Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework
Qiao Liu Ting Zhang Jihong Zhao
Discrete & Continuous Dynamical Systems - A 2016, 36(1): 371-402 doi: 10.3934/dcds.2016.36.371
In this paper, we consider the well-posedness of the Cauchy problem of the 3D incompressible nematic liquid crystal system with initial data in the critical Besov space $\dot{B}^{\frac{3}{p}-1}_{p,1}(\mathbb{R}^{3})\times \dot{B}^{\frac{3}{q}}_{q,1}(\mathbb{R}^{3})$ with $1< p<\infty$, $1\leq q<\infty$ and \begin{align*} -\min\{\frac{1}{3},\frac{1}{2p}\}\leq \frac{1}{q}-\frac{1}{p}\leq \frac{1}{3}. \end{align*} In particular, if we impose the restrictive condition $1< p<6$, we prove that there exist two positive constants $C_{0}$ and $c_{0}$ such that the nematic liquid crystal system has a unique global solution with initial data $(u_{0},d_{0}) = (u^{h}_{0}, u^{3}_{0}, d_{0})$ which satisfies \begin{align*} ((1+\frac{1}{\nu\mu})\|d_{0}-\overline{d}_{0}\|_{\dot{B}^{\frac{3}{q}}_{q,1}}+ \frac{1}{\nu}\|u_{0}^{h}\|_{\dot{B}^{\frac{3}{p}-1}_{p,1}}) \exp\left\{\frac{C_{0}}{\nu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{\frac{3}{p}-1}_{p,1}}+\frac{1}{\mu})^{2}\right\}\leq c_{0}, \end{align*} where $\overline{d}_{0}$ is a constant vector with $|\overline{d}_{0}|=1$. Here $\nu$ and $\mu$ are two positive viscosity constants.
keywords: Nematic liquid crystal flows well-posedness. Navier-Stokes equations
Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows
Jihong Zhao Qiao Liu Shangbin Cui
Communications on Pure & Applied Analysis 2013, 12(1): 341-357 doi: 10.3934/cpaa.2013.12.341
In this paper we consider a coupled hydrodynamical system which involves the Navier-Stokes equations for the velocity field and kinematic transport equations for the molecular orientation field. By applying the Chemin-Lerner's time-space estimates for the heat equation and the Fourier localization technique, we prove that when initial data belongs to the critical Besov spaces with negative-order, there exists a unique local solution, and this solution is global when initial data is small enough. As a corollary, we obtain existence of global self-similar solution. In order to figure out the relation between the solution obtained here and weak solutions of standard sense, we establish a stability result, which yields in a direct way that all global weak solutions associated with the same initial data must coincide with the solution obtained here, namely, weak-strong uniqueness holds.
keywords: Liquid crystal flow stability global existence blow up weak-strong uniqueness.
Global well-posedness for the dissipative system modeling electro-hydrodynamics with large vertical velocity component in critical Besov space
Jihong Zhao Ting Zhang Qiao Liu
Discrete & Continuous Dynamical Systems - A 2015, 35(1): 555-582 doi: 10.3934/dcds.2015.35.555
In this paper, we are concerned with a model arising from electro-hydrodynamics, which is a coupled system of the Navier-Stokes equations and the Poisson-Nernst-Planck equations through charge transport and external forcing terms. The local well-posedness and global well-posedness with small initial data to the 3-D Cauchy problem of this system are established in the critical Besov space $\dot{B}^{-1+\frac{3}{p}}_{p,1}(\mathbb{R}^{3})\times(\dot{B}^{-2+\frac{3}{q}}_{q,1}(\mathbb{R}^{3}))^{2}$ with suitable choices of $p, q$. Especially, we prove that there exist two positive constants $c_{0}, C_{0}$ depending on the coefficients of system except $\mu$ such that if \begin{equation*} \big(\|u_{0}^{h}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}+(\mu+1)\|(v_{0},w_{0})\|_{\dot{B}^{-2+\frac{3}{q}}_{q,1}} \big) \exp\Big\{\frac{C_{0}}{\mu^{2}}(\|u_{0}^{3}\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}^{2}+1)\Big\}\leq c_{0}\mu, \end{equation*} then the above local solution can be extended to the global one. This result implies the global well-posedness of this system with large initial vertical velocity component.
keywords: electro-hydrodynamics Navier-Stokes equations well-posedness Poisson-Nernst-Planck equations Besov space.

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