Why multiple scale analysis?

Many real world systems are interesting precisely because they exhibit different behaviors on different scales. This is certainly true for living organisms, geological and geophysical systems, materials and condensed matter systems and even social structures and hierarchies. Thus researchers across many disciplines grapple with the following two questions, which are the essence of multiple scale analysis:

Collective behavior :

What are the rules governing the small scale (microscopic) units in a large (macroscopic) system, and how does the macroscopic behavior emerge out of the collective microscopic behavior?

Small scale phenomena :

What are the rules governing the large scale behavior, and how does this influence the behavior on the small scale?

Here are two contexts in which multiple scale behaviors occur in mathematics and in physics.

Nonlinear PDE

This argument is adapted from Ref. [Eva90]. Assume that we are trying to solve a ``hard'' nonlinear PDE, written schematically as A[.] = 0. A natural approach is to set up an approximating sequence of problems A_k, that are easier to solve. Then, if u_k satisfies A_k[u_k] = 0, we would like to show that (for an appropriate subsequence) u_k $\rightarrow$ u^* and A_k $\rightarrow$ A in appropriate senses, so that A[u^*] = lim A_k[u_k] = 0, thereby giving a solution to the nonlinear PDE.

For this approach we need ``good'' uniform estimates on the sequence u_k to ensure a sufficiently strong convergence u_k $\rightarrow$ u^*. u_k may converge to u^* only weakly, and the nonlinearities in A are not in general compatible with weak convergence [Eva90], i.e. A[u^*] $\neq$ lim A_k[u_k]. It is therefore necessary to investigate the convergence of u_k to u^*, and characterize the possible failure of the sequence u_k to converge strongly.

There are two ways in which a weakly convergent sequence u_k can fail to converge strongly. Firstly, the sequence may develop rapid oscillations with an amplitude that does not go to zero. The sequence converges weakly but it doesn't converge pointwise. This is the problem of oscillation [Eva90]. Secondly, even if the functions converge pointwise and we can thus rule out oscillations, the functions could concentrate mass on sets of measure zero. This is the problem of concentration [Eva90].

Physical systems are often modeled by nonlinear PDE. Since physical systems cannot have arbitrarily small structures, the problems of oscillation and concentration for the model PDE are reflected by microstructure (oscillation on a small scale) and (near) singularities (concentration on a small scale). It is therefore of fundamental importance, in the study of nonlinear PDE [Tar79], and of physical systems modeled by PDE, to understand the nature of microstructure and singularities [Tar92,DKMO00].

Interacting Nonlinear Dynamical Systems

Extended dynamical systems are ubiquitous and often display a range of behaviors on many spatial and temporal scales. This is expected since extended systems typically have many physical processes, (e.g. surface tension, buoyancy, inertia and viscosity in two fluid flows with interface motion), each with different spatial/temporal scales. It is therefore very important to understand multiple scale behaviors that can occur in these systems.

Once class of such problems concerns pattern formation. Patterns result from the interplay of many factors including nonlinearities, external forcing and/or excitability of the medium, spatial interactions, and internal dissipation. They are therefore prototypical ``multiple scale'' phenomena.

The statistical mechanics of extended non-equilibrium systems is another outstanding open problem in physics. This is a difficult problem, since one has to account for multiple scale behavior coupled with random fluctuations. It often has to be treated outside a perturbation framework and the interactions between the fluctuations can give rise to large effects (fluctuation renormalization).

Multiple tools for Multiple scales

Multiple-scale behaviors arise in a variety of contexts, and are studied in a wide range of disciplines. Correspondingly, there is a multiplicity of methods and tools that are applied to the analysis of these problems.

The techniques for rigorous multiple-scale analysis of nonlinear PDE are functional analytic, and include Compensated Compactness [Tar79], Relaxation [KP91], Young Measures [You69,KP91,Mül99], Homogenization [Att84], H-measures [Tar95,Tar90] and $\Gamma$-convergence [DGDM83,DM93]. More recently, geometric techniques [Gro86,MŠ96] have been developed for the analysis of variational and under-determined nonlinear PDE, and the regularity theory for elliptic PDE.

In the applied mathematics and physics communities, the analysis of extended systems is conventionally done by the construction of approximate solutions using matched asymptotics [Hin91], similarity solutions [Bar79], scaling arguments [Bar79], averaging methods [LL92], renormalization and modeling/numerical simulations.

In my work, I have found it fruitful to exploit many of these approaches, and not restrict myself to particular ways of thinking about the problems. A combination of numerical simulations and ``non-rigorous'' analysis often provides useful conjectures and guides the rigorous analysis of a problem. Conversely, a rigorous understanding of the types of possible behaviors leads to ``good'' choices for approximate solutions or the appropriate method for numerical simulation. In addition, much of my work has been motivated by experiment, and in turn has motivated new experiments. My long term approach to research will be along these lines, integrating analytic, geometric and numerical techniques, and closely collaborating with experimentalists.