On an incompressible inertial nematic electrolyte model of liquid crystal flow

ABSTRACT

We study the Cauchy problem of the new model of the three dimensional inertial nematic electrolyte, which models the N-species charged particles diffuse and advect in the nematic liquid crystalline matrix. This new model was proposed by [Calderer MC, Golovaty D, Lavrentovich O, et al. Modeling of nematic electrolytes and nonlinear electroosmosis. SIAM J Appl Math. 2016;76:2260–2285. doi: 10.1137/16M1056225]. The inertial effect is such that the system involves a hyperbolic feature. We first derive the local existence in large initial data when the flow is isotropy, i.e. $\lambda =1$. For the anisotropy case ($\lambda >0$, $\lambda \ne 1$), we obtain small data local solutions. Then, by seeking some further dissipative mechanisms from the elliptic structures, the local solution can be extended globally near the constant states with electro-neutrality condition.

KEYWORDS:

• Inertial nematic electrolyte model
• well-posedness
• a priori estimate
• Parodi's relation
• Ericksen–Leslie theory

AMS SUBJECT CLASSIFICATIONS:

• 35B45
• 35A09
• 35B65
• 35Q35
• 76A15

Acknowledgments

The authors particularly thank Professor Ning Jiang for his constant support and his suggestions for this problem.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Starting Research Fund from Chongqing Jiaotong University [contract number 2020021022], the Scientific and Technological Research Program of Chongqing Municipal Education Commission [contract number KJQN202100709] and the grants from the National Natural Foundation of China [contract number 11971360].