On the formation of trapped surfaces

In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. In this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and, consistent with this, relaxing the corresponding propagation estimates just enough that a trapped surface still forms. We also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one. More importantly, the proof, which can be easily localized with respect to angular sectors, has the potential for further developments.     

On the role of energy in chip formation

The implicit code ABAQUS/Standard is used to simulate the formation of continuous and segmented chips. Using the described model, the idealized process of friction‐less machining of an elastic ideally‐plastic material is studied. It is shown that Merchant's classical shear angle relation does not hold, as chip formation does not minimize the energy as assumed by Merchant. The model is also used to study segmented chip formation using a realistic material law for the Titanium alloy Ti6Al4V at high cutting speeds.     

On pattern formation in the Gray-Scott model

In the paper, we investigate an elliptic system well-known as the Gray-Scott model and present some further results for positive solutions of this model. More precisely, we give the refined a priori estimates of positive solutions, and improve some previous results for the non-existence and existence of positive non-constant solutions as the parameters are varied, which imply some certain conditions where the pattern formation occurs or not.