Topics In Pattern Formation Problems and Related Questions

Abstract

Our thinking in pattern formation problems is more guided by the solution of simple models than by direct investigation of the basic equations for a well identified process. The clearest example of this situation is in instabilities of fluid flows: the Navier-Stokes equations are rather hopeless, but one can understand a lot of phenomena occuring in real flows by a reduced amplitude approach.

Those amplitude equations aim at describing the dynamics of weak fluctuations and are defined usually under restrictive assumptions: large aspect ratio (many typical wavelength), supercritical or weakly subcritical instabilities, control parameter near threshold. Dropping anyone of those assumptions leads to major difficulties linked to the summation of poorly controlled perturbation series. It seems better then to try to study simple amplitude models, with the most general structure one can think of, and to compare their predictions with real life experiments.

This is the line of thought followed here. To be a little more concrete, I shall consider first a now classical question, which is the connection between the subcritical character of the bifurcation in parallel flows and the occurence of localised patches of turbulence therein. This can be explained with a simple model of the reaction diffusion (RD) type. However this model is quite special in the sense that it has a variational structure. This entails some nongeneric qualitative properties, that are no more valid for more general system, as one expects the turbulent flows are. I shall consider some of those nongeneric properties, and refer briefly to other works on other ones. I will single out a remarkable theoretical discovery, that is the possibility of stable localised structures in a nonvariational system, although in gradient/variational systems those localised structures are always linearly unstable. Then I shall examine an application of this outside of the realm of fluid mechanics, that is the dynamics of a so-called Bloch wall in a ferromagnet. Those Bloch walls are between two magnetic domains of opposite polarization. Under the action of a convenient external magnetic field, those walls drift in a direction depending on their core structure. The time dependence in the external field breaks the variational structure of the system: in variational systems walls move to replace a state on one side of the wall by the other with a lower energy, and the direction of this motion cannot depend on the core structure of the defect.

Finally I will show how to analyse, again by using general arguments only, the dynamics of a periodic structure, as a set of Rayleigh-Bénard rolls under external stress. This makes appear a rather interesting mathematical structure, linked perhaps to the famous von Karman conjecture on the buckling of plates.

This is a written summary of a talk presented at the conference held at the NATO advanced scientific institute held in Como, Italy on “Chaos, Order and Patterns”, June 25-July 6, 1990.