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Research Accomplishments in 1999-2000: Reactive Flow and Transport Phenomena


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 up-to-date 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 mass-spring chain with non-convex 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 co-organier 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 14-16, 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 22-26, 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 Ginzburg-Landau 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 Navier-Stokes 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 Navier-Stokes. 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 self-similar singular solutions of CGL. (In fact, most of these solutiopns are completely new even for the Non-linear 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 "Non-Lipschitz 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 XX-th problems to the case of vector-values functions.)

Giovanni Zanzotto , Dipartimento di Metodi e Modelli Matematici, CNR-Universita 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 p-T phase diagram involves three phases with progressively reduced symmetries, tetragonal, orthorhombic, and monoclinic (t-o-m crystal). We have studied the details of the t-o-m transformation mechanisms, and have constructed an energy function for the t-o-m crystal, which reproduces well the experimental phase diagram of these materials. We are studying more in depth the case of zirconia (ZrO2), a well-known toughening agent for transformation-toughened ceramics (ZrO2 exhibits t-o-m phases in the range of not too-high 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 still-controversial aspects of the t-o-m 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.