Turbulence

Martina Hofmanová has been working as a professor at the
University of Bielefeld since October 2017. Previously, she was a
Junior Professor at TU Berlin from February 2016 onwards, and
before that an Assistant Lecturer, there.


She studied at the Charles University in Prague, and got her PhD
in 2013 at the École normale supérieure de Cachan in Rennes. Her
time in Germany started in 2013 when she moved to the Max Planck
Institute for Mathematics in the Sciences in Leipzig as a
postdoc.

Gudrun and Martina talk about randomness in the modeling of fluid
motion. This topic is connected to the study of turbulent flow.
Of course, we observe turbulence all around us, i.e. chaotic
behaviour of the pressure and the velocity field in fluid flow.
One example is the smoke pattern of a freshly extinguished
candle. Its first part is laminar, then we observe transitional
turbulent flow and fully turbulent one the further away the smoke
travels. A second example is Rayleigh Bénard convection. Under
the influence of a temperature gradient and gravity, one observes
convection rolls when the temperature difference between bottom
and top becomes large enough. If we look more closely, one can
prescribe the motion as a mean flow plus random fluctuations.
These fluctuations are difficult to measure but their statistical
properties are reproduced more easily.


A general procedure in physics and science is to replace
expensive time averages by ensemble averages, which can be
calculated together on a parallel computer. The concept why this
often works is the so-called ergodic hypothesis. To justify this
from the mathematical side, the main problem is to find the right
measure in the ensemble average.


In the model problem one can see that the solution is
continuously dependent on the initial condition and the solution
operator has a semigroup property. For random initial conditions,
one can construct the solution operator correspondingly.


Already with this toy problem one sees that the justification of
using ensemble averages is connected to the well-posedness of the
problem. In general, this is not apriori known. The focus of
Martina's work is to find the existence of steady solutions for
the compressible flow system, including stochastic forces with
periodic boundary conditions (i.e. on the torus).


At the moment, we know that there are global weak solutions but
only local (in time) strong solutions.


It turned out that the right setting to study the problem are
so-called dissipative martingale solutions: Unfortunately, in
this setting, the velocity is not smooth enough to be a
stochastic process. But the energy inequality can be proved. The
proof rests on introducing artificial dissipation in the mass
conservation, and a small term with higher order regularity for
the density. Then, the velocity is approximated through a
Faedo-Galerkin approximation and a lot of independent limiting
processes can be carried out successfully.


The project is a collaboration with Dominic Breit and Eduard
Feireisl.


References

M. Hofmanová: Stochastic partial differential equations,
Lecture notes, Technical University of Berlin, 2016.

D. Breit, E. Feireisl, M. Hofmanová, B. Maslowski: Stationary
solutions to the compressible Navier-Stokes system driven by
stochastic forces, preprint, 2016.

D. Breit, E. Feireisl, M. Hofmanová: Local strong solutions
to the stochastic compressible Navier-Stokes system, preprint,
2016.

D. Breit, E. Feireisl, M. Hofmanová: Compressible fluids
driven by stochastic forcing: The relative energy inequality and
applications, Comm. Math. Physics 350, 443-473, 2017.

D. Breit, M. Hofmanová: Stochastic Navier-Stokes equations
for compressible fluids, Indiana Univ. Math. J. 65 (4),
1183-1250, 2016.



Podcasts

N. Vercauteren: Lokale Turbulenzen, Gespräch mit S.
Ritterbusch im Modellansatz Podcast, Folge 144, Fakultät für
Mathematik, Karlsruher Institut für Technologie (KIT), 2017.
http://modellansatz.de/lokale-turbulenzen

B.Valsler, D. Ansell: The Science of Turbulence, The Naked
Scientists Podcast, 2010.