Singularity formation in three-dimensional vortex sheets

We study singularity formation of three-dimensional (3-D) vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contributions of the integral equation. By introducing an appropriate change of variables, we show that the leading order vortex sheet equation degenerates to a two-dimensional vortex sheet equation in the direction of the tangential velocity jump. This change of variables is guided by a careful analysis based on properties of certain singular integral operators, and is crucial in identifying the leading order singular behavior. Our result confirms that the tangential velocity jump is the physical driving force of the vortex sheet singularities. We also show that the singularity type of the three-dimensional problem is similar to that of the two-dimensional problem. Moreover, we introduce a model equation for 3-D vortex sheets. This model equation captures the leading order singularity structure of the full 3-D vortex sheet equation, and it can be computed efficiently using fast Fourier transform. This enables us to perform well-resolved calculations to study the generic type of 3-D vortex sheet singularities. We will provide detailed numerical results to support the analytic prediction, and to reveal the generic form of the vortex sheet singularity

Research on the Biomass Power Contract of “Company +Farmer” Under Default Risk

We research the biomass power generation problem in the perspective of farmer contracting supply chain. By establishing a mathematical model and under the random yield and demand, a multiple farmer contract with a biomass power generation company problem in a two-level supply chain is researched in this paper. We propose different mechanisms in two situations: under default risk and no default risk. And compare the optimal decision in different contracts. We find that the famers’ optimal input amount is always linear to the company’s order quantity; The company and farmers’ optimal decision quantity will both be improved under no default risk. However, the optimal input quantity by farmers will decrease with the increases of the compensation ratio in this situation

Holographic Schwinger effect with a moving D3-brane

We study the Schwinger effect with a moving D3-brane in a $\mathcal{N}$=4 SYM plasma with the aid of AdS/CFT correspondence. We discuss the test particle pair moving transverse and parallel to the plasma wind respectively. It is found that for both cases the presence of velocity tends to increase the Schwinger effect. In addition, the velocity has a stronger influence on the Schwinger effect when the pair moves transverse to the plasma wind rather than parallel.Comment: 8 pages, 6 figure