REGULARITY ANALYSIS OF DISCRETE MORSE FLOWS FOR ANARISYS OF GENERAL CRITICAL POINTS

1.在研究非光滑流形之间调和映射的变分问题时，对于具有上界曲率的目标流形，K.Hoshino证明了Alexandrov的NPC不等式在正曲流形上的一个松弛形式也成立，并应用它证明了离散Morse流的一个极大值原理和局部估计。这些结果假定初始数据半径的界，而局部估计的界是极大值原理的界的一半，并且我们在较小的界下构造了Morse流。我们准备发表结果，并将继续研究，以消除对初始数据界限的强烈限制。M.Misawa和Takayoshi Ogawa利用从流形到球面的调和热流的平均振荡证明了一个正则性判据，并发表在《变分和偏微分方程组》上。N.Kikuchi和G.Seregin构造了满足局部能量不等的非定常N-S方程的解。他们在一个特殊的Morrey空间中对Stokes方程的解进行构造和局部估计，并准备发表结果。

1. In study on variational problem of harmonic maps between nonsmooth manifolds, in the case of a target manifold with upper bounded curvature, K.Hoshino showed a relaxed version of Alexandrov's NPC inequality to hold also in positively curved manifolds, and apply it for proving a maximal principle and a local estimate for discrete Morse flows. These results assume bounds for radius of initial data, whereas a bound for the local estimate is half of that for the maximum principle, and we constructed Morse flow assuming the smaller bound. We prepare to publish the result and shall continue the research to remove the strong restriction on bounds for initial data.2. M. Misawa and Takayoshi Ogawa proved a regularity criteria by mean oscillation of harmonic heat flow from manifold into a sphere and published the result in Calculus of Variations and Partial Differential Equations.3. N. Kikuchi and G. Seregin constructed solutions to nonstationary Navier-Stokes equations satisfying the local energy inequality. They carry on construction and local estimate of solutions to Stokes equations in a special Morrey space and prepare to publish the result.