So I came exactly when I think Enrico was starting here in Erlangen in 2019 before COVID.

I was completely obsessed by a problem that was given to me or I talked or was suggested

by Charlie During. I was completely obsessed for years about this problem. It was about mixing

and the Bachelor of Scale and I came to Enrico because I was pretty desperate and trying to get

ideas on how to solve the problem. Today I'm not talking about the Bachelor of Scale conjecture

that is still open, it's a very difficult problem. I talk about something that in some sense can be,

has some connection with it. So I'm not talking about mixing, I'm talking about an important

problem which is an anticipation. Let me just mention from the start, so this topic I'm talking

about is a joint project with my PhD student Johannes Benthouse. Okay so let me start. Let's

start very easy. So this is a PDE talk, apologies there will be no machine learning, but in a sense

you would see that is a problem that you can recast in a control problem and it could be

interesting also for more applied questions. Okay so let me take the advection diffusion equation,

very simple. So you have a quantity which is a scalar quantity which diffuses and is

advected by a velocity field. The velocity field I assume to be incompressible and for

technical reason since the average is preserved of theta, then let me take without loss of

generality mean zero concentration theta. I am discussing now this problem on a torus,

it's not really a physical domain, but in a sense to illustrate this problem the torus is

doing is simplifying pretty much and still it will give you the ideas of an anticipation. So let me

take t2 or t3. The examples, so the advection diffusion equation can be found in many models

you're working with. That could be for example modeling the vorticity evolution for Navier-Stokes

equation in 2D, then it could be the temperature in the Boussinesq equation and so on. So you have

seen this equation many times I guess. There is the interplay between advection which is this bulk

motion and diffusion instead that happens at a molecular level and is the frantic motion of

molecules that goes from a region of high concentration to region of low concentration.

The application you can imagine from fluid dynamics, geophysical process, heat transfer

process, atmospheric convection and the pharmacine ethics. So and I will actually

talk about this towards the middle of this presentation. Okay so let me just start to

give you some idea on how we get to an anticipation and the definition of an anticipation itself.

So let me now take u equals 0, let's take the heat equation, you know it. Now I'm not specifying

the domain, you can take the torus, you can take about the domain with reasonable boundary

condition. Don't take Rn in this case because otherwise we have another type of decay. So

anyhow you do the classical energy estimates. There is no u, okay, so and then you see the

second line here so you have d dt of the L2 normal theta is equal minus kappa. Kappa is very

important in this talk. Kappa is the molecular diffusivity and typically kappa is very small in

the application and then you have the gradients appearing of theta, right? So it means already

this estimate that energy is decaying. If you want you can rewrite it by integrating in time,

you can write it like this and the reason why I write it like this is because you see just

as for a later observation in the talk, if kappa is very small, if kappa goes to 0, you expect

energy conservation, right? So this is what happens in the transport equation, okay? Okay so now let's

go from the d dt of theta in L2. Here now in fact you can apply the Poincare estimate very

easily. I took this theta average free also for this reason so that we can apply the Poincare

estimate very easily and you get to the exponential decay. This is how you show in the torus how you

the exponential decay for the heat equation, okay? The exponential decay apart the constant

minus kappa, kappa is the molecular diffusivity times d. Okay now this is the heat equation,

there is no u but now let's put u again and then you see what you have that if u is incompressible,

so it's divergence free, u drops out. You will not see u at all, you can run exactly the same

computation, you get to the same exponential decay. Now this exponential decay, I mean the

heat equation has the exponential decay which is very good, we know, but we can't believe that

this is the real exponential decay that we have. We imagine actually a faster decay because exactly

there is this bulk transport that is also creating this shift from so or the decay is moving energy