Nonhomogenous Fluids

In this episode Gudrun talks with her new colleague Xian Liao. In
November 2018 Xian has been appointed as Junior Professor (with
tenure track) at the KIT-Faculty of Mathematics. She belongs to
the Institute of Analysis and works in the group Nonlinear
Partial Differential Equations.


She is very much interested in Dispersive Partial Differential
Equations. These equations model, e.g., the behaviour of waves.
For that it is a topic very much in the center of the CRC 1173 -
Wave phenomena at our faculty.


Her mathematical interest was always to better understand the
solutions of partial differential equations. But she arrived at
dispersive equations through several steps in her carreer.
Originally she studied inhomogeneous incompressible fluids. This
can for example mean that the fluid is a mixture of materials
with different viscosities. If we have a look at the
Navier-Stokes equations for materials like water or oil, one main
assumption therein is, that the viscosity is a material constant.
Nevertheless, the equations modelling their flows are already
nonlinear and there are a few serious open questions. Studying
flows of inhomogneous materials brings in further difficulties
since there occur more and more complex nonlinearities in the
equations.


It is necessary to develop a frame in which one can characterise
the central properties of the solutions and the flow. It turned
out that for example finding and working with quantities which
remain conserved in the dynamics of the process is a good guiding
line - even if the physical meaning of the conserved quantitiy is
not always clear. Coming from classical theory we know that it
makes a lot of sense to have a look at the conservation of mass,
energy and momentum, which translate to conserved quantities as
combinations of velocity, its derivatives, pressure and density.
Pressure and density are not independent in these simplified
models but are independent in the models Xiao studies. In the
complex world of inhomogeneous equations we lose the direct
concept to translate between physics and mathematics but carry
over the knowledge that scale invarance and conservation are
central properties of the model.


It is interesting to characterize how the complex system develops
with a change of properties. To have a simple idea - if it is
more developing in the direction of fast flowing air or slow
flowing almost solid material. One number which helps to see what
types of waves one has to expect is the Mach number. It helps to
seperate sound waves from fluid waves. A mathematical/physical
question then is to understand the process of letting the Mach
number go to zero in the model. It is not that complicated to
make this work in the formulae. But the hard work is done in
proving that the solutions to the family of systems of PDEs with
lower and lower Mach number really tend to the solutions of the
derived limit system. For example in order to measure if
solutions are similar to each other (i.e. they get nearer and
nearer to each other) one needs to find the norms which measure
the right properties.


Xian was Undergraduate & Master student at the Nanjing
University in China from 2004 to 2009, where she was working with
Prof. Huicheng Yin on Partial Differential Equations. She
succeeded in getting the scholarship from China Scholarship
Council and did her PhD within the laboratory LAMA (with Prof.
Raphaël Danchin on zero-Mach number system). She was member of
the University Paris-Est but followed many master courses in the
programs of other Parisian universities as well.


In 2013 she spent 8 months at the Charles University in Prague as
Postdoc within the research project MORE. There she collaborated
with Prof. Eduard Feireisl and Prof. Josef Málek on understanding
non-Newtonian fluids better.


After that period she returned to China and worked two years at
the Academy of Mathematics & Systems Science as Postdoc
within the research center NCMIS. With Prof. Ping Zhang she was
working on density patch problems. Before her appointment here in
Karlsruhe she already returned to Europe. 2016-2018 she was
Postdoc at the University Bonn within the CRC 1060. She was
mainly working with Prof. Herbert Koch on Gross-Pitaevskii
equations - a special topic within dispersive equations.

References

Short Interview with the CRC 1173 Wave phenomena

X. Liao, R. Danchin: On the wellposedness of the full
low-Mach number limit system in general Besov spaces. Commun.
Contemp. Math.: 14(3), 1250022, 2012.

X. Liao: A global existence result for a zero Mach number
system. J. Math. Fluid Mech.: 16(1), 77-103, 2014.

X. Liao, E. Feireisl and J. Málek: Global weak solutions to a
class of non-Newtonian compressible fluids. Math. Methods Appl.
Sci.: 38(16), 3482-3494, 2015.

X. Liao: On the strong solutions of the nonhomogeneous
incompressible Navier-Stokes equations in a thin domain.
Differential Integral Equations: 29, 167-182, 2016.

X. Liao, P. Zhang: Global regularities of 2-D density patches
for viscous inhomogeneous incompressible flow with general
density: high regularity case, 2016.