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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

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

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 $u^3$.

keywords:
Navier-Stokes equations
,
regularity criteria
,
global well-posedness
,
axisymmetric
,
swirl

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.

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