The 1977’ Acta paper by Lawson-Osserman studied the Dirichlet problem for minimal surfaces of high codimensions. Several astonishing results essentially distinct from the case of codimension 1 were obtained there. In particular, they found Lipschitz but non-C1 solutions to the problems associated to Hopf maps between unit spheres. Recently we made systematic developments and discovered certain interesting new phenomena on the existence, non-uniqueness, non-minimizing and minimizing properties of solutions to related Dirichlet problems. This talk is based on joint works with Xiaowei XU and Ling YANG.

In this talk i will present a general theory of mathematical modeling for irreversible processes. By mathematical modeling, I mean to establish some relations among the apparently unrelated time-space functions characterizing a given non-equilibrium thermodynamic system. This theory is initiated with an observation that many classical mathematical models (hyperbolic PDEs) share certain common properties. Because of this, it is natural to require the constructed models to possess the properties when modeling an irreversible phenomenon with PDEs. Models constructed with this theory are hyperbolic balance laws and fulfill some fundamental requirements.

Recently there is some progress on the rationality of compact Kahler manifolds with positive holomorphic sectional curvature, for example the work of Heier-Wong and Xiaokui Yang. In this talk, we present a differential-geometric result on such manifolds, and exhibit new examples on Kahler and Hermitian manifolds.

Minkowski problem is one of a central problem in convex geometry. In this talk, we will present our recent work on Lp Minkowski problem for capacity.

Consider weak solutions of the instationary Navier-Stokes system in a three-dimensional bounded smooth domain $/Omega$. It is well known that any solenoidal initial value $u_0$ in $L^2(/Omega)$ with a vanishing normal component on the boundary admits a global in time weak solution. Moreover, if $u_0 /in H^1$ or even only $u_0 /in /mathcal{D}(A^{1/4})/subset L^3$, where $A =-P/Delta$ denotes the Stokes operator, then $u_0$ admits a unique local in time regular (strong) solution in Serrin’s class $L^s(0,T;L^q (/Omega))$ where $2/s + 3/q = 1$ for some $T = T(u_0)/leq/infty$.

题目：Stability Analysis for Imcompressible Navier-Stokes Equations with Navier Boundary Conditions报告人：丁时进 教授（华南师范大学必赢国际bwin登录）地点：宁静楼108室时间：2017年11月15日（周三） 下午16：00-17:00欢迎广大师生参加

We develop the Lawson-Osserman's works on minimal graphs. Firstly, we construct a constellation of uncountably many Lawson-Osserman spheres, which are minimal in Euclidean spheres and therefore generate Lawson-Osserman cones that correspond to Lipschitz but non-differentiable solutions to the minimal surface system. Then, by the theory of autonomous systems in plane, we find for each Lawson-Osserman cone an entire minimal graph having it as tangent cone at infinity.

In this talk, we shall discuss the global stability of prey-taxis model with a variety of predator-prey interactions. In the previous limited results, various assumptions are imposed to ensure the boundedness of classical solutions. In our work, we shall first remove those conditions, and then establish the asymptotic behavior of solutions and identify the conditions under which the predator-prey asymptotic outcomes including coexistence, exclusion and extinctions will be achieved. Applications of our results will be discussed and some open questions will be discussed.