Upper bound on the coarsening rate for an epitaxial growth model
Upper bound on the coarsening rate for an epitaxial growth model
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DOI：
10.1002/cpa.10103
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发表时间：
2003-11
影响因子：
3
通讯作者：
R. Kohn;Xiaodong Yan
中科院分区：
文献类型：
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作者：
R. Kohn;Xiaodong Yan
我们研究了二维空间维度中能量驱动粗化的具体例子。能量为 ∫|∇∇u|2 + (1 ‐ | ∇u|2)2;演化是代表最陡下降的四阶偏微分方程。该方程已被提出作为具有斜率选择的系统的外延生长模型。数值模拟和启发式论证表明，u 的标准差以 t1/3 的方式增长，单位面积的能量以 t-1/3 的方式衰减。我们证明了后一种说法的一个弱的、片面的版本：每单位面积的时间平均能量衰减不快于 t-1/3。我们的论证遵循 Kohn 和 Otto 在相分离背景下引入的策略，结合了 (i) 耗散关系、(ii) 等周不等式和 (iii) ODE 引理。插值不等式是新的并且相当微妙；我们的证明是通过矛盾来证明的，依赖于最近阿维莱斯-千兆能量的紧致性结果。 © 2003 Wiley 期刊公司。
We study a specific example of energy‐driven coarsening in two space dimensions. The energy is ∫|∇∇u|2 + (1 ‐ | ∇u|2)2; the evolution is the fourth‐order PDE representing steepest descent. This equation has been proposed as a model of epitaxial growth for systems with slope selection. Numerical simulations and heuristic arguments indicate that the standard deviation of u grows like t1/3, and the energy per unit area decays like t‐1/3. We prove a weak, one‐sided version of the latter statement: The time‐averaged energy per unit area decays no faster than t‐1/3. Our argument follows a strategy introduced by Kohn and Otto in the context of phase separation, combining (i) a dissipation relation, (ii) an isoperimetric inequality, and (iii) an ODE lemma. The interpolation inequality is new and rather subtle; our proof is by contradiction, relying on recent compactness results for the Aviles‐Giga energy. © 2003 Wiley Periodicals, Inc.