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.

CPAA

Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes
equations are obtained. The results show that a weak solution $u$
becomes regular if the gradient of velocity component
$\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional
conditions in the class of $L^{q}(0,T;
\dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where
$\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal
gradient operator. Besides, we also consider the anisotropic
regularity criterion for the weak solution of Navier-Stokes
equations in $\mathbb{R}^3$. Finally, we also get a further
regularity criterion, when give the sufficient condition on
$\partial_3u_3$.

DCDS

This paper is devoted to the study of the inhomogeneous
hyperdissipative Navier-Stokes equations on the whole space
$\mathbb{R}^N,N\geq3$. Compared with the classical inhomogeneous
Navier-Stokes, the dissipative term $- Δ u$ here is replaced by
$D^2u$, where $D$ is a Fourier multiplier whose symbol is
$m(\xi)=|\xi|^{\frac{N+2}{4}}$. For arbitrary small positive
constants $ε$ and $δ$, global well-posedness is showed for the
data $(\rho_0, u_0)$ such that $(ρ_{0} - 1, u_0)∈
H^{\frac{N}{2}+ε} × H^{δ}$ with $\inf_{x\in
\mathbb{R}^N}\rho_0>0$. To our best knowledge, this is the first
result on the inhomogeneous hyperdissipative Navier-Stokes
equations, and it can also be viewed as the high-dimensional
generalization of the 2D result for classical inhomogeneous
Navier-Stokes equations given by Danchin [Local and global
well-posedness results for flows of inhomogeneous viscous fluids.
Adv. Differential Equations 9 (2004), 353--386.]

CPAA

In this paper, we will prove a $L^2$-concentration result of
Zakharov system in space dimension two, with initial data
$(u_0,n_0,n_1)\in H^s\times L^2\times H^{-1}$ ($\frac
{1 2}{1 3} < s < 1$), when blow up of the solution happens, by resonant
decomposition and I-method, which is an improvement of [13].

CPAA

Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of
equations admits a unique, global mild solution provided only the horizontal components of the initial
data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in
[1,\infty)$.

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

We consider the Cauchy problem of the $N$-dimensional incompressible viscoelastic fluids with $N≥2$. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group $e^{± it\Lambda}$. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations.