On well-posedness of Ericksen-Leslie's paraboloc-hyperbolic liquid crystal model

TL;DR

This paper proves well-posedness results for Ericksen-Leslie's liquid crystal model, showing conditions for local and global solutions depending on initial data size and model coefficients.

Contribution

It establishes new well-posedness results for the parabolic-hyperbolic liquid crystal model under various coefficient and initial data conditions.

Findings

Solution lifespan depends on initial energy and dissipation coefficients.

Global solutions are achievable under specific coefficient constraints.

The relation between Lagrangian multiplier and geometric constraint is crucial in the proof.

Abstract

We establish the following well-posedness results on Ericksen-Leslie's parabolic-hyperbolic liquid crystal model: 1, if the dissipation coefficients \beta = \mu_4 - 4 \mu_6 > 0, and the size of the initial energy E^{in} is small enough, then the life span of the solution is at least -O(\ln E^{in}); 2, for the special case that the coefficients \mu_1 = \mu_2 = \mu_3 = \mu_5 = \mu_6 = 0, for which the model is the Navier-Stokes equations coupled with the wave map from \mathbb{R}^n to \mathbb{S}^2, the same existence result holds but without the smallness restriction on the size of the initial data; 3, with further constraints on the coefficients, namely \alpha = \mu_4 - 4 \mu_6 - \tfrac{ (|\lambda_1| - 7 \lambda_2)^2 }{\eta} - \tfrac{ 2 ( 7 |\lambda_1| - 2\lambda_2 )^2 }{ |\lambda_1| } > 0 and \mu_2 < \mu_3, the global classical solution with small initial data can be established. A relation between the Lagrangian multiplier and the geometric constraint |d|=1 plays a key role in the proof.