organizing this conference it's always a pleasure to be here in Erlangen Nuremberg

and my last word before I'm coming to mathematics is some advice and for on my

own account because Santa's evening this year will be a particular evening at

least in Erlangen because the Gauss lecture the second Gauss lecture this

year takes place on December 6 here in Erlangen and we are happy that Erlangen

is hosting the Gauss lecture and I'm looking forward meeting many of you

again for the Gauss lecture. Now coming to mathematics I would like to report on

some results for a particular particular class of moving boundary problems the

so-called musket problem. So yeah let me start by the following I for many many

years in Hanover we were interested in the phenomenon of fingering and at a

certain point we came across with this article of a physicist Sharp is his name

and he gave an overview of the Rayleigh-Taylor instability and he said and this is in a sense

of a report because this was known to Rayleigh many many years ago and then Taylor in the 50s

so if if the heavy fluid pushes the light fluid so we are talking about two phase fluids the

interface between the two fluids is stable and the other way around the interface is unstable and he

in this paper he also gave the following in some sense experiment so if you put a layer of water

on the ceiling he said the air pressure would be strong enough to hold the water by far what's

happening is what you can see here now is there a point here there's a pointer so what's happening

and in a quite a fast time the interface between water and air becomes and stay unstable and later

on there are bubbles and spikes and everything falls down but in principle if you think of other

fluids for example a painting the situation is quite differently and again in Sharpe's paper he

he presented a list of physical relevant quantities for this Rayleigh-Taylor instability here is the

density ratio we will talk about that quite quite in detail also a surface tension is involved then

he said viscosity is important in this process and he gave other other impacts which which I will not

address today and of course also this list is not not complete so here is a simple setup in what

I'm interested in so we do have two fluids so gamma minus is an impermeable layer above this

impermeable layer gamma minus one there is a fluid called minus and the fluid domain is denoted

by omega minus then in on top there is a fluid plus omega plus for the its domain and there is

an interface separating these two fluids now what about the modeling we are not using first

principles of hydrodynamics neither Euler neither Navier-Stokes equations what we are doing for

the for the hydrodynamics is applying so-called Darcy's law saying that the velocity field is

proportional in the simplest case proportional to the the gradient of a potential or of the pressure

sometimes you may be interested to have here symmetric matrix to to have a more accurate

modeling but in most of what I'm going to talk here K will be just the identity okay then we do

have boundary conditions on the upper layer so gamma H is the upper layer where the fluid plus

is separated from the air there we have a so-called a dynamic boundary condition it reads like that if

the interface is given by a function H depending on T and variable X then we have a time derivative

here we have the derivative of the potential for the for the fluid plus U and there is the viscosity

for for the fluid plus and this boundary condition is simply guarantee guaranteeing that particles

on the upper surface stay on the upper surface then there is a so-called dynamic boundary condition

so you measure forces and it turns out that the potential that this dynamic boundary condition gives

you the relation that the velocity of the potential is given by the height the density times the

gravitational force and there this is the situation on the upper interface on on the middle interface

which is described by a function F we have a similar similar kinematic boundary condition

and here because it's in the in have here the two fluids involved we have here a jump condition

that's the difference of the potential is given by more or less the difference of the of the

gravitational force produced by rho plus and rho minus so this these are the equations and if we

impose that the that the fluids are not incompressible Darcy's law just gives you that the potential both

potentials have to be harmonic in the corresponding domains so this is Darcy's law plus plus

incompressibility these are the boundary conditions I talked to you already and finally at gamma minus