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).

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

A direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations. The new method provides a more systematical and convenient handling of the solution process of nonlinear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, the mapping method, and the F -expansion type methods. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. As an application, the construction problem of exact solutions to the 3 + 1 dimensional Jimbo–Miwa equation is treated, together with a Backlund transformation.

https://shell.cas.usf.edu/~wma3/MaL-CSF2009.pdf

A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation

In this article, we give some mathematical results for an isothermal model of capillary compressible fluids derived by Dunn and Serrin in [1]Dunn JE, Serrin J. On the thermodynamics of interstitial working. Arch Rational Mech Anal. 1985; 88(2):95–133), which can be used as a phase transition model. We consider a periodic domain Ω = T  d (d = 2 ou 3) or a strip domain Ω = (0,1) × T  d −1. We look at the dependence of the viscosity μ and the capillarity coefficient κwith respect to the density ρ. Depending on the cases we consider, different results are obtained. We prove for instance for a viscosity μ(ρ) = νρ and a surface tension the global existence of weak solutions of the Korteweg system without smallness assumption on the data. This model includes a shallow water model and a lubrication model. We discuss the validity of the result for the shallow water equations since the density is less regular than in the Korteweg case.

On Some Compressible Fluid Models: Korteweg, Lubrication, and Shallow Water Systems

https://www.math.mcgill.ca/~mei/YJDEQ_5804.pdf

Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity

This article is devoted to the low Mach number limit of weak solutions to the compressible Navier-Stokes equations for polytropic fluids with periodic boundary conditions and ill-prepared data. We derive formally the equation satisfied by the mean value of the velocity and the equations governing the dynamics of the nonlinear acoustic waves in dimension d = 2 or 3.

Low Mach Number Limit of Viscous Polytropic Flows: Formal Asymptotics in the Periodic Case

This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies $1 e$, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay $r$ or the wave speed $c$ is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when $e c_*>0$ are exponentially stable, where $c_*>0$ is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed $c>c_*$ and any size of the time-delay $r>0$; however, when $\frac{p}{d}> e^2$ with a smal...

/pdf/exponential-stability-of-nonmonotone-traveling-waves-for-3yjhzuu82v.pdf

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

This paper concerns the well-posedness and semiclassical limit of nonlinear Schrodinger-Poisson systems. We show the local well-posedness and the existence of semiclassical limit of the two models for initial data with Sobolev regularity, before shocks appear in the limit system. We establish the existence of a global solution and show the time-asymptotic behavior of a classical solutions of Schrodinger-Poisson system for a fixed re-scaled Planck constant.

http://www.kurims.kyoto-u.ac.jp/EMIS/journals/EJDE/Volumes/2003/93/li.pdf

Chi-Kun Lin

Papers

On Some Compressible Fluid Models: Korteweg, Lubrication, and Shallow Water Systems

Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity

Low Mach Number Limit of Viscous Polytropic Flows: Formal Asymptotics in the Periodic Case

Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Semiclassical limit and well-posedness of nonlinear Schrodinger-Poisson systems