Global strong solutions to the frame hydrodynamics for biaxial nematic phases

TL;DR

This paper proves the global existence of strong solutions for the frame hydrodynamics model of biaxial nematic phases in 2D and 3D, based on a coupled system involving orthonormal frames and Navier-Stokes equations.

Contribution

It establishes the first global well-posedness results for the frame hydrodynamics of biaxial nematic phases with small initial data.

Findings

Global well-posedness of strong solutions in 2D and 3D

Estimates of nonlinear rotational derivatives on SO(3)

Identification of dissipative structure in the model

Abstract

In this article, we consider the frame hydrodynamics of biaxial nematic phases, a coupled system between the evolution of the orthonormal frame and the Navier--Stokes equation, which is derived from a molecular-theory-based dynamical tensor model about two second-order tensors. In two and three dimensions, we establish global well-posedness of strong solutions to the Cauchy problem of frame hydrodynamics for small initial data. The key ingredient of the proof relies on estimates of nonlinear terms with rotational derivatives on $SO(3)$, together with the dissipative structure of the frame hydrodynamics.