Applied mathematics, the Hans van Duijn way

This is a former PhD student's take on his teacher's scientific philosophy. I describe a set of `principles' that I believe are conducive to good applied mathematics, and that I have learnt myself from observing Hans van Duijn in action

Generalized monotonicity from global minimization in fourth-order ODEs

We consider solutions of the stationary Extended Fisher-Kolmogorov equation with general potential that are global minimizers of an associated variational problem. We present results that relate the global minimization property to a generalized concept of monotonicity of the solutions. This monotonicity can be described as the lack of intersections of the solution curve when projected onto the $(u,u')$--plane. Our method is based on applying a cut-and-paste argument in the space H^2( {mathbb{R ) to intersections in the $(u,u')$--plane. The statements and proofs are presented for the Extended Fisher-Kolmogorov equation, but the method can be directly extended to a wide class of fourth-order ordinary differential equations that derive from minimization problems

Non-existence and uniqueness results for the extended Fisher-Kolmogorov equation

We give a classification of all bounded solutions of the equation [ u'''' + p u'' + F'(u) = 0, qquad -infty < t< infty, ] in which $F$ is a general quartic polynomial and $p$ is restricted to various subsets of $(-infty,0]$. These results are obtained by combining an a priori estimate with geometric arguments in the $(u,u'')$-plane

Kink band instability and propagation in layered structures

A recent two-dimensional prototype model for the initiation of kink banding in compressed layered structures is extended to embrace the two propagation mechanisms of band broadening and band progression. As well as interlayer friction, overburden pressure and layer bending energy, the characteristics of transverse layer compressibility and foundation stiffness are now included. Experiments on constrained layers of paper show good agreement with the predictions of angle of orientation, kink band width and post-kink load-deflection response obtained from the model

Quadratic and rate-independent limits for a large-deviations functional

We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers’ law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via ‘L log L’ gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process

Spatial localization for a general reaction-diffusion system

We use a local energy method to study the spatial localization of the supports of the solutions of a reaction-diffusion system with nonlinear diffusion and a general reaction term. We establish finite speed of propagation and the existence of waiting times under a set of weak assumptions on the structural form of the system. These assumptions allow for additive and multiplicative reaction terms and space-and time-dependence of the coefficients, as well as a divergence-free convection term