主要针对自由流体与多孔介质耦合问题数值方法中非线性问题、多物理问题耦合与多变量耦合系统等难点问题展开讨论：对于整个多物理耦合系统，我们主要采用高效解耦方法使得耦合问题求解化为各自物理域问题求解。进一步，针对耦合问题数值方法，我们分别有：1.对非线性问题针对稳态问题采取非奇异解束理论和非稳态问题采用small data假设来解决；2.对于Robin-Robin区域分解对稳态问题使用多物理迭代解耦方法，对非稳态问题采用多物理非迭代方法解耦方法；3.利用裂解算法求解复杂自由流Stokes问题，降低计算存储和规模，使得整个多变量耦合系统转化为拟椭圆问题或泊松问题求解、大规模的科学计算化为小规模计算。最后，我们针对现场油藏开采数值模拟进行简要汇报。

High order nonlinear weighted essentially non-oscillatory finite difference (WENO) schemes, which have the capability of capturing shocks essentially non-oscillatory while re-solving small scale structures efficiently, are popular for solving hyperbolic conservation laws. However, the WENO scheme is fairly complex to implement, computationally expen-sive and too dissipative for certain classes of problems. A natural way to alleviate some of these difficulties is to construct a hybrid scheme conjugating a nonlinear WENO scheme in the non-smooth stencils with a linear method in the smooth stencils. In this talk, I will briefly introduce the difficulties and recent development in the hybrid schemes.

报告人将介绍在高维唯一性问题领域的最新工作，包括三个方面：一是对于一般的可能线性退化的亚纯映射，在一个新类型的条件下得到的唯一性结果；二是分担 2n+2 个超平面的唯一性结果；三是分担较少超曲面的退化性和唯一性结果。

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure (M; g). This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure J has small energy(depending on the norm |/nabla J|), then the flow exists for all time and converges to a Kaehler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no Kaehler structure in the homotopy class. A main technical tool is a version of monotonicity formula, similar as in the theory of the harmonic map heat flow. We also construct an almost complex structure on a flat four tori with small energy such that the harmonic heat flow blows up at finite time with such an initial data.

题目：Numerical Method for the Maxwell Equations with Random Interfaces报告人：张凯 教授（吉林大学）地点：腾讯会议室时间：2020年6月9日 上午10:00-11:0

We solve the Gursky-Streets equations with uniform C^1,1 estimates for . An important new ingredient is to show the concavity of the operator which holds for all . Our proof of the concavity heavily relies on Garding's theory of hyperbolic polynomials and results from the theory of real roots for (interlacing) polynomials. Together with this concavity, we are able to solve the equation with the uniform C^1,1 a priori estimates for all the cases . Moreover, we establish the uniqueness of the solution to the degenerate equations for the first time.

We shall first give a brief introduction for Alexandrov geometry, which is a class of singular metric spaces with curvature bounded from below by triangle comparisons, and then we introduce a result about regularity of harmonic maps from an Alexandrov space to a compact Riemannian manifold, which is based on the joint works with Prof. Huabin Ge and Prof. Wenshuai Jiang.

In this talk, I first introduce mean curvature flow briefly, and then mainly consider closed hypersurfaces immersed in a space of constant sectional curvature evolving in direction of its outer unit normal vector with speed given by a general curvature function of principal curvatures, such that the initial hypersurface is pinched in the sense that the ratio of the biggest and smallest principal curvatures of the hypersurface is close enough to 1 everywhere. We prove that the pinching is preserved as long as the flow exists, and the flow shrinks to a point in finite time. Especially, if the speed is a high order homogeneous function, the normalized flow exists for all time and converges smoothly and exponentially to a round sphere in Euclidean space.