in domain and you consider a weight function then you suppose that the initial data is greater than 0 and satisfies this condition so you have an exponential lambda t here so you are weighting the initial condition and incoming condition through this weight function omega you suppose that all this is less than delta for a positive delta and a lambda that is positive then using two al-infinity estimates through the function that is explicit definition of the function that I gave you before he was able to prove that there exists a solution that actually weighted by this exponential function is in al-infinity and satisfies this estimate so it's like decaying with an exponential weight here to the equilibrium remember that this half capital F was always the capital F divided by square root of mu so you are considering that you are converging to the equilibrium in this way so whenever actually you have to reduce to a smaller lambda so lambda prime here that is smaller than lambda because of some computation that I'm just hiding inside actually you don't have the maxian here I was saying something this is not true sorry I was looking at the equilibrium but no no no this is not the case actually but you have this exponential here that actually is yes sorry sorry you are right it's not true I was confusing with the next estimate that are coming afterwards so okay now whenever you suppose that omega is strictly convex then whenever you have an initial datum that is a continuous and an inflow boundary condition continuous then F is continuous as well but you have to remove the grazing set so except on the grazing set the function will be continuous because the regularity of the initial condition and of the boundary condition will propagate now maybe it's better to stop here because like stating these slides alone is no sense though so maybe we will start again afterwards on this side