[论文解读] The small Deborah number limit of the Doi-Onsager equation to the Ericksen-Leslie equation

Wei Wang, Pingwen Zhang+1 more|arXiv (Cornell University)|Jun 24, 2012

Numerical methods in inverse problems参考文献 15被引用 7

一句话总结

该论文通过Hilbert展开，在小德博拉数极限下严格推导出Ericksen-Leslie方程。通过建立线性化Doi-Onsager算子的谱稳定性，并引入一种新颖的坐标变换和Maier-Saupe空间，作者证明了余项的统一有界性，并展示了能量耗散，从而验证了具有复杂分子相互作用的向列液晶的流体动力学极限。

ABSTRACT

We present a rigorous derivation of the Ericksen-Leslie equation starting from the Doi-Onsager equation. As in the fluid dynamic limit of the Boltzmann equation, we first make the Hilbert expansion for the solution of the Doi-Onsager equation. The existence of the Hilbert expansion is connected to an open question whether the energy of the Ericksen-Leslie equation is dissipated. We show that the energy is dissipated for the Ericksen-Leslie equation derived from the Doi-Onsager equation. The most difficult step is to prove a uniform bound for the remainder in the Hilbert expansion. This question is connected to the spectral stability of the linearized Doi-Onsager operator around a critical point. By introducing two important auxiliary operators, the detailed spectral information is obtained for the linearized operator around all critical points. However, these are not enough to justify the small Deborah number limit for the inhomogeneous Doi-Onsager equation, since the elastic stress in the velocity equation is also strongly singular. For this, we need to establish a precise lower bound for a bilinear form associated with the linearized operator. In the bilinear form, the interactions between the part inside the kernel and the part outside the kernel of the linearized operator are very complicated. We find a coordinate transform and introduce a five dimensional space called the Maier-Saupe space such that the interactions between two parts can been seen explicitly by a delicate argument of completing the square. However, the lower bound is very weak for the part inside the Maier-Saupe space. In order to apply them to the error estimates, we have to analyze the structure of the singular terms and introduce a suitable energy functional.

研究动机与目标

• 在小德博拉数范围内，严格证明Doi-Onsager方程向Ericksen-Leslie方程的流体动力学极限。
• 通过证明源自Doi-Onsager模型的解满足能量耗散，解决所推导的Ericksen-Leslie方程中能量耗散的开放问题。
• 建立Hilbert展开余项的统一有界性，这需要对线性化Doi-Onsager算子进行详细的谱分析。
• 通过为双线性形式构造精确的下界，克服速度方程中弹性应力项的强奇异性。
• 开发一种新的能量泛函和坐标变换，以处理线性化算子中核部分与非核部分之间的复杂相互作用。

提出的方法

• 对Doi-Onsager方程的解进行Hilbert展开，按德博拉数的幂次展开。
• 引入两个辅助算子，以分析线性化Doi-Onsager算子在临界点附近的谱性质。
• 定义五维Maier-Saupe空间，以显式分解线性化算子中核部分与非核部分之间的相互作用。
• 利用坐标变换和配方法分析与线性化算子相关的双线性形式。
• 构造一个定制的能量泛函，以控制奇异项并估计Hilbert展开中的余项。
• 通过利用谱分解导出的系数的正性，证明双线性形式的统一下界。

实验结果

研究问题

• RQ1 能否在小德博拉数极限下，严格从Doi-Onsager方程推导出Ericksen-Leslie方程？
• RQ2 所推导的Ericksen-Leslie方程的能量是否如物理一致性所要求的那样耗散？
• RQ3 尽管弹性应力中存在强奇异性，能否建立Hilbert展开余项的统一有界性？
• RQ4 如何显式控制线性化Doi-Onsager算子中核部分与非核部分之间的相互作用？
• RQ5 线性化算子的何种谱结构使得能够为双线性形式构造下界？

主要发现

• 从Doi-Onsager方程推导出的Ericksen-Leslie方程的能量是耗散的，确认了该极限的物理一致性。
• 建立了Hilbert展开余项的统一有界性，这对流体动力学极限的有效性至关重要。
• 线性化Doi-Onsager算子的谱分析揭示了所有临界点附近的详细信息，从而支持了所需能量估计的构造。
• 与线性化算子相关的双线性形式具有精确的下界，该下界通过Maier-Saupe空间中的坐标变换和分解实现。
• 能量估计中的系数被证明为正，确保了二次型的正性，从而保证了展开的稳定性。
• 所推导的Ericksen-Leslie方程继承了Doi-Onsager模型的耗散结构，验证了其作为向列液晶流体宏观描述的有效性。