The problems that I'm going to discuss have very deep connections with the German scientific

community and many German scientists such as Pronto, Sommerfeld, and actually Heisenberg

all made very fundamental contributions to this field so it's not too far from the I think the

man's spirit if not the spirit if not the content of this conference. Okay so let me start the talk

right now with that being said so today I'm going to discuss some problems in and the results in the

dynamics of two-dimensional Euler and Navier-Stokes equations

and I'm particularly interested in the case when the Reynolds number is very high.

So in this field in the heterodynamic stability area so we would like to ultimately understand

what is from the PDE point of view how this type of behavior or this type of behavior influence

could develop so you see that this is a flow past the ball where you have very high turbulent flow

and here you have boundary layer which already separated from the boundary and becomes turbulent

and we also see these vertices when you have large scale fluids and you can see a lot of

vertices and they often dominate in 2D flows and these are a pair of vertices generated by

airflow so real fluids of course are three-dimensional but in some situation

2D flows are also very interesting and in fact mathematically there are still many very important

questions in given for 2D flow and not only mathematically in fact in even in physics

literature you see many papers still devoted to this topic they are particularly interesting

in vertices. So actually 2D but 2D fluids and 3D fluids are quite different in terms of dynamics

so this is a numerical simulation of a highly is a slightly viscous 2D fluid in a square torus

so you start with some sort of a random distribution of vorticity and you evolve the

2D navier stocks with very small viscosity what you see is not turbulence okay it's not turbulence

rather you see this spontaneous emergence of vortices so where the vorticity field is

radio function and they interact with each other in fact you see a simplification in the distribution

of vorticity so you have this coalesce of vertices small vertices

coalesce if they have different signs and they co-rotate if they have the same sign so on the

left you have the vorticity field the right hand side is a corresponding stream function field

so in the end you end up with two large vertices which co-rotate with each other and this is a

picture that's been intensely studied in physics literature but still like a rigorous mathematical

understanding from the perspective of navier stocks in high Reynolds number limit so I think

we can say that for the you so you have you have this kind of two different types of processes

one is this symmetrization axis symmetrization of vortices so if you have a large vertex with a small

perturbation small perturbation will simply be symmetrized around the becoming to become part of

the large vertex and the two vertices if they have the same sign would emerge so for the first process

the symmetrization what is is I think we are beginning to see a pass to the rigorous mathematical

proof but for the second part of the merging of vertices when they have the same sign that's still

very far away from a PDE explanation okay so you know we which we want to understand

automatically this phenomenon so we'll start we start with the two dimension navier stocks equation

with very small viscosity so you have so this is the navier stocks written in the vorticity

formulations you have positive omega minus nu laplacian omega plus u dot grain omega is equal to zero

so this equation is from the regularity perspective is not very interesting because we know all

solutions to this equation are smooth and exist for all time but if nu is very small then and in

fact you could you could take it as zero so you have a transition from a parabolic equation to

have a vol equation and all sorts of interesting phenomena occur when you have this type of

of singular perturbation so the actually the general as we can see as we have already seen

from numerical simulation the general dynamics are actually quite quite complicated if you if

you consider the especially if you consider dynamics up to the so-called diffusion time which is y over

nu nu is extremely small so because after after the diffusion time since would become simpler

because eventually it becomes a heady question everything disappears but the interesting dynamics

still occur over a very very long time the time scale which is y over nu and nu is very small

so we the first step a first step in understanding rigorously this kind of dynamics