On rigidity of the steady Ericksen-Leslie system
Jeaheang Bang, Changyou Wang
TL;DR
The paper establishes Liouville-type rigidity for the steady simplified Ericksen-Leslie system in ${\mathbb{R}^n\setminus\{0\}}$, showing that small scaling-invariant bounds force trivial or highly restricted behavior. In 2D, it fully classifies self-similar solutions, revealing three explicit families and proving that any self-similar solution in the plane falls into one of these classes. For $n\ge 3$, the authors prove rigidity under scaling-invariant bounds: in $n\ge 4$ one has $u\equiv 0$ and $d$ constant, while in $n=3$ either $u\equiv 0$ or $u$ is a Landau solution with $d$ constant; in the self-similar setting, these conclusions hold under refined assumptions. In 4D, the smallness assumptions can be weakened for self-similar solutions, and a precise derivative-estimate/energy argument yields the same rigidity, with a note on the necessity of the smallness constant due to harmonic maps. The results connect fluid dynamics, harmonic map theory, and Liouville-type rigidity to constrain long-range behavior and potential singularity profiles of the Ericksen-Leslie system.
Abstract
We study solutions, with scaling-invariant bounds, to the steady simplified Ericksen-Leslie system in $\mathbb{R}^n\setminus \{0\}$. When $n=2$, we construct and classify a class of self-similar solutions. When $n\ge 3$, we establish the rigidity asserting that if $(u,d)$ satisfies a scaling-invariant bound with a small constant, then $u\equiv 0$ and $d=$ constant for $n\geq 4$ or $u$ is a Landau solution and $d=$ constant for $n=3$. Such a smallness condition can be weaken when $n=4$ or the solutions are self-similar.