Abstract:
For the 3-D incompressible Magneto-hydrodynamics equations, whether the limiting case $u\in L^\infty(0,T;L^3(R^3))$ implies the regularity of weak solution is unknown. Here we prove that the solution $(u,b)$ of the 3-D Magneto-hydrodynamics equations is regular in $(0,T]\times R^3$ if $u\in L^\infty(0,T;L^3(R^3))$ and $b_h$ satisfies the Ladyzhenskaya-Prodi-Serrin condition, where $b_h$ is the horizontal components of the magnetic field $b$.

Abstract:
We prove the global existence of weak solution for two dimensional Ericksen-Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. We also prove the local well-posedness of the Ericksen-Leslie system in two and three spatial dimensions.

Abstract:
We obtain the $C^{\a}$ regularity for weak solutions of a class of non-homogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent $C^{\a}$ regularity results of homogeneous ultraparabolic equation.

Abstract:
We obtained the $C^{\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form ${\ptl_t u}= \sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j u)+X_0 u$. The result is proved by simplifying and generalizing our earlier arguments for the $C^{\a}$ regularity of homogeneous ultraparabolic equations.

Abstract:
The backward uniqueness of the Kolmogorov operator $L=\sum_{i,k=1}^n\partial_{x_i}(a_{i,k}(x,t)\partial_{x_k})+\sum_{l=1}^m x_l\partial_{y_l}-\partial_t$, was proved in this paper. We obtained a weak Carleman inequality via Littlewood-Paley decomposition for the global backward uniqueness.

Abstract:
Let $u=(u_h,u_3)$ be a smooth solution of the 3-D Navier-Stokes equations in $\R^3\times [0,T)$. It was proved that if $u_3\in L^{\infty}(0,T;\dot{B}^{-1+3/p}_{p,q}(\R^3))$ for $3

Abstract:
It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes and MHD equations are H\"older continuous near boundary provided that either $r^{-3}\int_{B_r^+}|u(x)|^3dx$ or $r^{-2}\int_{B_r^+}|\nabla u(x)|^2dx$ is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by Dong-Strain \cite{DS}.

Abstract:
We obtained the $C^{\a}$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form $${\ptl_t u}= \ptl_x(a(x,y,t)\ptl_x u)+b_0(x,y,t)\ptl_x u+b(x,y,t)\ptl_y u,$$ which generalized our recent results on KFP equations.

Abstract:
Consider the system $|\partial_tu+\Delta u|\leq M(|u|+|\nabla u|)$, $|u(x,t)|\leq Me^{M|x|^2}$ in $\mathcal{C}_{\theta}\times[0,T]$ and $u(x,0)=0$ in $\mathcal{C}_{\theta}$, where $\mathcal{C}_{\theta}$ is a cone with opening angle $\theta$. L. Escauriaza constructed an example to show that such system has a nonzero bounded solution when $\theta<90^\circ$, and it's conjectured that the system has only zero solution for $\theta>90^\circ$. Recently Lu Li and V. \v{S}ver\'{a}k \cite{LlS} proved that the claim is true for $\theta>109.5^\circ$. Here we improve their result and prove that only zero solution exists for this system when $\theta>99^\circ$ by exploring a new type of Carleman inequality, which is of independent interest.

Abstract:
We establish some interior regularity criterions of suitable weak solutions for the 3-D Navier-Stokes equations, which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin's criterion and Escauriza-Seregin-\v{S}ver\'{a}k's criterion. We also show that if weak solution $u$ satisfies $$ \|u(\cdot,t)\|_{L^p}\leq C(-t)^{\frac {3-p}{2p}} $$ for some $3