55 research outputs found
Inviscid Limit for Vortex Patches in A Bounded Domain
In this paper, we consider the inviscid limit of the incompressible
Navier-Stokes equations in a smooth, bounded and simply connected domain
. We prove that for a vortex patch initial
data the weak Leray solutions of the incompressible Navier-Stokes equations
with Navier boundary conditions will converge (locally in time for and
globally in time for ) to a vortex patch solution of the incompressible
Euler equation as the viscosity vanishes. In view of the results obtained in
[1] and [19] which dealt with the case of the whole space, we derive an almost
optimal convergence rate for any small
in
An Extension of Riesz Transform
In this paper, we consider the following singular integral \begin{equation*}
T_jf(x)=K_j*f(x), K_j(x)=\frac{x_j}{|x|^{n+1-\beta}}, \end{equation*} where
. When , it corresponds to
the Riesz transform. We will make an estimate the norm of
, which holds uniformly for . In particular,
when , the strong type estimate of the Riesz transform for
is recovered from the obtained estimate.Comment: 12 page
On One-dimensional Compressible Navier-Stokes Equations with Degenerate Viscosity and Constant State at Far fields
In this paper, we are concerned with the Cauchy problem for one-dimensional
compressible isentropic Navier-Stokes equations with density-dependent
viscosity and pressure
. We will establish the global existence and
asymptotic behavior of weak solutions for any and under
the assumption that the density function keeps a constant state at far fields.
This enlarges the ranges of and and improves the previous
results presented by Jiu and Xin. As a result, in the case that
, we obtain the large time behavior of the strong solution
obtained by Mellet and Vasseur when the solution has a lower bound (no vacuum).Comment: 25 page
On possible time singular points and eventual regularity of weak solutions to the fractional Navier-Stokes equations
In this paper, we intend to reveal how the fractional dissipation
affects the regularity of weak solutions to the 3d
generalized Navier-Stokes equations. Precisely, it will be shown that the
dimensional Hausdorff measure of possible time singular
points of weak solutions on the interval is zero when
. To this end, the eventual regularity for the weak
solutions is firstly established in the same range of . It is worth
noting that when the dissipation index varies from to ,
the corresponding Hausdorff dimension is from to . Hence, it seems that
the Hausdorff dimension obtained is optimal. Our results rely on the fact that
the space is the critical space or subcritical space to this
system when .Comment: 24 pages. We improve the results of the first version. We obtain the
optimal dimensional Hausdorff estimate of possible time singular points of
weak solutions to the fractional Navier-Stokes equation
Global Regularity of 2D Generalized MHD Equations with Magnetic Diffusion
This paper is concerned with the global regularity of the 2D
(two-dimensional) generalized magnetohydrodynamic equations with only magnetic
diffusion . It is proved that when there exists
a unique global regular solution for this equations. The obtained result
improves the previous known one which requires that .Comment: 6 page
On Liouville Type of Theorems to the 3-D Incompressible Axisymmetric Navier-Stokes Equations
Liouville type of theorems play a key role in the blow-up approach to study
the global regularity of the three-dimensional Navier-Stokes equations. In this
paper, we will prove Liouville type of theorems to the 3-D axisymmetric
Navier-Stokes equations with swirls under some suitable assumptions on swirl
component velocity which are scaling invariant. It is known that
satisfies the maximum principle. The assumptions on will
be natural and useful to make further studies on the global regularity to the
three-dimensional incompressible axisymmetric Navier-Stokes equations.Comment: 22 page
A Remark On Global Regularity of 2D Generalized Magnetohydrodynamic Equations
In this paper we study the global regularity of the following 2D
(two-dimensional) generalized magnetohydrodynamic equations \begin{eqnarray*}
\left\{\begin{array}{llll} u_t + u \cdot \nabla u & = & - \nabla p + b \cdot
\nabla b - \nu (-\triangle)^{\alpha} u b_t + u \cdot \nabla b & = & b \cdot
\nabla u - \kappa (-\triangle)^{\beta} b
\end{array}\right. \end{eqnarray*} and get global regular solutions when ,
which improves the results in \cite{TYZ2013}. In particular, we obtain the
global regularity of the 2D generalized MHD when and .Comment: 9 page
Decay of solutions to the three-dimensional generalized Navier-Stokes equations
In this paper, we first obtain the temporal decay estimates for weak
solutions to the three dimensional generalized Navier-Stokes equations. Then,
with these estimates at disposal, we obtain the temporal decay estimates for
higher order derivatives of the smooth solution with small initial data. The
decay rates are optimal in the sense that they coincides with ones of the
corresponding generalized heat equation. These results improve the previous
known results to the classical Navier-Stokes equations.Comment: 16 page
Local well-posedness and blow up criterion for the Inviscid Boussinesq system in H\"{o}lder spaces
We prove the local in time existence and a blow up criterion of solution in
the H\"{o}lder spaces for the inviscid Boussinesq system in ,
under the assumptions that the initial values , with
.Comment: 20 page
Global classical solution of the Cauchy problem to 1D compressible Navier-Stokes equations with large initial data
In this paper, we prove that the 1D Cauchy problem of the compressible
Navier-Stokes equations admits a unique global classical solution if the viscosity with . The initial
data can be arbitrarily large and may contain vacuum.
Some new weighted estimates of the density and velocity are obtained when
deriving higher order estimates of the solution
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