On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals
On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals
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DOI：
10.3934/dcdsb.2018115
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发表时间：
2016-06
期刊：
影响因子：
--
通讯作者：
Yuning Liu;Wei Wang
中科院分区：
文献类型：
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作者：
Yuning Liu;Wei Wang
研究了三维欧氏空间中有界区域上的液晶流动的Navier-Stokes/$Q$-张量耦合方程组的可解性，该方程组的序参量为强锚定边界条件.证明了具有各向异性弹性能量的系统在时间上局部强解的存在性。证明主要是基于两个成分：首先，我们证明了Euler-Lagrange算子对应的Landau-de Gennes自由能与一般弹性系数满足强Legendre条件。这一结果与高阶能量估计一起导致线性化系统的适定性，然后通过不动点参数得到原系统在时间上的局部解，该解在时间变量上是正则的。其次，耦合系统的流体动力学部分可以转化为一个准定常的Stokes型方程，广义Stokes系统的正则性理论，然后可以应用自助参数来增强局部时间解的空间正则性。
This work is concerned with the solvability of a Navier-Stokes/$Q$-tensor coupled system modeling the nematic liquid crystal flow on a bounded domain in three dimensional Euclidian space with strong anchoring boundary condition for the order parameter. We prove the existence of local in time strong solution to the system with the anisotropic elastic energy. The proof is based on mainly two ingredients: first, we show that the Euler-Lagrange operator corresponding to the Landau-de Gennes free energy with general elastic coefficients fulfills the strong Legendre condition. This result together with a higher order energy estimate leads to the well-posedness of the linearized system, and then a local in time solution of the original system which is regular in temporal variable follows via a fixed point argument. Secondly, the hydrodynamic part of the coupled system can be reformulated into a quasi-stationary Stokes type equation to which the regularity theory of the generalized Stokes system, and then a bootstrap argument can be applied to enhance the spatial regularity of the local in time solution.