Partial regularity of solutions to the 3D chemotaxis-Navier-Stokes equations at the first blow-up time

In this note, we investigate partial regularity of weak solutions of the three dimensional chemotaxis-Navier-Stokes equations, and obtain the $\frac53$-dimensional Hausdorff measure of the possible singular set is vanishing at the first blow-up time. The new ingredients are to establish certain type of local energy inequality and deal with the non-scaling invariant quantity of $n\ln n$, which seems to be the first description for the singular set of weak solutions of the chemotaxis-fluid model, which is motivated by Caffarelli-Kohn-Nirenberg's partial regularity theory \cite{CKN}

Capacity sharing, product differentiation and welfare

This article constructs a duopoly market with product differentiation and analyses profits, consumer surplus and social welfare under three conditions: (a) two enterprises have sufficient capacity; (b) one enterprise has insufficient capacity, and another enterprise has excess capacity that is not shared; and (c) one enterprise has insufficient capacity, and another enterprise has excess capacity and engages in capacity sharing. Through comparison, the implementation conditions for and effects of capacity sharing and the role of product differentiation are revealed. The results show that capacity sharing helps increase producer surplus and social welfare. Capacity constraints reduce social welfare but can be solved by capacity sharing. Capacity sharing can only be realised when both enterprises are profitable, and the charge for capacity sharing should not be too high or too low. Product differentiation has impacts on output, profit, consumer surplus and social welfare, and these impacts are restricted by the existence of capacity constraints and capacity sharing

Global existence of suitable weak solutions to the 3D chemotaxis-Navier-Stokes equations

In 2004, Dombrowski et al. showed that suspensions of aerobic bacteria often develop flows from the interplay of chemotaxis and buoyancy, which is described as the chemotaxis-Navier-Stokes model, and they observed self-concentration occurs as a turbulence by exhibiting transient, reconstituting, high-speed jets, which entrains nearby fluid to produce paired, oppositely signed vortices. In order to investigate the properties of these vortices (singular points), one approach is to follow the partial regularity theory of Caffarelli-Kohn-Nirenberg to study the singularity properties of suitable weak solutions. In this paper, we established the existence of suitable weak solution for the three dimensional chemotaxis-Navier-Stokes equations, where the main difficulty is to establish appropriate local energy inequalities

Teacher Efficacy, Work Engagement, and Social Support Among Chinese Special Education School Teachers

This paper investigates the relationship between teacher efficacy and sociodemographic factors, work engagement, and social support among Chinese special education school teachers. The sample comprised 1,027 special education school teachers in mainland China. The Teachers’ Sense of Efficacy Scale, the Multi-Dimensional Scale of Perceived Social Support, and the Utrecht Work Engagement Scale were used for data collection. Correlation analysis revealed that social support, work engagement, and teacher efficacy were significantly correlated with each other. Additionally, gender, years of experience, and monthly salary were significant predictors of teacher efficacy. Furthermore, structural equation modeling analysis showed that social support exerted its indirect effect on teacher efficacy through the mediation of work engagement. The findings of this study provide a new perspective on the complex association between social support and teacher efficacy. The explanations and limitations of these findings are discussed