arXiv 0912.02192009-12math.APmath-phmath.MP

关于Ohta-Kawasaki方程对非局部Mullins-Sekerka定律运动的收敛性

On the convergence of the Ohta-Kawasaki Equation to motion by nonlocal Mullins-Sekerka Law

Nam Q. Le

备注：: 40 pages, to be published in SIAM. J. Math. Analysis; final version incorporating comments/suggestions from referee reports
发表信息：: SIAM J. Math. Anal. 42 (2010), no. 4, 1602-1638

摘要：

本文利用非局部Mullins-Sekerka定律，在空间维数$N\leq 3$中的任意光滑区域上建立了Ohta-Kawasaki方程对运动的收敛性.这些方程是在模拟二嵌段共聚物中的微相分离时产生的。保证收敛结果的唯一假设是初始数据的$(i)$充分准备和极限界面的$(ii)$光滑性。我们的方法利用了Sandier和Serfaty提出的梯度流格式的“Gamma收敛性”以及由于其光滑性而使极限界面的多重性保持不变。对于无准备的径向对称初始数据情形，我们给出了M.所有空间维度的亨利。最后，我们建立运输估计的解决方案的Ohta-Kawasaki方程表征其运输机制。

英文摘要：

In this paper, we establish the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law on any smooth domain in space dimensions $N\leq 3$. These equations arise in modeling microphase separation in diblock copolymers. The only assumptions that guarantee our convergence result are $(i)$ well-preparedness of the initial data and $(ii)$ smoothness of the limiting interface. Our method makes use of the "Gamma-convergence" of gradient flows scheme initiated by Sandier and Serfaty and the constancy of multiplicity of the limiting interface due to its smoothness. For the case of radially symmetric initial data without well-preparedness, we give a new and short proof of the result of M. Henry for all space dimensions. Finally, we establish transport estimates for solutions of the Ohta-Kawasaki equation characterizing their transport mechanism.

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