Technica Federico Santa Maria in Chile.

And he's also a visiting scientist who is at the moment still a digitally visiting scientist.

And he will give a talk on inverse problems in gravity water waves.

Please go ahead.

Okay. Well, thank you. Thank you, the organizer. Thank you, Enrique.

And thank you everybody for coming today, this presentation.

You see the presentation on the screen?

Yes, perfect.

Okay. Well, today I was talking about the gravity waves equation and the inverse problems in that equation.

Well, first, let me say this work is a collaboration with a lot of persons,

Fontelos, Lopez Rio, Ortega, and also Sebastian Zamorano.

And well, let me introduce some motivation for the question.

What is the gravity waves?

What actually is when we try to modelize the fluid where it has a free boundary,

and this free boundary, well, is free more or less because in this free boundary,

is actually in the fluid is under the gravity force.

And the idea is in this free surface, the fluid produce a wave.

This is the mathematical model.

Mathematical model here, you can see where this is a cross section of the fluid.

Then we have the batimetry, which is the surface of the bottom.

And we have the free surface on the top.

And in the middle, we have the fluid.

Of course, we have this fluid is under the gravity force in the vertical axis.

And of course, the domain is changing time because the free surface is changed.

And we assume the free surface is possible to parameterize by a function,

function with respect to the x, x is the other variables except of the vertical value.

The bottom, we also assume is parameterized by this variable.

With this parameterization, it's possible to describe the

normal component of the surface.

And well, the introduction of this equation is more or less the,

we consider for the fluid, the Euler equation.

This is the classical Euler equation.

Here is the derivative time and here is the nonlinearity of the Euler equation.

And here we have the component of the pressure and the external force.

And we assume, of course, the version of the velocity zero.

This is the conservation of the mass.

And we assume is irrotational.

And with this, the classical assumption is assumed the velocity is the gradient of other function.

In this case, we say this function phi is the potential velocity.

And then we obtain the Laplacian of this function is zero.

This is harmonic function.

And the first equation with this, the conservation of the moment,

is simply this red equation, which is called the Bernoulli equation.

This is the conservation of the energy in basically.

And of course, here we have a constant, but this constant is supposed to

delete when you fix the place when you put the zero for the potential energy.

And with this, well, this is the equation in the fluid.

But for the free boundary, we have this condition.

This condition says the y represents here the size of the column of the water.

And we say this, the water is exactly in the top, it's given by the parameterization of the free surface.