The framework of crystalline motion is explained. We introduced the defor-mation process of a vapor figure through Nakayaʼs experiment, and showed the area-preserving crystalline curvature flow could be regarded as a simple model of the process. We here mentioned two results: one is the “convexity phenomena” and the other is the asymptotic behavior “convergence to the Wul. shape”. These results lead us to understand the all deformation pro-cess of the vapor .gure and give the mathematical justi. cation of Nakayaʼs observation.
Internal gravity wave beams propagate in a form of ʻbeamʼ from a wave source in the atmosphere and the ocean. This beam solution is superposition of monochromatic wave solutions for a given propagation direction, and is an exact solution of Euler equations. In the present review article, we consider ʻ3-wave interactionʼ of these beams. The 3-wave interaction among monochromatic waves is well-known to occur, but their dynamics among wave beams has not been elucidated so far. According to recent theoretical studies, occurrence of instability due to 3-wave interaction among wave beams depends not only on their amplitudes but also on their ʻwidthʼ. We demonstrate this fact by numerical simulation of Euler equations.
We describe recent development of finite element error analysis. In finite element error analysis, estimating errors of Lagrange interpolation on triangles is very important. To obtain error estimations of Lagrange interpolation, it is believed that triangles or triangulations must satisfy a certain geometric conditions such as minimum or maximum angle conditions. The authors have found that the circumradius condition of triangles or triangulations is more essential than the minimum and maximum angle conditions. To show our claim, the authors extended techniques presented by Babuška-Aziz and Liu-Kikuchi. Our method can be applied to the case of Lagrange interpolation on tetrahedrons and Crouzeix-Raviart interpolation on triangles.