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The global weak martingale solution is built through a four-level approximation scheme to stochastic compressible active liquid crystal system driven by multiplicative noise in a smooth bounded domain in $mathbb{R}^{3}$ with large initial data. The coupled structure makes the analysis challenging, and more delicate arguments are required in stochastic case compared to the deterministic one cite{11}.
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We study the three-dimensional compressible Navier-Stokes equations coupled with the $Q$-tensor equation perturbed by a multiplicative stochastic force, which describes the motion of nematic liquid crystal flows. The local existence and uniqueness of strong pathwise solution up to a positive stopping time is established where ``strong is in both PDE and probability sense. The proof relies on the Galerkin approximation scheme, stochastic compactness, identification of the limit, uniqueness and a cutting-off argument. In the stochastic setting, we develop an extra layer approximation to overcome the difficulty arising from the stochastic integral while constructing the approximate solution. Due to the complex structure of the coupled system, the estimates of the high-order items are also the challenging part in the article.
In this paper, we study the active hydrodynamics, described in the Q-tensor liquid crystal framework. We prove the existence of global weak solutions in dimension two and three, with suitable initial datas. By using Littlewood-Paley decomposition, we also get the higher regularity of the weak solutions and the uniqueness of weak-strong solutions in dimension two.
This paper is devoted to establishing the optimal decay rate of the global large solution to compressible nematic liquid crystal equations when the initial perturbation is large and belongs to $L^1(mathbb R^3)cap H^2(mathbb R^3)$. More precisely, we show that the first and second order spatial derivatives of large solution $(rho-1, u, abla d)(t)$ converges to zero at the $L^2-$rate $(1+t)^{-frac54}$ and $L^2-$rate $(1+t)^{-frac74}$ respectively, which are optimal in the sense that they coincide with the decay rates of solution to the heat equation. Thus, we establish optimal decay rate for the second order derivative of global large solution studied in [12,18] since the compressible nematic liquid crystal flow becomes the compressible Navier-Stokes equations when the director is a constant vector. It is worth noticing that there is no decay loss for the highest-order spatial derivative of solution although the associated initial perturbation is large. Moreover, we also establish the lower bound of decay rates of $(rho-1, u, abla d)(t)$ itself and its spatial derivative, which coincide with the upper one. Therefore, the decay rates of global large solution $ abla^2(rho-1,u, abla d)(t)$ $(k=0,1,2)$ are actually optimal.
We propose lyotropic chromonic liquid crystals (LCLCs) as a distinct class of materials for organic electronics. In water, the chromonic molecules stack on top of each other into elongated aggregates that form orientationally ordered phases. The aligned aggregated structure is preserved when the material is deposited onto a substrate and dried. The dried LCLC films show a strongly anisotropic electric conductivity of semiconductor type. The field-effect carrier mobility measured along the molecular aggregates in unoptimized films of LCLC V20 is 0.03 cm^2 V^(-1) s^(-1). Easy processibility, low cost, and high mobility demonstrate the potential of LCLCs for microelectronic applications.
Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the Oseen-Frank model), free interfaces and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations. In this paper, through a study of the phase transition between the isotropic and nematic states of liquid crystal based on the Ericksen model, we can show, when the size of droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a $Gamma$-convergence theorem for minimizers. This $Gamma$-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter $varepsilon$, corresponding to the transition layer width, goes to zero. This limiting process not only provides a geometric description of the shape of the droplet as one would expect, and surprisingly it also gives the anchoring conditions for the orientations of liquid crystals on the surface of the droplet depending on material constants. In particular, homeotropic, tangential, and even free boundary conditions as assumed in earlier phenomenological modelings arise naturally provided that the surface tension, Frank and Ericksen constants are in suitable ranges.
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