Okay, good morning. We started dealing with the Navier-Stokes equation and the Navier-Stokes

equation is now our first real nonlinear problem in some sense and it's actually not so easy

to deal with so we have to go through a bit of technical considerations. So what are the

major problems? Let's first start with a stationary version and then later on extended to the

in-stationary version. We have seen, so what do we need? As we have found out with dealing

with linear problems, we need approximate solutions where we can be sure that those

solutions exist. Linear problems are basically either problems we know already or even more

simple finite dimensional problems and because of the linearity existence was no problem.

So also there we have now a sort of a problem. We will finally then deal with finite dimensional

nonlinear problems and have to see which guarantees us an existence for this problem but this

is a minor point. Then we need convergence and typically we get convergence by a priori

bounds. We had already a little bit of a look in this direction and we saw as far as spatial

regularity is required, that is if we aim for H1 regularity and H1 bounds, this seems

not to be so difficult because there the nonlinear convective part goes away. So we are basically

in the linear Stokes equation. On the other hand, what about the time derivative? We saw

also that apparently it is not so easy to test with the time derivative and to get an

estimate for the time derivative which is L2 in time. So this we will not get. Okay,

that is the side of the a priori estimates. Now what about convergence? Up to now we only

argued with weak convergence and we could do this because our formulations were linear

formulations for formulation in terms of functionals. But now we have a nonlinear term and the whole

thing. Nonlinear convergence times nonlinear convergence does not lead to anything as you

might remember from functional analysis. So we need at some place strong convergence.

So we need compact embeddings of spaces which gives us a strong convergence. We had had

one theorem, the one of Aubin-Léonce which gives us such a result which tells us if we

have a Gelfand triple and we are in L2, the function is L2 in time in the good space and

the time derivative is L2 in time in the larger space then we can get a compact embedding

in between, a spatial space in between. But we do not have an L2 estimate for the time

derivative. So we need weaker notions of time derivatives. So we need fractional time derivatives.

That is one technical point and then we need more refined compact embedding theorems. That

is the technical program we would like to look at today. So we start with a stationary

equation as I said. So the stationary equation and if we now, so y is our underlying space,

so in our case is the H1, 0, omega, Rn. So we have Dirichlet boundary conditions vector

field V equals to 0 at the boundary and what we look at, we look at the divergence free

vector fields. So V is the subspace of those vector fields for which the divergence is

0 and we know already if we work within those tests, so that is the space where we are looking

for, that is the space where the solution lives in, we are looking for and if we take

the same space as a test function then we get the pressure free stationary Navier-Stokes

equation in the sense now our time derivative is gone, now I test with a test function phi

from this space, the vector Laplacian becomes gradient times gradient. Now we get here as

we defined last time the nonlinear term coming from the convective part where we have V

giving the vector field which drives the quantity and the quantity is the vector field and here

we get the test function. So actually we have a tri-linear form, a product of three terms

including the fact that written point wise we have a product form that is we have a nonlinear

term and then of course the right hand side with some f phi and the pressure term, the

gradient P term is gone by partial integration taking advantage of the divergence freeness

of the test function. So if we can solve this problem this is quite ok because then most

of the further work we did already in conjunction with the Stokes problem, namely what does

that mean if we now define let's say a functional which is just minus Laplacian of u, so let's

call the solution of this problem u. Now assume we have a solution and we look at minus Laplacian

of u and so this is then something in a this is in the dual space of y and correspondingly