Henning Struchtrup,
Department of Mathematics, Arizona State University,
struchtr@math.la.asu.edu http://math.la.asu.edu/~struchtr/
I was as a postdoc at the IMA during the program
on Reactive flows and Transport phenomena (Sept. 1, 1999  June
15th, 2000).
While the large number of workshops allowed
me to learn a lot about different uptodate problems in my
and related areas, I consider as most remarkable for my stay
at the IMA the scientific contacts I made with other researchers:
workshop visitors, long term visitors, faculty of the University
of Minnesota, and fellow postdocs.
I met experts from my field of interest (kinetic
theory of gases), again or for the first time, and we had numerous
discussions about our field.
Even more valuable are the scientific collaborations
with longterm visitors and U of M faculty, which broadend my
spectrum of research considerably. In particular, I worked with
John W. Dold (longterm visitor from UMIST, Manchester) on surface
tension in reacting binary mixtures. In another collaboration
with Micheal R. Zachariah and Mitch Luskin (U of M faculty)
we developed a model for coagulation of aerosol droplets with
coagulating enclosures. The papers about these works were just
submitted. However, both topics are not closed, but raise many
questions which will be addressed in the future. For the aerosol
project we have outlined the next steps just before I left Minneapolis,
and included Yalchin Efendiev (IMA postdoc) into the team. I
also met regularly with Yalchin and Lev Truskinovsky (U of M
faculty) to discuss Yalchin's simulations of a massspring chain
with nonconvex potentials.
Moreover, I continued my own research on moment
equations in kinetic theory. My results on the solution of boundary
value problems were presented at the IMA workshop on transition
regimes.
The IMA postdoc seminar gave me the opportunity
to learn about the fields of interest of the other postdocs,
and to present and discuss my own work. I served as it's coorganier
in the fall, and gave three talks during the year.
Last but not least, I like to mention the excellent
atmosphere at the IMA, which is particularly due to the friendly
and helpful staff! Below a list of papers and presentations
relevant for my time at the IMA.
Papers:
H. Struchtrup & J.W. Dold: Surface Tension
in a Reactive Binary Mixture of Incompressible Fluids, submitted
to Interfaces and Free Boundaries, 2000
H. Struchtrup: Positivity of Entropy Production
and Phase Density for Approximate Solutions of the Boltzmann
Equation, submitted to J. Thermophys. Heat Trans., 2000
M. Luskin, H. Struchtrup & M. Zachariah: A
Model for Kinetically Controlled Internal Phase Segregation
during Aerosol Coagulation, submitted, 2000
H. Struchtrup: Heat Transfer in the Transition
Regime: Solution of Boundary Value Problems for Moment Equations
via Kinetic Schemes, in preparation
Presentations:
Kinetic Schemes and Boundary Conditions for
Moment Equations, IMA Postdoc Seminar, October 19, 1999
Kinetic Schemes and Boundary Conditions for
Moment Equations, Dept. of Aerospace Engineering and Mechanics,
University of Minnesota, February 4, 2000
Surface Tension in a Reactive Binary Mixture
of Incompressible Fluids, IMA Postdoc Seminar, March 7, 2000
Surface Tension in a Reactive Binary Mixture
of Incompressible Fluids, 2000 Midwest Thermodynamics and Statistical
Mechanics Conference, Minneapolis, May 1416, 2000
Heat Transfer in the Transition Regime: Solution
of Boundary Value Problems for Grad's Moment Equations via Kinetic
Schemes, IMA workshop on Simulation of Transport in Transition
Regimes, Minneapolis, May 2226, 2000
Coagulation of Aerosol Droplets with Coagulating
Enclosures IMA Postdoc Seminar, Minneapolis, June 13th, 2000
Heat Transfer in Rarefied Gases: Temperature
Jumps and Boundary Layers computed with Grad's Moment Method,
Department of Mechanical Engineering, University of Victoria
(BC), June 16th, 2000
Vladimir
Sverak,
Department of Mathematics, University
of University
My main result during the year is the investigation of singularities
in the Complex GinzburgLandau Equation. This is a version on
the Nonlinear Schroedinger Equation, where a viscosity term
is added. The equation is one of the important models for studying
turbulence in PDEs, and is also used in describing physical
phenomena related to focusing of waves. One of the major open
questions related to this equation is the existence of singularities
arising from smooth initial data. This problem is in fact similar
to the well known question about smoothness of 3D NavierStokes
equations. For the Complex Ginzburg Landau (CGL) equation one
can prove the existence and partial regularity of weak solutions,
which are analogous to Leray's weak solutions for the NavierStokes.
