Okay, so everyone, welcome to our first seminar in the new semester.

And today we have Professor Anna Mascato from Pennsylvania State University.

She is an expert in analysis for nonlinear PDEs with applications, as I know, to avoid

mechanics, elasticity, and some inference problems.

So in the coming one hour, we will get to know her recent work on global existence for

2D Cremodo-Svazinski equation.

Okay, now I give the floor to Anna.

Thank you for that kind invitation for the introduction.

It's really a pleasure to speak here.

And so what I will talk about, as you said, it's some recent work on the Cremoto-Svazinski

equation.

I don't assume that you know what the Cremoto-Svazinski equation is, so we'll spend some time introducing

the problem.

And I would like to acknowledge the support of the United States National Science Foundation

and also I'm supported by Simon's Foundation Fellowship this year, so support also from

the Simon's Foundation.

So as I said, I want to spend a little bit of time introducing the problem, the equation,

and why the equation is important and why there are still several basic open questions

that are left, expect higher dimensions.

So I will study the two-dimensional, many of the results I will speak about generalized

to higher dimensions as well, but I will tell you why two-dimension is particularly important.

And then I will spend some time reviewing the new results for the 2D Cremoto-Svazinski

equation, which are actually not that many.

And again, I will explain why.

And I will present my results in this area for the Cremoto-Svazinski equation.

And then I will discuss results where we modify the equation a little bit by adding advection.

And here we're using this phenomenon of enhanced dissipation when you combine dissipation with

advection.

So I will explain what an enhanced dissipation is.

And then I will present two results, one or two depending on time, for the Cremoto-Svazinski

equation with advection.

So the advective Cremoto-Svazinski equation, one where we add a flow that's mixing and

the other one where we add a flow that's a shear flow.

And I will explain all of this.

So what is the Cremoto-Svazinski equation?

Well, it's a model for long wavelength instabilities in dissipative system.

It was derived by Cremoto-Svazinski from a coordinate free model for flame front propagation.

So if you do a certain composition, the highest order term gives you the Cremoto-Svazinski

equation.

But in fact, it's been used in other contexts, for example, for in reaction diffusion equation

and other models.

And I will tell you why in models the long wavelength instability.

So I will set the problem in two-space dimension in the periodic box.

So think of a torus, but the periods are allowed to be different.

So I will denote them by L1 and L2.

I will use the notation for the standard torus to represent it.

But keep in mind that the periods in one direction and the other could be different.

And there are two forms of the equation.

One is an integral form for a scalar potential.

And the other one is a derivative form.