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

•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 → ∞).

•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.
Keywords
Lyapunov Function Strong Convergence Transition Layer Weak Sense Sharp InterfacePreview
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