G.J. van den Berg

Stable patterns for fourth-order parabolic equations

We consider fourth-order parabolic equations of gradient type. For the sake of simplicity, the analysis is carried out for the specific equation u

Numerical Path Following as an Analysis Method for Electrostatic MEMS

This paper aims to find all static states, including stable and unstable states, of electrostatically actuated microelectromechanical systems (MEMS) device models. We apply the numerical path-following technique to solve for the curve connecting the static states. We demonstrate that device models with 2 DOF can already exhibit symmetry-breaking bifurcations in the curve of static states and can have multiple disjoint solution paths. These features are also found in a finite-element method (FEM) model for a flexible beam suspended by a torsion spring. We have observed multiple hysteresis loops in measurements of a capacitive RF-MEMS device and have captured the qualitative features of these measurements in a model with 5 DOF. Numerical procedures for determining stability of solutions and finding bifurcation points are provided. Numerical path following is shown to be an efficient technique to find the curve of static states both in low-dimensional models and in FEM models.

(In-)Stability of singular equivariant solutions to the Landau-Lifshitz-Gilbert equation

In this paper we use formal asymptotic arguments to understand the stability proper- ties of equivariant solutions to the Landau-Lifshitz-Gilbert model for ferromagnets. We also analyze both the harmonic map heatflow and Schrodinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic. Solutions permitted to deviate from radial symmetry remain global for all time but may, for suitable initial data, approach arbitrarily close to blowup. A careful asymptotic analysis of solutions near blowup shows that finite-time blowup corresponds to a saddle fixed point in a low dimensional dynamical system. Radial symmetry precludes motion anywhere but on the stable manifold towards blowup. A similar scenario emerges in the equivariant setting: blowup is unstable. To be more precise, blowup is co-dimension one both within the equivariant symmetry class and in the unrestricted class of initial data. The value of the parameter in the Landau-Lifshitz-Gilbert equation plays a very subdued role in the analysis of equivariant blowup, leading to identical blowup rates and spatial scales for all parameter values. One notable exception is the angle between solution in inner scale (which bubbles off) and outer scale (which remains), which does depend on parameter values. Analyzing near-blowup solutions, we find that in the inner scale these solution quickly rotate over an angle {\pi}. As a consequence, for the blowup solution it is natural to consider a continuation scenario after blowup where one immediately re-attaches a sphere (thus restoring the energy lost in blowup), yet rotated over an angle {\pi}. This continuation is natural since it leads to continuous dependence on initial data.

Rigorous Computation of a Radially Symmetric Localized Solution in a Ginzburg–Landau Problem

We present a rigorous numerical method for proving the existence of a localized radially symmetric solution for a Ginzburg--Landau-type equation. This has a direct application to the problem of finding spots in the Swift--Hohenberg equation. The method is more generally applicable to finding radially symmetric solutions of stationary PDEs on the entire space. One can rewrite such a problem in the form of a singular ODE. We transform this ODE into a finite domain and use a Greens function approach to formulate an appropriate integral equation. We then construct a mapping whose fixed points coincide with solutions to the ODE, and we show via computer-aided analytic estimates that the mapping is contracting on a small neighborhood of a numerically determined approximate solution.

Multiplicity and stability of travelling wave solutions in a free boundary combustion-radiation problem

We study travelling wave solutions of a one-dimensional two-phase Free Boundary Problem, which models premixed flames propagating in a gaseous mixture with dust. The model combines diffusion of mass and temperature with reaction at the flame front, the reaction rate being temperature dependent. The radiative effects due to the presence of dust account for the divergence of the radiative flux entering the equation for temperature. This flux is modelled by the Eddington equation. In an appropriate limit the divergence of the flux takes the form of a nonlinear heat loss term. The resulting reduced model is able to capture a hysteresis effect that appears if the amount of fuel in front of the flame, or equivalently, the adiabatic temperature is taken as a control parameter.

Gargoylism (mucopolysaccharidosis type II). accumulation of glycosaminoglycans, gangliosides and glycoproteins and activity of some related glycolytic enzymes in liver, spleen and brain

Abstract For a case of gargoylism (Hunters disease, mucopolysaccharidosis, type II) the effect of enzyme deficiencies on the sugars of glycosaminoglycans, gangliosides and glycoproteins were compared. To study this the glycoproteins and glycosaminoglycans after lipid extraction, were first solubilized by proteolytic digestion with pronase or papain. Subsequently both classes of compounds were separated on DEAE-Sephadex A-50. Compared to the normal values the glycosaminoglycans were elevated about 100-fold, 10-fold and 3-fold in liver, spleen and brain grey matter, respectively. The gangliosides, as measured by their sialic acid content, were much less affected. In liver, spleen, brain grey and white matter respectively 2.90-, 3.98-, 1.16- and 2.16fold values were obtained. The glycoproteinic sugars were hardly influenced, except for sialic acid from liver and brain white matter (1.87- and 1.92-fold respectively). The lipid content of brain white matter was about half the normal value. Both for grey and white matter the ganglioside pattern was abnormal. In both cases G 1 and G 2 were diminished. In grey matter G 3 a , G 4 , G s5 and G 6 were elevated. In white matter G 3 and G 3 a . β-Galactosidase activity was very low in liver (4.7%) and spleen (11.0%) and moderately affected in brain grey (41%) and white (24%) matter. β-Galactosaminidase, β-glucosaminidase and β-glucuronidase activity were enhanced, especially in liver.

Travelling waves in a radiation-combustion free-boundary model for flame propagation

We study a combustion-radiation model which models premixed flames propagating in a gaseous mixture with inert dust. This model combines diffusion of mass and temperature with reaction at the flame front. We choose a free boundary model to describe the propagating flames and take a linearized approximation to model the radiation, but we keep a nonlinear reaction term which is temperature dependent. The radiative transfer of thermal energy emitted and absorbed by dust is modelled using the Eddington equation. We analyse the bifurcation diagram of the travelling wave solution curve. In a specific parameter plane, travelling waves are given by a single smooth curve which is parameterized by the flame temperature.