Rigorous biaxial limit of a molecular-theory-based two-tensor hydrodynamics

TL;DR

This paper rigorously establishes the local existence, uniqueness, and the connection between molecular-theory-based two-tensor hydrodynamics and biaxial frame hydrodynamics through a Hilbert expansion.

Contribution

It provides the first rigorous proof of the local well-posedness and the limit transition from molecular-based two-tensor hydrodynamics to biaxial frame hydrodynamics.

Findings

Proved local existence and uniqueness of smooth solutions.

Established the convergence of two-tensor hydrodynamics to frame hydrodynamics.

Validated the molecular-theory-based model through rigorous mathematical analysis.

Abstract

We consider a two-tensor hydrodynamics derived from the molecular model, where high-order tensors are determined by closure approximation through the maximum entropy state or the quasi-entropy. We prove the existence and uniqueness of local in time smooth solutions to the two-tensor system. Then, we rigorously justify the connection between the molecular-theory-based two-tensor hydrodynamics and the biaxial frame hydrodynamics. More specifically, in the framework of Hilbert expansion, we show the convergence of the solution to the two-tensor hydrodynamics to the solution to the frame hydrodynamics.