On the existence of weak martingale solution for stochastic non-homogeneous penalised nematic liquid crystal system

Abstract

<p style='text-indent:20px;'>The existence of a global weak martingale solution to the two and three-dimensional stochastic non-homogeneous penalised nematic liquid crystal system with multiplicative noise is considered in a smooth bounded domain. The proof relies on the classical finite-dimensional approximation and stochastic compactness argument. Particularly, we will design two-level approximation scheme to overcome the difficulties arising from the density <inline-formula><tex-math id="M1">\begin{document}$ \rho $\end{document}</tex-math></inline-formula> and the random element <inline-formula><tex-math id="M2">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>.</p>