Poiseuille flow of hyperbolic Ericksen-Leslie system in dimension two

TL;DR

This paper investigates the existence, regularity, and potential singularity formation in Poiseuille laminar flow within a tube for the hyperbolic Ericksen-Leslie system, revealing conditions for finite-time blowup and solution behavior.

Contribution

It establishes the global existence of weak solutions and analyzes their regularity and singularity formation, which is novel for this coupled hyperbolic-parabolic system in two dimensions.

Findings

Global weak energy solutions exist for the system.

Solutions may develop discontinuities at the origin.

Blowup sequences converge to non-constant harmonic maps.

Abstract

In this paper, we study the Poiseuille laminar flow in a tube for the full Ericksen-Leslie system. It is a parabolic-hyperbolic coupled system which may develop singularity in finite time. We will prove the global existence of energy weak solution, and the partial regularity of solution to system. We first construct global weak finite energy solutions by the Ginzburg- Landau approximation and the fixed-point arguments. Then we obtain the enhanced regularity of solution. Different from the solution in one space dimension, the finite energy solution of Poiseuille laminar flow in a tube may still form a discontinuity at the origin. We show that at the first possible blowup time, there are blowup sequences which converge to a non-constant time-independent (axisymmetric) harmonic map.