# Well-posedness of the Ericksen-Leslie System for the Oseen-Frank Model in L^3_{uloc}(\mathbb{R}^3)

## TL;DR

This paper proves the existence and uniqueness of solutions to the Ericksen-Leslie system for the Oseen-Frank model in three-dimensional space with initial data in a localized L^3 space, using new estimates and invariance properties.

## Contribution

It introduces a novel approach to establish local L^3 estimates and removes restrictions on elastic constants for solution uniqueness in the Ericksen-Leslie system.

## Key findings

- Existence of solutions with small L^3_{uloc} initial data.
- A new method for local L^3 estimates via interpolation and covering.
- Uniqueness of solutions without restrictions on elastic constants.

## Abstract

We investigate the Ericksen-Leslie system for the Oseen-Frank model with unequal Frank elastic constants in $\mathbb{R}^3$. To generalize the result of Hineman-Wang \cite{HW}, we prove existence of solutions to the Ericksen-Leslie system with initial data having small $L^3_{uloc}$-norm. In particular, we use a new idea to obtain a local $L^3$-estimate through interpolation inequalities and a covering argument, which is different from the one in \cite{HW}. Moreover, for uniqueness of solutions, we find a new way to remove the restriction on the Frank elastic constants by using the rotation invariant property of the Oseen-Frank density. We combine this with a method of Li-Titi-Xin \cite{LTX} to prove uniqueness of the $L^3_{uloc}$-solutions of the Ericksen-Leslie system assuming that the initial data has a finite energy.