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 $\Omega \subset \mathbb{R}^d, d=2,3$. 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 $d=3$ and globally in time for $d=2$) 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 $(\nu t)^{\frac34-\varepsilon}$ for any small $\varepsilon>0$ in $L^2$

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 $x\in R^n, 0\le \beta<n, j=1,2,\cdots, n$. When $\beta=0$, it corresponds to the Riesz transform. We will make an estimate the $L^q (1<q<\infty)$ norm of $T_jf$, which holds uniformly for $0\le\beta<\frac{n(q-1)}{q}$. In particular, when $\beta=0$, the strong $(q,q)$ type estimate of the Riesz transform for $1<q<\infty$ 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 $\mu(\rho)=\rho^\alpha (\alpha>0)$ and pressure $P(\rho)=\rho^{\gamma}\ (\gamma>1)$. We will establish the global existence and asymptotic behavior of weak solutions for any $\alpha>0$ and $\gamma>1$ under the assumption that the density function keeps a constant state at far fields. This enlarges the ranges of $\alpha$ and $\gamma$ and improves the previous results presented by Jiu and Xin. As a result, in the case that $0<\alpha<\frac12$, 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 $(-\Delta)^{\alpha}$ affects the regularity of weak solutions to the 3d generalized Navier-Stokes equations. Precisely, it will be shown that the $(5-4\alpha)/2\alpha$ dimensional Hausdorff measure of possible time singular points of weak solutions on the interval $(0,\infty)$ is zero when $5/6\le\alpha< 5/4$. To this end, the eventual regularity for the weak solutions is firstly established in the same range of $\alpha$. It is worth noting that when the dissipation index $\alpha$ varies from $5/6$ to $5/4$, the corresponding Hausdorff dimension is from $1$ to $0$. Hence, it seems that the Hausdorff dimension obtained is optimal. Our results rely on the fact that the space $H^{\alpha}$ is the critical space or subcritical space to this system when $\alpha\geq5/6$.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 $\Lambda^{2\beta} b$. It is proved that when $\beta>1$ there exists a unique global regular solution for this equations. The obtained result improves the previous known one which requires that $\beta>\frac32$.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 $u_\theta$ which are scaling invariant. It is known that $ru_\theta$ satisfies the maximum principle. The assumptions on $u_\theta$ 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 $0\leqslant\alpha 3$, which improves the results in \cite{TYZ2013}. In particular, we obtain the global regularity of the 2D generalized MHD when $\alpha=0$ and $\beta>\frac 32$.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 $R^{N},N\geq2$, under the assumptions that the initial values $\theta_{0},u_{0}\in C^{r}$, with $r>1$.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 $(\rho,\rm u)$ if the viscosity $\mu(\rho)=1+\rho^{\beta}$ with $\beta\geq0$. 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