In connection with the recent proposal for possible singularity formation at the boundary for solutions of three-dimensional axisymmetric incompressible Euler’s equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite-time blowup from smooth data.

Japan’s lunar probe satellite “Kaguya”, China’s lunar probe satellite “Chang’ e-1 lunar probe” left the earth on September 14th, 2007 and October 24th, 2007, began the exploration to the moon. The launching of the China’s lunar satellite is the proud to the global Chinese. However, at the same time as we are proud of the event, we must admit that the pictures published which were sent back by Japanese are clearer and more beautiful than Chinese. While we are saying that the Japan has a better technology, maybe we also need to say that the Japanese can hold better chance than Chinese when taking the pictures. The following two pictures were sent back by the satellite. Picture one is sent back by Japan’s satellite, Picture two is sent back by China’s. It can be seen that there is a long way for China if we want to become stronger at science technology. We also hope that we can have some contribution in the prosperity of China now and in the future.

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The$H$^{$p$}corona problem is the following: Let$g$_{1}, ...,$g$_{$m$}be bounded holomorphic functions with 0<δ≤Σ‖$g$_{$i$}‖. Can we, for any$H$^{$p$}function ϕ, find$H$^{$p$}functions$u$_{1}, ...,$u$_{$m$}such that Σ$g$_{$i$}$u$_{$i$}=ϕ? It is known that the answer is affirmative in the polydisc, and the aim of this paper is to prove that it is in non-degenerate analytic polyhedra. To prove this, we construct a solution using a certain integral representation formula. The$H$^{$p$}estimate for the solution is then obtained by localization and some harmonic analysis results in the polydisc.

No abstract uploaded!