So today we have the pleasure to welcome to our seminar Professor Weiwei Wu.

So she is the Associate Professor of Mathematics at the University of Georgia in the US.

Before joining the University of Georgia, Dr. Wu held a tenure-track position in mathematics at Oklahoma State University from August 2016 to July 2019.

She also held a postdoctoral fellowship at the Institute for Mathematics and its Applications, University of Minnesota, on the program Control Theory and its Applications, from September 2015 to August 2016.

She is also a non-tenure-track assistant professor position in mathematics at the University of Southern California from August 2012 to May 2015.

She received her doctorate in Applied Mathematics at Virginia Tech University in May 2012.

Dr. Wu's current research interests include mathematical control theory of partial differential equations, control and estimation of flow transport systems, and computational methods for optimal control design.

She has been awarded various grants from the NSF and the DOD. She was also awarded the Wimbledon Return Fellowship for Experience Researches in 2004.

So, Professor Weiwei, the floor is yours.

Thanks, Andreas, for your kind introduction.

Today I'm going to give an introduction, actually, on control design for mixing and compressing flows, the topic I have been working on in the past few years.

So, this project has been supported by NSF and the Aero Force.

So, of course, first of all, I'm going to explain what's so-called mixing and how we're going to achieve good mixing and how we construct our control for possible optimal mixing.

So, let's start.

All right.

So, first, so this is my outline for today's talk. So, I think I have two hours, right?

I'll try to keep it within 15 minutes. So, I will start with mixing phenomena. I'll explain what's so-called good mixing.

And then we mainly will focus on objective mechanisms for mixing.

And then, of course, here, mathematically, we have to be rigorously talking about what we mean by good mixing, what would be a mathematical measure to quantify mixing.

And then in the end, I'm going to focus on the control design, both open and closed control designs.

So, mixing.

So, basically mixing is to disperse one material or field in another medium. It occurs many natural phenomena and industrial applications.

Actually, we say this every day in daily life, like mixing in painting, mixing in baking. I guess everyone has this experience about this phenomenon.

And from large scales, such as spreading the pollutant in the atmosphere and mixing of temperature salt and then the

And then controllable and fast mixing is corrective in practice, actually, for microfluidic devices, especially with light on chip devices.

So, basically mixing, as you've already sort of had a basic idea about the application I'm showing. So, mixing means a basic process where both are sterile.

So, here, sterile is especially referred to as the objective mechanism. Later on, I'm going to come back to this more on this term.

And then the diffusion occurs simultaneously. So, now, what do we mean by sterile? Sterile means the advection of materials, material blobs subject to mixing without diffusion action.

So, then let's recall what's called diffusion. Diffusion is the average effect of small scale random particle motion.

So, in this case, we learned this in high school physics. And then based on this cartoon, actually, let's say, if you look at the figure A,

so, say we have two fluids in contact with each other. And then the exchange molecules across an interface, next to have black and white,

which stands for different fluids. So, the molecules crossing the interface between these different fluids actively divide the molecule's random action.

So, figure A somehow shows before starting the exchange, the black and white blobs start to exchange.

And then here, figure B shows its entire near state during the exchange. And then, as you can imagine,

like, although this random motion occurs everywhere in the two fluids, but it does not really cause much difference for the same fluids which are far away from the interface.

However, in the region near the interface, the molecules on both sides have different properties than random motion.

The random molecule motion results in permission. If you look at the figure B, right, and then from one side to another side.

So, such permission is so-called diffusion. And then if you recall the so-called the fake losses,

the flux of one species through the interface is proportional to the gradient of the concentration of those two fluids.

And this proportional constant is then defined as the diffusivity constant.

So, the diffusion mixing can be made effective if there are sufficiently small blobs of one fluid immersed in the region of the other.

And the length scale of these blobs reaches the diffusion length scales, then both blobs diffusive to the other fluids without including mixing.

If you want diffusive mixing to occur. So, the diffusive scale of the blob itself is quite important.

Otherwise, the diffusive mixing is inefficient.

Then in summary, wouldn't mixing basically, especially here I consider like low diffusivity materials occur in two stages.

The first sterium in the first stage, and then diffusion in the second stage.

So, we'd like to sort of chew the property of the vector structure such as chaotic stretching and folding,

which lead to fluid blobs or fine phalanenins, and then the diffusion is able to take in.

But here in this part, we're mainly focused on advection dominant test.

So, in this case, I'm going to assume molecular diffusivity is negligible.