Analyticity for the (generalized) Navier-Stokes equations with rough initial data

We study the Cauchy problem for the (generalized) incompressible Navier-Stokes equations \begin{align} u_t+(-\Delta)^{\alpha}u+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0. \nonumber \end{align} We show the analyticity of the local solutions of the Navier-Stokes equation ($\alpha=1$) with any initial data in critical Besov spaces $\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$ with $1< p<\infty, \ 1\le q\le \infty$ and the solution is global if $u_0$ is sufficiently small in $\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$. In the case $p=\infty$, the analyticity for the local solutions of the Navier-Stokes equation ($\alpha=1$) with any initial data in modulation space $M^{-1}_{\infty,1}(\mathbb{R}^n)$ is obtained. We prove the global well-posedness for a fractional Navier-stokes equation ($\alpha=1/2$) with small data in critical Besov spaces $\dot{B}^{n/p}_{p,1}(\mathbb{R}^n) \ (1\leq p\leq\infty)$ and show the analyticity of solutions with small initial data either in $\dot{B}^{n/p}_{p,1}(\mathbb{R}^n) \ (1\leq p<\infty)$ or in $\dot{B}^0_{\infty,1} (\mathbb{R}^n)\cap {M}^0_{\infty,1}(\mathbb{R}^n)$. Similar results also hold for all $\alpha\in (1/2,1)$.Comment: 31 page

The Basic Situation of Regional Grain Trade in China

This paper is designed first to evaluate the changes of China's regional grain trade policy and second to examine the changes of the pattern of regional grain trade from the 1950s to the late 1990s

RESEARCH OF REGION WATER SECURITY EVALUATION BY SET PAIR ANALYSIS

Abstract The set pair analysis theory was applied in the water security evaluation due to the existing uncertain relationship in the system. Connection degree is used to describe this sort of uncertain problem. In the process, the indexes information of the region water security was reflected and the difficulty of weight allocation was avoided. The water security of Shandong province was evaluated by this method. The result was compared with what is got by fuzzy comprehensive evaluation and matter-element analysis. It proves that SPA analysis is simple and practical and can be a useful evaluating tool for region water security

Improvement of resistance to rice blast and bacterial leaf streak by CRISPR/Cas9-mediated mutagenesis of Pi21 and OsSULTR3;6 in rice (Oryza sativa L.)

Rice (Oryza sativa L.) is a staple food in many countries around the world, particularly in China. The production of rice is seriously affected by the bacterial leaf streak and rice blast, which can reduce rice yield or even cause it to fail to be harvested. In this study, susceptible material 58B was edited by CRISPR/Cas9, targeting a target of the Pi21 gene and a target of the effector-binding element (EBE) of the OsSULTR3;6 gene, and the mutants 58b were obtained by Agrobacterium-mediated method. The editing efficiency of the two targets in the T0 generation was higher than 90.09%, the homozygous mutants were successfully selected in the T0 generation, and the homozygous mutation rate of each target was higher than 26.67%. The expression of the edited pi21 and EBE of Ossultr3;6 was significantly reduced, and the expression of defense responsive genes was significantly upregulated after infected with rice blast. The lesion areas of rice blast and bacterial leaf streak were significantly reduced in 58b, and the resistance of both was effectively improved. Furthermore, the gene editing events did not affect the agronomic traits of rice. In this study, the resistance of 58b to rice blast and bacterial leaf streak was improved simultaneously. This study provides a reference for using Clustered Regularly Interspaced Short Palindromic Repeats/Cas9 (CRISPR/Cas9) to accelerate the improvement of rice varieties and the development of new materials for rice breeding

Tumor-suppressor activity of RRIG1 in breast cancer

<p>Abstract</p> <p>Background</p> <p>Retinoid receptor-induced gene-1 (RRIG1) is a novel gene that has been lost in several types of human cancers. The aim of this study was to determine whether RRIG1 plays a role in breast cancer, such as in the suppression of breast cancer cell growth and invasion.</p> <p>Methods</p> <p>Immunohistochemistry was used to detect RRIG1 expression in breast tissue specimens. Gene transfection was used to restore or knock down RRIG1 expression in breast cancer cell lines for analysis of cell viability, colony formation, and migration/invasion potential. Reverse-transcription polymerase chain reaction and western blot assays were used to detect the changes in gene expression. The RhoA activation assay was used to assess RRIG1-induced inhibition of RhoA activity.</p> <p>Results</p> <p>The immunohistochemical data showed that <it>RRIG1 </it>expression was reduced in breast cancer tissues compared with normal and atypical hyperplastic breast tissues. <it>RRIG1 </it>expression was inversely correlated with lymph node metastasis of breast cancer but was not associated with the status of hormone receptors, such as estrogen receptor, progesterone receptor, or HER2. Furthermore, restoration of <it>RRIG1 </it>expression inhibited proliferation, colony formation, migration, and invasion of breast cancer cells. Expression of RRIG1 also reduced phosphorylated Erk1/2 and Akt levels; c-Jun, MMP9, and Akt expressions; and RhoA activity. In contrast, knockdown of RRIG1 expression promoted breast cancer cell proliferation, colony formation, migration, and invasion potential.</p> <p>Conclusion</p> <p>The data from the current study indicated that <it>RRIG1 </it>expression was reduced or lost in breast cancer and that restoration of RRIG1 expression suppressed breast cancer cell growth and invasion capacity. Future studies will determine the underlying molecular mechanisms and define RRIG1 as a tumor-suppressor gene in breast cancer.</p