Global solution to the n-dimensional non-resistive MHD system with damping in magnetic field

The global existence is obtained for the solution to the viscous non-resistive MHD system with magnetic damping in ${\mathbb{R}}^n(n\ge 2)$. This study is inspired by the recent examinations obtained by Fefferman et al. (2014) and Chemin et al. (2016) on the local well-posedness of the viscous non-resistive MHD system.     

Characterizations of perturbations of spectra of 2×2 upper triangular operator matrices

When $A\in B\left(H\right)$ and $B\in B\left(K\right)$ are given, we denote by $M_C$ the operator acting on the infinite dimensional separable Hilbert space $H\oplus K$ of the form $M_C=\left(\begin{array}{cc}\hfill A\hfill & \hfill C\hfill \\ \hfill 0\hfill & \hfill B\hfill \end{array}\right)$. In this paper, we first give some necessary and sufficient conditions for $M_C$ to be a left invertible operator (an upper semi-Weyl, upper semi-Fredholm) operator for some $C\in B(K,H)$, which extend the corresponding results in Cao et al. (2006) [4], Cao and Meng (2005) [5], Hwang and Lee (2001) [12] and Li and Du (2006) [15]. Then we present some counter-examples.     

Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function

Let M be the Hardy–Littlewood maximal function and b be a locally integrable function. Denote by $M_b$ and $[b,M]$ the maximal commutator and the (nonlinear) commutator of M with b. In this paper, the author considers the boundedness of $M_b$ and $[b,M]$ on Lebesgue spaces and Morrey spaces when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.     

Cohomology over Fiber products of local rings

Let S and T be local rings with common residue field k, let R be the fiber product $S{\times }_{k}T$, and let M be an S-module. The Poincaré series $P_M^R$ of M has been expressed in terms of $P_M^S$, $P_k^S$ and $P_k^T$ by Kostrikin and Shafarevich, and by Dress and Krämer. Here, an explicit minimal resolution, as well as theorems on the structure of ${\mathrm{Ext}}_R(k,k)$ and ${\mathrm{Ext}}_R(M,k)$ are given that illuminate these equalities. Structure theorems for the cohomology modules of fiber products of modules are also given. As an application of these results, we compute the depth of cohomology modules over a fiber product.