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.