DCDS

We study the limit of vanishing ratio of the electron mass to the
ion mass (zero-electron-mass limit) in the scaled Euler-Poisson
equations. As the first step of this justification, we construct the
uniform global classical solutions in critical Besov spaces with the
aid of ``Shizuta-Kawashima" skew-symmetry. Then we establish
frequency-localization estimates of Strichartz-type for the equation
of acoustics according to the semigroup formulation. Finally, it is
shown that the uniform classical solutions converge towards that of
the incompressible Euler equations (for *ill-prepared*initial data) in a refined way as the scaled electron-mass tends to
zero. In comparison with the classical zero-mach-number limit in
[7,23], we obtain different dispersive estimates due to
the coupled electric field.

CPAA

Local solutions of the multidimensional Navier-Stokes equations for
isentropic compressible flow are constructed with spherically
symmetric initial data between a solid core and a free boundary
connected to a surrounding vacuum state. The viscosity coefficients
$\lambda, \mu$ are proportional to $\rho^\theta$,
$0<\theta<\gamma$, where $\rho$ is the density and $\gamma >
1$ is the physical constant of polytropic fluid. It is also proved
that no vacuum develops between the solid core and the free
boundary, and the free boundary expands with finite speed.

CPAA

In this paper, we consider the one-dimensional compressible
Navier-Stokes equations for isentropic flow connecting to vacuum
state with a continuous density when viscosity coefficient depends
on the density. Precisely, the viscosity coefficient $\mu$ is
proportional to $\rho^\theta$ and $0<\theta<1/2$, where $\rho$ is
the density. The global existence of weak solutions is proved.

DCDS

Considering the stochastic 3-D incompressible anisotropic Navier-Stokes equations, we prove the local existence of strong solution in $H^2(\mathbb{T}^3)$. Moreover, we express the probabilistic estimate of the random time interval for the existence of a local solution in terms of expected values of the initial data and the random noise, and establish the global existence of strong solution in probability if the initial data and the random noise are sufficiently small.

DCDS

In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair $(\frac{ω^{r}}{r},\frac{ω^{θ}}{r})$, we get several Prodi-Serrin type regularity criteria based on the angular velocity, $u^θ$. Moreover, we obtain the global well-posedness result if the initial angular velocity $u_{0}^{θ}$ is appropriate small in the critical space $L^{3}(\mathbb{R}^{3})$. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say $ω^3$ or $u^3$.

CPAA

In this paper, we consider the free boundary problem of the
spherically symmetric compressible isentropic Navier--Stokes
equations in $R^n (n \geq 1)$, with density--dependent
viscosity coefficients. Precisely, the viscosity coefficients $\mu$
and $\lambda$ are assumed to be proportional to $\rho^\theta$,
$0 < \theta < 1$, where $\rho$ is the density. We obtain the global
existence, uniqueness and continuous dependence on initial data
of a weak solution, with a Lebesgue initial velocity $u_0\in
L^{4 m}$, $4m>n$ and $\theta<\frac{4m-2}{4m+n}$. We weaken the regularity requirement
of the initial velocity, and improve
some known results of the one-dimensional system.

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.

JIMO

In this paper, duals for standard semidefinite programming problems from both the primal and dual sides are studied. Explicit expressions of the minimal cones and their dual cones are obtained under closeness assumptions of certain sets. As a result, duality formulations resulting from regularizations for both primal and dual problems can be expressed explicitly in terms of equality and inequality constraints involving three vector and matrix variables under such assumptions. It is proved in this paper that these newly developed duals can be cast as the Extended Lagrange-Slater Dual (ELSD) and the Extended Lagrange-Slater Dual of the Dual (ELSDD) with one reduction step. Therefore, the duals formulated in this paper guarantee strong duality, i.e., a zero duality gap and dual attainment.

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.

DCDS

In this work we consider a class of degenerate
analytic maps of the form
\begin{eqnarray*}
\left\{
\begin{array}{l}
\bar{x} =x+y^{m}+\epsilon f_1(x,y,\theta,\epsilon)+h_1(x,y,\theta,\epsilon),\\
\bar{y}=y+x^{n}+\epsilon f_2(x,y,\theta,\epsilon)+h_2(x,y,\theta,\epsilon),\\
\bar{\theta}=\theta+\omega,
\end{array}
\right.
\end{eqnarray*}
where $mn>1,n\geq m,$ $h_1 \ \mbox{and} \ h_2$ are of order $n+1$ in $z,$ and $\omega=(\omega_1,\omega_2,\ldots,\omega_{d})\in \Bbb{R}^{d}$ is a
vector of rationally independent frequencies. It is shown that, under
a generic non-degeneracy condition on $f$, if
$\omega$ is Diophantine and $\epsilon>0$ is small enough, the map has
at least one weakly hyperbolic invariant torus.