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Archive for Rational Mechanics and Analysis

, Volume 133, Issue 3, pp 199–247 | Cite as

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

  • G. Friesecke
  • J. B. McLeod
Article

Abstract

We study the dynamics of pattern formation in the one-dimensional partial differential equation
$$u_u - (W'(u_x ))_x - u_{xxt} + u = 0{\text{ (}}u = u(x,t),{\text{ }}x \in (0,1),{\text{ }}t > 0)$$
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 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:
$$E[u,u_t ] = \int\limits_0^1 {(\frac{{u_t^2 }}{2} + W(u_x ) + \frac{{u^2 }}{2})dx.}$$
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]:
  • •While minimizing the above energy predicts infinitely fine patterns (mathematically: weak but not strong convergence of all minimizing sequences (unvn) of E[u,v] in the Sobolev space W1p(0, 1)×L2(0,1)), solutions to the evolution equation of ball et al. typically develop patterns of small but finite length scale (mathematically: strong convergence in W1p(0,1)×L2(0,1) of all solutions (u(t),ut(t)) with low initial energy as time t → ∞).

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 ux 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.

Keywords

Lyapunov Function Strong Convergence Transition Layer Weak Sense Sharp Interface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Verlag 1996

Authors and Affiliations

  • G. Friesecke
    • 1
    • 2
  • J. B. McLeod
    • 1
    • 2
  1. 1.Mathematisches InstitutUniversität FreiburgFreiburg
  2. 2.Department of Mathematics and StatisticsUniversity of PittsburghPittsburgh

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