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

- Accepted:

- 23 Citations
- 99 Downloads

## Abstract

*W*is a double-well potential like 1/4((

*u*

_{x})

^{2}−1)

^{2}. 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:

•While minimizing the above energy predicts infinitely fine patterns (mathematically:

*weak but not strong convergence*of all minimizing sequences (*u*_{n}v_{n}) of E[u,v] in the Sobolev space*W*^{1}^{p}(0, 1)×L^{2}(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically:*strong convergence*in*W*_{1}^{p}(0,1)×L^{2}(0,1) of all solutions (u(t),u_{t}(t)) with low initial energy as time*t*→ ∞).

*u*

_{x}has a transition layer structure; our analysis includes proofs that

•at low energy, the number of phases is in fact exactly preserved, that is, there is no nucleation or coarsening

•transition layers lock in and steepen exponentially fast, converging to discontinuous stationary sharp interfaces as time

*t*→ ∞•the limiting patterns — while not minimizing energy globally — are ‘relative minimizers’ in the weak sense of the calculus of variations, that is, minimizers among all patterns which share the same strain interface positions.

## Preview

Unable to display preview. Download preview PDF.