to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The flow of homogeneous fluids through porous media: analogies with other physical problems

The Heat Equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. In this monograph, aimed at research students and academics in mathematics and engineering, as well as engineering specialists, Professor Vazquez provides a systematic and comprehensive presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises for the reader.

/pdf/the-porous-medium-equation-mathematical-theory-42juib4cgk.pdf

The Porous Medium Equation: Mathematical Theory

WITH the ever-widening scope of modern mathematical analysis and its many ramifications, it is quite impossible to include, in a single volume of reasonable size, an adequate and exhaustive discussion of the calculus in its more advanced stages. It therefore becomes necessary, in planning a thoroughly sound course in the subject, to consider several important aspects of the vast field confronting a modern writer. The limitation of space renders the selection of subject-matter fundamentally dependent upon the aim of the course, which may or may not be related to the content of specific examination syllabuses. Logical development, too, may lead to the inclusion of many topics which, at present, may only be of academic interest, while others, of greater practical value, may have to be omitted. The experience and training of the writer may also have, more or less, a bearing on both these considerations.Advanced CalculusBy Dr. C. A. Stewart. Pp. xviii + 523. (London: Methuen and Co., Ltd., 1940.) 25s.

Advanced Calculus I

dynamics. We have arrived at an interesting concept, seeing solutions as continuous curves moving around in an infinite-dimensional metric space X (here, the function space L1(Ω)). Viewing solutions as continuous curves in a general space is the starting point of the abstract theory of differential equations, a way that we will travel quite often. In the so-called Abstract Dynamics it is typical to forget the variable x in the notation and look at the map t 7→ u(t) ∈ X, where u(t) is the abbreviated form for u(·, t). Remarks. (1) Note that the theorem allows to define the value u(t) of a limit solution (in particular, of a weak solution) u at any time t > 0 as a well-defined element of L1(Ω). Actually, in many cases, as when Φ is superlinear and f is bounded, it is an element of L∞(Ω). (2) If u0 and f are bounded the initial regularity is better. In that case the initial data are taken in the Lp sense: ũ(t) → ũ(0) in Lp(Ω), for every p 0; if u0 is continuous, then the convergence takes place uniformly in x as t → 0, see Section 7.5.1. (3) Unfortunately, there are no equivalent L1 estimates for the Dirichlet Problem with nonhomogeneous data g 6= 0. We end this subsection with a simple but very useful consequence. Corollary 6.3 Let u be a limit solution with data u0 ∈ L1(Ω) and f ∈ L1(Q). If t1 > 0, then ũ(x, t) = u(x, t + t1) is the limit solution with data ũ0(x) = u(x, t1) and forcing term f(x, t) = f(x, t + t1). This important result is immediate for the approximations. We leave the details to the reader. Remark. Let us note that any concept of limit solution depends on the type of admissible approximations and on the functional setting in which limits are taken. The definition we propose applies in the L1 setting. If needed, these solutions will be called L1-limit solutions. For an extension see Section 6.6. 6.2 Theory of very weak solutions The continuous dependence with respect to the L1-norm is a powerful property. It has allowed us to extend the existence result for weak solutions of the preceding section and

The Porous Medium Equation

http://eaton.math.rpi.edu/Faculty/Schwendeman/Research/mpJCPRevised2.pdf

The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow

http://eaton.math.rpi.edu/faculty/schwendeman/Research/mog.pdf

Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow

The effects of the variation of chemical mechanical planarization (CMP) process parameters on slurry hydrodynamics and removal rate are studied using physically based models. The two models which are developed to describe and fundamentally understand the CMP process are (i) the lubrication model for slurry flow and ( ii) the mass transport model for material removal. The mass transport model is developed for copper CMP. Conditions for stable operation and reduced wafer scratching are identified from the lu brication model. The mass transport model takes into account the chemical reaction at the wafer surface, the slurry flow hydrodynam ics, and the presence of abrasive particles. The polish rates predicted by the model agree well with those measured experimental ly.

Two‐Dimensional Wafer‐Scale Chemical Mechanical Planarization Models Based on Lubrication Theory and Mass Transport

An adaptive numerical scheme for high-speed reactive flow on overlapping grids

Emergence of a detonation in a homogeneous, exothermically reacting medium can be deemed to occur in two phases. The first phase processes the medium so as to create conditions ripe for the onset of detonation. The actual events leading up to preconditioning may vary from one experiment to the next, but typically, at the end of this stage the medium is hot and in a state of nonuniformity. The second phase consists of the actual formation of the detonation wave via chemico-gasdynamic interactions. This paper considers an idealized medium with simple, rate-sensitive kinetics for which the preconditioned state is modelled as one with an initially prescribed linear gradient of temperature. Accurate and well-resolved numerical computations are carrried out to determine the mode of detonation formation as a function of the size of the initial gradient. For shallow gradients, the result is a decelerating supersonic reaction wave, a weak detonation, whose trajectory is dictated by the initial temperature profile,...

Donald W. Schwendeman

Papers

The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow

Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow

Two‐Dimensional Wafer‐Scale Chemical Mechanical Planarization Models Based on Lubrication Theory and Mass Transport

An adaptive numerical scheme for high-speed reactive flow on overlapping grids

Mechanisms of detonation formation due to a temperature gradient