At least I had hoped that we could finalize our discussion of fluids today.

Then we have a break of one week.

Next week there will be no lecture and then we go over the calculus of variations

and on the basis of this discussion of nonlinear elasticity.

So what I have in mind for today is, first of all, to come to an end

first with the Navier-Stokes equation and what is missing is a uniqueness proof in 2D.

As we discussed already, if we would have something like that in 3D we would be richer.

In 2D it is possible and maybe we can ask ourselves what is the reason

for not being able to do a uniqueness proof.

The problem was that in the end what we would like to do is we would like to test

we would look at the difference of two solutions and of course we would like to derive an equation

so our inequality such that we can conclude that this differential inequality can only have one solution

namely a zero solution meaning that there is uniqueness.

I will only sketch it a little bit because otherwise I will not have enough time to discuss

the Euler equations finally.

So what is important is the following interpolation lemma.

This is really specific for two dimension.

It is an interpolation lemma about something strange namely the L4.

Let me first write it down that we see what the assertion is.

We have a domain, an open domain in R2 and the assertion is that we can interpolate for a 4 function

we can interpolate between the L2 and the gradient of L2.

That is we can estimate up to a well known constant square root of 2 here with the L2 norm

so we go below the level of space.

Of course an estimate of this type could not hold without anything further

and we compensate this by further knowledge of the gradient.

Also in L2 that means this is an estimate for

Oh god what is going on with the turf?

It can move.

It cannot move, not in my case.

Then we have to live with a little bit restricted blackboard space.

Of course this is for then corresponding H1 functions, actually H1 0 functions.

I am not going to prove this, it is really a specific proof related to 2D.

Because of this we can now, the problem is if we do this procedure

maybe we start with the uniqueness proof here and let's see what we would like to see.

Let's first formulate the uniqueness statement.

So uniqueness for Navier-Stokes in 2D.

So the statement is the following, n equals 2.

We don't need too much regularity, we only need open and bounded.

Nothing more, nothing about the boundary, which is also for a uniqueness proof quite natural.

And the assertion is, we know already there is a solution

and the assertion is that this is a unique solution, a solution is unique

and the meaning of the weak solution is to be that the velocity is in L2 0 T.

V is the small space, the H1 space as we have used it all the time

and we have in addition boundedness in time if we go here to the H space.

So more precisely we work with the divergence free formulation

so these are the divergence free subspaces of the H1 and of the L2 correspondingly.

So let's start with this proof and let's see what we still would need that we could do this proof.

So abstractly written, a solution is a solution of the equation.

So we have the time derivative and all the rest we put to the right hand side.

So we have the solstice, we have the linear part and then we have the nonlinear part