Okay, let's start. So last time we were concerned with dealing with fractional derivatives.

The background is that in general we will not get an L2 estimate for the time derivative.

So as we will see we will only get something weaker, at least in the general case. And

therefore we introduce the notion of a fractional derivative by means of the Fourier transform

and have seen two things. First of all we have seen basically one thing which kept us

busy. We have now a new more refined compactness result which only I maybe I refer it again

because this is what we will need. This is what we have shown last time. That was the

theorem 24.2 which dealt, I make it short. So what we had is we had here as usual so

to speak concerning the space regularity the L2v, v being the small space, let's say

the h1, 0 space in time we only had h gamma 0, 0t, v0 where the Galvan triple was like

that we embedded into x continuously and compact. x embedded continuously into v0 so v0 could

be a larger space or not. And then what we have seen is there is a continuous embedding

into now Lp, L2, 0t, x the middle space. The middle space will be typically the L2 omega

space and the important thing is this embedding is compact. That means if we have boundedness

in these two spaces we not only have reconvergence which is good for linear terms in the equations

but not sufficient for non-linear terms but we also have strong convergence than in this

L2 space. And this will help us and will be enough to go to the limit what we finally

want to do. Okay now the question is how to get this information about the h gamma and

I give one theorem maybe I don't prove it because we will not use it directly and because

then in our proof we will prove really this result directly by really looking at the Fourier

transform and giving an a priori estimate for the corresponding term in the form of

the Fourier transform. So how to show that a function has this fractional derivative

in time. And this is the next proposition as I said I'm not going to prove that. The

next proposition which basically again starts with a Gelfand triple. Now the spaces are

denoted by H and V prime so now we requiring a little bit more because we require that

we have Hilbert spaces and we require really so to speak everything what is related to

a Gelfand triple so we have here not only continuous but compact in pettings here and

here and we have in addition what we called the consistent dual space structure which

basically means that the duality pairing between V and V prime that V prime is really the dual

space of V and that the duality pairing is really an extension of the scalar product

of the middle space H. So that is really so to speak the full program with respect of

the Gelfand triple and in this situation we can show the following. Let me first write

it down and then we indicate what this new space means so if we have an alpha between

0 and 1 over 4 then the embedding L2 0 TV so that is what we typically have from the

typical energy estimate with a solution and assume that we now concerning the time derivative

we know the following. We have a time derivative with values in V prime so that is the typical

thing but we do not require that this time derivative is an L2 function but we require

that is only an L1 function in time. So again so what is this so these are this W11 are

those functions which are L1 functions and the time derivative the weak time derivative

is an L1 function and the 0 means that at the boundary of the time interval so at those

two points the function vanishes in the trace sense. So if you want to apply this is a bit

of a technical problem because typically we will not have this here we will have an initial

data and here we will have no requirement at the final time. So to apply this theorem

technically we have to expand a little bit the time interval prolongate the function

by okay for example constant prolongation but of course 0 the 0 prolongation which has

problems as we will see in a second so it is not so 100 percent the technicalities are

a little bit given. So but anyhow if we have this L1 estimate of the time derivative then

we have a continuous embedding into such a fractional derivative space with values in

age. So then we could use then this corresponding compact embedding theorem above. So these

are now the technical preparations and now we would like to start with the existence