Uniqueness of global weak solutions to the frame hydrodynamics for biaxial nematic phases in $\mathbb{R}^2$

TL;DR

This paper proves the uniqueness of global weak solutions for the two-dimensional frame hydrodynamics model describing biaxial nematic phases, using advanced energy estimates and Littlewood--Paley analysis techniques.

Contribution

It establishes the first rigorous proof of uniqueness for weak solutions in this biaxial nematic hydrodynamics model in 2D.

Findings

Uniqueness of global weak solutions in 2D for the model.

Development of energy estimates using Littlewood--Paley analysis.

Effective handling of nonlinear rotational derivatives on SO(3).

Abstract

We consider the hydrodynamics for biaxial nematic phases described by a field of orthonormal frame, which can be derived from a molecular-theory-based tensor model. We prove the uniqueness of global weak solutions to the Cauchy problem of the frame hydrodynamics in dimensional two. The proof is mainly based on the suitable weaker energy estimates within the Littlewood--Paley analysis. We take full advantage of the estimates of nonlinear terms with rotational derivatives on $SO(3)$, together with cancellation relations and dissipative structures of the biaxial frame system.