Dynamics as a mechanism preventing the formation of finer and finer microstructure.

We study the dynamics of pattern formation in the one-dimensional partial differential equation proposed recently by Ball, Holmes, James, Pego & Swart [BHJPS] as a mathematical 'cartoon' for the dynamic formation of microstructures observed in various crystalline solids. Here W is a double-well potential like 1/4(( u) −1). What makes this equation interesting and unusual is that it possesses as a Lyapunov function a free energy (consisting of kinetic energy plus a nonconvex 'elastic' energy, but no interfacial energy contribution) which does not attain a minimum but favours the formation of finer and finer phase mixtures: Our analysis of the dynamics confirms the following surprising and striking difference between statics and dynamics, conjectured in [BHJPS] on the basis of numerical simulations of Swart & Holmes [SH]: Moreover, in order to understand the finer details of why the dynamics fails to mimic the behaviour of minimizing sequences and how solutions select their limiting pattern, we present a detailed analysis of the evolution of a restricted class of initial data - those where the strain field u has a transition layer structure; our analysis includes proofs that [ABSTRACT FROM AUTHOR]