In a joint work with Petr Plechac we addressed the problem of
singularities for CGL. Using a combination of rigorous results
and numerical computations, we described a countable family
of selfsimilar singular solutions of CGL. (In fact, most of
these solutiopns are completely new even for the Nonlinear
Schroedinger equation.) Our method is not based on direct numerical
simulations. (These have been done by other authors and the
results concerning singularities coming from these calculations
are inconclusive.) A crucial part of our analysis are rigorous
results, which enable us to reduce the problem to an ODE on
a finite interval, which is then solved numerically. Our calculation
establish the existence of singularities, and, moreover, show
some unexpected and very interesting new features in the behavior
of the singularities. We believe that the significance of our
results goes beyond the CGL. For example, our results point
out some pitfalls one may want to avoid when trying to address
the problem of singulaties for the NSE. A version of our paper
on the subject is available from http://xxx.lanl.gov/ as preprint
math.AP/0007149, and a revised version will be submitted to
the IMA preprint series shortly.
Apart from these results, Xiaodong Yan and I wrote a paper "NonLipschitz
Minimizers of Strongly Convex Functionals", which appeared as
an IMA preprint No. 1675. In this paper we solve some old problems
in the regularity theory of multiple integrals in the Calculus
of Variation. We construct singular minimizers of certain smooth
variational integrals, which show that minimizers can have even
less regularity than was expected. (These question can be thought
of as an extension of the Hilbert's XIX and XXth problems to
the case of vectorvalues functions.)
Giovanni Zanzotto
, Dipartimento di Metodi e Modelli Matematici, CNRUniversita
di Padova
While at the IMA in July 2000 I have continued my work with
Lev Truskinovsky of the Dept. of Aerospace Engineering and Mechanics
(U of MN) on the constitutive theory of multiphase elastic crystals,
also in collaboration with Giuseppe Fadda, who is also visiting
that Department.
'Active' crystalline substances exhibiting martensitic transformations
are of growing importance in applications. We are investigating
the energy functions that exhibit minimizers and phase diagrams
that reproduce the behavior of the polymorphs of such multiphase
crystalline materials.
We have investigate in detail the kinematics of
a fairly typical case, that is, of a crystal whose pT phase
diagram involves three phases with progressively reduced symmetries,
tetragonal, orthorhombic, and monoclinic (tom crystal). We
have studied the details of the tom transformation mechanisms,
and have constructed an energy function for the tom crystal,
which reproduces well the experimental phase diagram of these
materials. We are studying more in depth the case of zirconia
(ZrO2), a wellknown toughening agent for transformationtoughened
ceramics (ZrO2 exhibits tom phases in the range of not toohigh
pressures and temperatures). Our analysis so far suggests that
for zirconia there are enough available experimental data so
as to allow for a completely explicit determination of its energy
function, which will be a very interesting part of the work
proposed here. If this will indeed be possible, we will be able
to study some stillcontroversial aspects of the tom transformations
in zirconia. If the available data are insufficient, we should
be able to make proposals for suitable data to be collected
in order to have an explicit zirconia energy. We also initiated
the investigation of the role of the crystal motif in the phase
transitions of zirconia, that is, of the different behavior
of the O and Zr atoms of the ZrO2 lattice during the transitions
(breaking of the coordinations, etc.). The study of the implications
of these phenomena regarding the structure of the energy function
is an almost completely unexplored aspect of the question.
