# Existence, uniqueness and regularity for the stochastic Ericksen-Leslie equation

## TL;DR

This paper establishes existence, uniqueness, and partial regularity results for stochastic liquid crystal flows driven by colored noise on a 2D torus, using advanced probabilistic and analytical techniques.

## Contribution

It provides the first local and global well-posedness results for the stochastic Ericksen-Leslie equations in critical function spaces.

## Key findings

- Proved local solvability in $L^p$ spaces for $p>2$.
- Established existence of partially regular global solutions.
- Developed a novel combination of bootstrap and compactness methods.

## Abstract

We investigate existence and uniqueness for the stochastic liquid crystal flow driven by colored noise on the two-dimensional torus. After giving a natural uniqueness criterion, we prove local solvability in $L^p$-based spaces, for every $p>2.$ Thanks to a bootstrap principle together with a Gy\"ongy-Krylov-type compactness argument, this will ultimately lead us to prove the existence of a particular class of global solutions which are partially regular, strong in the probabilistic sense, and taking values in the "critical space" $L^2\times H^1.$