Okay, apparently the meeting is being recorded.

So first I want to motivate this object for you.

And the way that I'll motivate it for you is I'll really just show you how to use it.

And so once we kind of understand why we even want to include entropy, then I'll construct

it.

And then I think we'll be basically out of time.

And so I'll quickly summarize our results and say a couple of words about future directions.

Okay.

So first of all, well, what are cross-diffusion systems?

Cross-diffusion systems look like this.

So it's a system of strongly coupled, quasi-linear, possibly degenerate parabolic PDEs.

So we use lambda to denote the space-time domain.

And so for the purposes of this talk, we assume that lambda is a bounded domain in RD.

And we assume that D is bigger than or equal to 2.

So here the unknown u is a vector with little n components.

And usually we look at the system with some no-flex boundary data and some initial condition.

I mean, you don't have to use no-flex here.

You could also use some sort of mixed, traditionally Neumann condition.

But for simplicity, we just take no-flex.

And so systems of this type are quite ubiquitous in nature.

So they can be used to model many different kinds of physical phenomena.

So in particular, you can use them to model gas or fluid mixtures.

So this would be the Maxwell-Stefan model, which goes back to the 1800s.

You can also use systems of this type to model population dynamics.

So that would be the SKT model, which I'll talk a lot about in this talk.

You can also use systems of this type to model the flow of electrons and holes through semiconductors.

So I think that this is a less well-known model than, say, Maxwell-Stefan or SKT.

But I'm including it here because this specific model falls within the models we can treat

with our regularity theory.

And this model was derived by Reznik in 95.

There's also a well-studied model for tumor growth by Jackson and Brin.

And there is many more examples of cross-diffusion systems.

So just to be clear, so let's say that we're using this cross-diffusion system to model

some sort of chemical reaction.

Then the unknown u, so the components of u, we would think of them as chemical densities.

And so their interaction would be governed by the diffusion coefficients and the reaction

terms.

And of course, if you were modeling population dynamics as opposed to some sort of chemical

reaction, then the components, you would think of them as relation densities.

And so before I jump into the details, I want to first give you a heuristic description

of our main results.

So what are we heading towards?

So first of all, the existence, uniqueness, and longtime behavior of weak solutions of

cross-diffusion systems have all been quite well studied in recent years.

So there's tons of open questions to be had in these areas, but there's also, I think,

quite a few works addressing these issues.

But there's not many works out there that are really looking at the regularity of these

weak solutions.

And so the main result that we were after has the following form.

So we assume that A is uniformly continuous and F is continuous.