The small Deborah number limit of the Doi-Onsager equation without hydrodynamics
The small Deborah number limit of the Doi-Onsager equation without hydrodynamics
复制标题
无流体动力学的 Doi-Onsager 方程的小 Deborah 数极限
DOI：
10.1016/j.jfa.2018.07.013
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发表时间：
2018
影响因子：
1.7
通讯作者：
Wang Wei
中科院分区：
文献类型：
--
作者：
Liu Yuning;Wang Wei
在忽略流体动力学影响的情况下，研究了Doi-Onsager方程的小Deborah数极限。这是一个Smoluchowsk型方程，通过描述数密度函数的演化，在分子水平上描述向列相液晶的动力学，该数密度函数依赖于粒子位置x∈Rd(d=2，3)和取向矢量m∈S 2(单位球)。证明了当Deborah数趋于零时，初值在局部平衡点附近的解族将强收敛到由进入S 2的调和映射热流的弱解所规定的局部平衡分布.该热流是向列相液晶中著名的Oseen-Frank能量泛函的梯度流的特例.关键是要证明数密度函数族的强紧性。证明依赖于相应Q张量的强紧性(即二阶矩)、对极限局部平衡分布附近的线性化算子的详细分析以及能量耗散估计。
We study the small Deborah number limit of the Doi–Onsager equation in the case when hydrodynamic effects are neglected. This is a Smoluchowski-type equation that describes the dynamics of nematic liquid crystals at a molecular level, by characterizing the evolution of a number density function, depending upon both particle position x∈ R d (d= 2, 3) and orientation vector m∈ S 2 (the unit sphere). We prove that, when the Deborah number tends to zero, the family of solutions with rough initial data near local equilibria will converge strongly to a local equilibrium distribution prescribed by a weak solution of the harmonic map heat flow into S 2. This flow is a special case of the gradient flow of the well known Oseen–Frank energy functional for nematic liquid crystals. The key ingredient is to show the strong compactness of the family of number density functions. The proof relies on the strong compactness of the corresponding Q-tensor (namely the second moment), a detailed analysis of the linearized operator near the limit local equilibrium distribution, and an energy dissipation estimate.