Narrow Results

In this paper, we consider the embeddings of weighted Besov spaces into Besov-type spaces with w being a (local) Muckenhoupt weight, and give sufficient and necessary conditions on the continuity and the compactness of these embeddings. As special cases, we characterize the continuity and the compactness of embeddings in case of some polynomial or exponential weights. The proofs of these conclusions strongly depend on the geometric properties of dyadic cubes.

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