标题： Ericksen-Leslie模型中向列液晶定常剪切流的分岔与稳定性 显示英文标题

标题： Bifurcation and stability of stationary shear flows of Ericksen-Leslie model for nematic liquid crystals

摘要： 在本工作中，针对向列相液晶剪切流的一个关键情况，我们通过抛物型Ericksen-Leslie系统研究了定态解的多重性和稳定性。我们建立了定态解集合与代数方程解集合之间的一一对应关系。通过在代数方程临界剪切速度处的可数多个鞍点分支，确立了多个定态解的存在性。然后，我们为定态解建立了类似的局部分支；更准确地说，(i) 对于每个临界剪切速度，存在唯一的定态解，当剪切速度较小时定态解消失，当剪切速度较大时附近有两个定态解分支出来；(ii) 更重要的是，在一个一般条件下，剪切流在临界定态解处线性化的算子恰好有一个零特征值，该零特征值对于两个定态解中的一个分支为负特征值，而对于另一个定态解分支为正特征值。这些结果扩展并受到Dorn和Liu [{\em J. 微分方程 {\bf 253} (2012), 3184-3210}]以及Jiao等 [{\em J. Diff. Dyn. Syst. {\bf 34} (2022), 239-269}]工作的启发。 显示英文摘要

摘要： In this work, focusing on a critical case for shear flows of nematic liquid crystals, we investigate multiplicity and stability of stationary solutions via the parabolic Ericksen-Leslie system. We establish a one-to-one correspondence between the set of the stationary solutions with the set of solutions of an algebraic equation. The existence of multiple stationary solutions is established through countably many saddle-node bifurcations at critical shear speeds for the algebraic equation. We then establish similar local bifurcations for the stationary solutions; more precisely, (i) for each critical shear speed, there is a unique stationary solution, the stationary solution disappears for smaller shear speed, and two stationary solutions nearby bifurcate for larger shear speed; (ii) more importantly, under a generic condition, there is exactly one zero eigenvalue for the linearization of the shear flow at the critical stationary solution, and the zero eigenvalue bifurcates to a negative eigenvalue for one of the two stationary solutions and to a positive eigenvalue for the other stationary solution. These results extend and are motivated by the works of Dorn and Liu [{\em J. Differential Equations {\bf 253} (2012), 3184-3210}] and of Jiao, et. al. [{\em J. Diff. Dyn. Syst. {\bf 34} (2022), 239-269}].