So I hope you are all registered for the exam.

So if not, contact Yanni Elear.

I hope you got the email all safely if there are problems with the exam registration.

Just tell us and we hope we can still register you.

It's not so clear because starting yesterday they have switched to the new system.

So we'll see what happens.

So just to remind you, we had looked at equations of this form.

We had some function of the gradient Q. We will now switch to stick to this form with

a function of gradient Q squared because it's a bit easier to write.

And we started to look into energies and their derivative.

So for example, we had the heat equation with the energy gradient U squared integrated on

the image domain.

And we have defined first of all the directional derivative in the direction phi, which was

integral grad U grad phi.

And then we've seen if it's a linear operator, like in this case, we can write it as the

Gatot derivative, so it's a linear operator applied to phi.

And the question was how to represent this or how to write this in which scalar product

if you are in the Hilbert space, if you are in some Hilbert space X, somehow the form

of this one will depend on X.

Somehow you always get the same, it's always also the same as the directional derivative

for each phi.

But of course, if you want to have the same thing here and you change the scalar product

here and keep the same phi, you have to change the form of the E prime.

Okay, we've seen for X equals L2, we had E prime was minus the Laplacian of U.

And for X equals W12, the Sobolev space, which the scalar product being integral Uv plus

gradient U gradient V.

Then we just get some equation for E prime of U.

Or if you formally write this, we have E prime of U is minus Laplacian plus identity inverse

Laplacian of U.

And there is another minus here.

Okay.

So we somehow in the L2 form, we just compute two derivatives of U in the H1 form.

We compute also these two derivatives, but then we kind of integrate twice.

Okay, so it's the inverse of a derivative.

So it's like, it's a bit like integrating twice.

And if you put this into the scalar product and integrate by parts, you see you get the

same directional derivative, of course.

Okay.

We also see for the heat equation.

So this is Laplacian of U.

So this is minus E prime of U.

Okay, this case in L2, the L2 derivative.

Then in general, we say if we have a form like this, we call this a gradient flow.

So what's happening here, so this is somehow the negative gradient of the energy and we

move in the direction of the negative gradient.

So we make a simple picture if you would have an energy like this.

Then of course E prime goes into this direction and minus E prime always goes downward.

So we always decrease the energy.

That's the interesting point here.

We can easily see what's happening.