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

1. 在研究非光滑流形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给出了Alexandrov NPC不等式在正弯曲流形中也成立的一个放宽版本，并将其应用于证明离散莫尔斯流的极大原理和局部估计。这些结果假设初始数据半径的边界，而局部估计的边界是最大值原理的一半，并且我们假设较小的边界来构造莫尔斯流。我们准备发表结果，并将继续研究，以消除对初始数据边界的强烈限制。M. Misawa和Takayoshi Ogawa用从流形到球体的调和热流的平均振荡证明了一个正则性准则，并在《变分学与偏微分方程》中发表了结果。N. Kikuchi和G. Seregin构造了满足局部能量不等式的非平稳Navier-Stokes方程的解。他们在一个特殊的Morrey空间中对Stokes方程的解进行了构造和局部估计，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.