I'm going to talk about Lagrangian controllability of the Navier-Stokes equation in 2D.

So I start with explaining what is the equation to record you what is the equation and the

basic results we have.

So we are going to consider the Navier-Stokes equation in the incompressible city which

means that there is also the divergence free condition and you see the other instance in

particular let me just emphasize if you are not so used to this equation but here there

is a black-glass term.

In particular if you drop if you discount distance then it is the Euler equation.

So that is another one with some diffusive effect and here I will focus on the 2D setting

which is a little bit more simple than the 3D setting.

In particular considering the Cauchy problem I'm going to record first what is known as

the Cauchy program it's a pretty nice event for quite irregular initial data let's say

L2 data so there is a theorem which is well known which is due to Jean Loret so more

than 100 years no it should be less than 100 years ago.

However the theorem is that let's say that you consider a bounded domain let's say in

2D and then initial data which is only L2 let's say U0 which is L2 and divergence free

to say that it's divergence free I have reduced this notation sigma it has to satisfy the

constraint at the beginning and for such a data you have existence and uniqueness of

solution which is continuous in time so it's a global solution so continuous in time with

values in L2 and it is also in L2 with values in H1 so this is due to the diffusive part

or some regularization of the data and the uniqueness actually the continuity as well

are really linked to the 2D setting I emphasize that and I believe that actually the work

of Loret contains the existence the uniqueness it's rather due to a 3D ambience so you have

this nice setting for the Cauchy problem and you have also some extra properties which

I want to give to you about the...

So what is sigma?

It's incompressible it's just to say that divergence of U0 is U0 it's a solenoid add

to L2 with a constraint of the divergence free so there is an extra part for that theorem

regarding the corresponding motion of the particle so the second part is the following

one moreover if you consider the associated ODE so you consider capital X the solution

associated with this vector field so U of t and evaluated in that capital X with the

initial data at time zero which is in X so this is the motion of the particle you have

a particle which is in X and you follow the trajectory in time so this is well defined

it's well posed meaning that you have a uniqueness an existence and a uniqueness it's even more

precise in the sense that you also have some properties which are somehow global in X so

it's something like there exists a unique flow a flow map let's say which is infinity in

time and in space it's quite okay in the sense that it is older so you have some older regularity

actually locally so you see that you have existence and uniqueness of trajectories thanks

to that quite two regularities enough to define the trajectories of the particles and actually

I also here but you don't need anything on the regularity of the domain for this part

but then what you have is only inside so you may have something quite bad but you know you

look at the particle X and for some time you will be able to look at the

you can look at the trajectories so that is the first thing you may be interested in the motion

of the particles in such a setting and here what I want to do is not so much about the koshi

problem it's about the confiability issue so if the regularity is so low how do you guarantee

this existence and you think this for the EU is not lifted right yeah but you can use the regularization

in time in order to have that u is actually not legit but it's just up to half a log

and then you can define so there are various statements there are various ways to prove the

existence of this flow there is one which is due to schoeman and then where they use as you may

imagine a nice base of species and so on and there is recently another approach by robinson