| Abstract: |
| The Ericksen-Leslie equations describe the hydrodynamics of nematic liquid crystals, coupling a fluid velocity with a director field constrained to the unit sphere. This non-convex constraint poses significant analytical challenges. The Ginzburg-Landau approximation relaxes the constraint via a penalization term, providing a tractable framework.
In this talk, we consider the stochastic Ginzburg-Landau system and prove that, as the penalization vanishes, its solutions converge to a local, strong martingale solution of the original stochastic Ericksen-Leslie equations. The convergence is established for initial data in the energy space $\mathbb{H}^1 \times \mathbb{H}^2$, improving upon existing results that required higher regularity. Key ingredients include uniform estimates independent of the penalization, tightness arguments, and the Jakubowski-Skorokhod representation theorem. This work rigorously justifies the Ginzburg-Landau approximation in the stochastic setting and lays the foundation for studying noise effects in liquid crystal dynamics. |
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