So last time we discussed the Vlasov equation which is given in strong form as a first order

PDE where we have a force that consists in our case of a damping part and of this external

interaction if you have some other forces in general we would have an equation like

this here with the interaction forces and then maybe also some external force that may

depend on x and v. We've seen that using the method of characteristics

we can basically analyze still the Vlasov equation like we did for the particles but

we already can have a continuum solution in the sense that f is really a smooth density.

Then we have seen that we go back to Euler or Navier-Stokes by looking at appropriate

moments in the Vlasov equation so the obvious zero order moment is an integral of f with

respect to velocity and then we get the flux rho times u as the first moment integral f

times v. And we have seen when we derived the Euler

equation or Navier-Stokes there was a remaining part which was okay which we called the stress

tensor t which is basically the divergence of x of rho u times u so the usual term we

are used in Euler and Navier-Stokes minus the corresponding mesoscopic integral and

we have seen if f is concentrated so what we call monokinetic so f would be rho times

delta at u in the v space then the stress tensor t vanishes and we obtain the Euler

equation. If this is not the case we will have some

remaining stress tensor that may depend on the mesoscopic solution so the simplest thing

we could expect is for example if we have some kind of Gaussian distribution in the

velocity space which is called a Maxwellian in kinetic theory for some reasons then basically

this u times u and also this part here they will differ only by a multiple so u times

u is more or less the product of the expected value this is the expected value of the product

so basically here you have the variance of this Maxwellian or more precisely the covariance

matrix and this if it's isotropic is a multiple of the identity so in this case you will get

rho times the Gaussian in velocity space then in this case the density vector is equal to

a constant times well the rho is still there times the identity so this is when you get

just a pressure term in this case a linear pressure term so if this constant is d times

gradient rho so this is the simplest case when you get some pressure and if you have

more complicated mesoscopic solutions you can have all kinds of other laws for the stress

tensor okay and as I said last time either you go for solving an equation somehow mesoscopically

so either you try to get the stress tensor from some mesoscopic simulation or you go

on with an infinite system or you have some appropriate closure relation that is very

often just some ad hoc or heuristic ansatz that you use for this equation okay we are

going to look a bit more into some other version and into some well a bit more rigorous way

to get some kind of closure or some of these closed laws and the other option to get some

closure is to have some kind of small parameter limits and in order to understand this we

will look at so-called diffusive and convective scaling okay.

So this applies to the system if we are interested in some of very macroscopic pictures so we

look from reasonably far above and we go to a large time scale so what we do if you have

a convective scaling we introduce a new time variable

t tilde which is epsilon times t so this means t tilde is of order 1 if t is of order 1 or

epsilon so we go to very large time scales if epsilon is small and we also introduce

x tilde is epsilon times x we think about epsilon a very small parameter okay so let's

first look at the oil equations and see what we get there.

We start in the t variables and the original x variable with the continuity equation and

use the momentum conservation which should be minus or plus sin u plus k star rho okay

and now we do a simple chain rule.

It's very easy to figure out that dt is epsilon times the derivative with respect to t tilde

and spatial gradient with respect to x is epsilon times the spatial gradient with respect

to x tilde okay.