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### Open Access Journals

DCDS

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.

CPAA

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.

DCDS

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.

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