Analysis and Applications

This paper deals with a study on an initial-boundary value problem for incompressible non-Newtonian fluids of degree two, in a bounded and simply connected open set $\mathrm{\Omega }$ of ${ℝ}^3$. It will be shown that the global-in-time weak solution is uniformly stable (i.e. the behavior of the solution changes continuously with the data), and consequently this solution is unique. Furthermore, an exponential decay rate for the energy of the weak solution will also be established.

In this paper, we study an unconstrained $l_{2,p}$ minimization and its associated iteratively reweighted least squares algorithm (UBIRLS) for recovering block sparse signals. Wang et al. [Y. Wang, J. Wang and Z. Xu, On recovery of block-sparse signals via mixed $l_2/l_q$ $(0<q\le 1)$ norm minimization, EURASIP J. Adv. Signal Process. 2013(76) (2013) 76] have used numerical experiments to show the remarkable performance of UBIRLS algorithm for recovering a block sparse signal, but no theoretical analysis such as convergence and convergence rate analysis of UBIRLS algorithm was given. We focus on providing convergence and convergence rate analysis of UBIRLS algorithm for block sparse recovery problem. First, the convergence of UBIRLS is proved strictly. Second, based on the block restricted isometry property (block RIP) of linear measurement matrix $A$, we give the error bound analysis of the UBIRLS algorithm. Lastly, we also characterize the local convergence behavior of the UBIRLS algorithm. The simplicity of UBIRLS algorithm, along with the theoretical guarantees provided in this paper, will make a compelling case for its adoption as a standard tool for block sparse recovery.

In this paper, we study the following biharmonic elliptic problem involving slightly subcritical non-power nonlinearity

In this paper, we study the inhomogeneous reaction–diffusion equation involving the infinity Laplacian:

The aim of this study was to establish a new regularity criterion for local-in-time smooth solutions for the 3D nematic liquid crystal flows in terms of the horizontal gradient of two horizontal velocity components in the framework of the homogeneous Besov ${Ḃ}_{\infty ,\infty }^{-1}$ space and one directional derivative of molecular orientations in the framework of the homogeneous Morrey–Campanato ${Ṁ}_{2,\frac{3}{r}}$ space. More precisely, we show that the smooth solution $(u,d)$ can be extended beyond $T$, provided that