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For any smooth domain Ω ⊂ R 3 , we establish the existence of a global weak solution (u, d, θ) to the simplified, non-isothermal Ericksen-Leslie system modeling the hydrodynamic motion of nematic liquid crystals with variable temperature for any initial and boundary data (u0, d0, θ0) ∈ H × H 1 (Ω, S 2 ) × L 1 (Ω), with d0(Ω) ⊂ S 2 + (the upper half sphere) and ess infΩθ0 > 0.

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In this paper, we will establish the global existence of a suitable weak solution to the Erickson–Leslie system modelling hydrodynamics of nematic liquid crystal flows with kinematic transports for molecules of various shapes in R 3 , which is smooth away from a closed set of (parabolic) Hausdorff dimension at most 15 7 .

<p style='text-indent:20px;'>We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen–Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg–Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and then the Skorokhod compactness theorem, which is built upon uniform energy estimates.</p>