We will start with the lecture modeling and analysis

in continuum mechanics 2.

We will discuss several topics that partly extend what we did

in the first part of the lecture and partly

our new techniques that we learned.

So particularly, we will discuss with the transition

from microscopic to macroscopic models.

So we will start with Newton's equation of motion,

then go to so-called kinetic equations,

like the Lasov or the Boltzmann equation,

and finally arrive back at the equations,

the continuum equations of fluid dynamics.

So we will particularly learn some new methods

how to formulate interactions of particles

and their macroscopic limits that will go, again,

back to the fluid dynamics.

And we'll get some new insights on things

that were a bit maybe heuristic or unclear in the first part,

like where does the stress tensor or things

like this come from.

Another part, we'll deal with the coupling

with electrostatic interactions, something

that we encounter if you, for example,

have the flow of electrons in semiconductors

or the flow of ions, for example,

through cell membranes in biology or nowadays

an important field of applications

are also batteries where you have very similar effects.

Then finally, we also have some part

that is mainly dealing with methods, namely

some kind of asymptotics with respect

to some parameters in the system.

We had some of these examples already in the first part

in a very formal setting.

Remember, for example, the transition from Navier-Stokes

to Stokes where we had the limit of viscosity

to infinity or typically we formulate things

in a small parameter going to 0.

So the viscosity would be 1 over epsilon and epsilon

converging to 0.

And there are several important techniques

that we can have and examples that we will discuss.

Is the convergence of variational problems,

which is typically called, the concept that we use,

called gamma convergence.

Which is very important for the static situations,

like in nonlinear elasticity or in other problems

when you have a small parameter going to infinity,

you may converge to a new optimization problem.

And of course, you want to have the minimizers also converge