I would like to thank the organizer, especially Nico for the invitation.

It's always a great pleasure to give a talk.

And I should not forget to acknowledge my co-sus, so Geng Chen is from New York, Kansas,

and Mujin Kong is from CAISD in South Korea.

And basically I will talk about presenting the equation I work on, but the equation is

pretty simple.

It's isentropic oil and Navier stocks are associated with the distribution.

And the only thing I will do today is one.

And finally, so the main result I want to present is in this limit from the Navier stocks

to Euler, but showing that we can recover as a limit the theory of Breisan and Gerdkine

Breisan based on a small data solution.

But I will show also that actually from this theory we can recover for this kind of solutions

of some sort of weak-beavis stability of Euler, what I call weak-beavis stability, in

the sense of the result of the thermos and the perna with strong.

So now basically saying that's why we have a weak solution, then it is actually, it has

to be a good solution that you know exists.

Except that here we don't compare to only a limited solution, they will compare also to

a weak-beavis solution.

And all this work is actually based on energy based message.

So it's really you can really see this as a continuation of the result of the thermos

and the perna.

But now trying to push it even to stability of this solution, which was actually a program

that was already in place a long time ago by the perna, so it's really pushing it, putting

things that we called a contraction with shift and I will explain where this name comes

from.

And finally give a couple of ideas for the proof and I think awfully we'd be done when

I reach this point.

So the equation for today is about tropic, I'm being in natural by the Navier Stokes

equation here is a biotropic, so it's a biotropic equation.

So it's as I say, well I'll consider only one D. So you have the continuous equation,

D2 plus Dx1U equals zero.

And of course you have the I-Crabble port here with a pressure that I take dependent

only on the density, that's what I call it, biotropic, gamma.

And we have Navier Stokes port, which is a diffusion new.

You can see that here I will assume that actually the viscosity term depends on the density

and I will say some words in the next slide about that.

But when this is fixed, then I consider this cosetic coefficient new here, which is just

a positive number and the gamma is to see what's happening when the new converged zero.

And basically what we want to investigate is when you converge to zero, can we recover

the series of small BV solution for the Euler equation, and recover the Euler equation,

which is the same equation, but of course we know a viscosity term.

And in this case we can call it biotropic or even isotropic, right?

What should I say?

Well, I will say more later, but basically this idea that you can recover the Euler from

Nagier Stokes, I mean it's very old idea, is basically saying that well, when you refine

your model, even if you're interested in, for instance in shocks, so discontinuous solution

at the Euler level, if you refine your model, then actually this model makes sense because

the viscosity is super small, but the physical model is really this one.

And so we should, any good solution there should be, we should be able to find them as

the new converged zero.