Speaker: 苗长兴

Title: Hamilton型发展方程的散射理论。

Abstract:

本报告主要介绍Hamilton型发展方程的散射理论。基于Fourier限制性理论的 Strichartz估计不仅为我们提供在能量空间及在低正则空间中求解Hamilton型发展方程，同时也为在能量空间中研究Hamilton型发展方程的散射理论提供了研究框架。 本报告拟从物理不变量（能量、质量、动量、共形与拟共形等式）与变换群出发，讨论经典Morawetz估计、相互作用的Morawetz估计在散射理论研究中的作用，给出经典散射理论的新观点与简单证明，昭示临界散射理论研究的困难与挑战性。与此同时，简要介绍报告人与合作者通过发展与具反平方位势的Laplace算子相对应的调和分析方法，解决的具有反平方位势的Schrodinger方程的散射猜想及“临界模猜想”等工作。

Speaker: ChangYou Wang

Title: A minimizing problem involving nematic liquid crystal droplets.

Abstract: In this talk, we will describe an energy minimizing problem arising from seeking the optimal configurations of a class of nematic liquid crystal droplets, which was initiated by Lin-Poon back in 1996. More precisely, the general problem seeks a pair $(/Omega, u)$ that minimizes the energy functional:

$$E(u,/Omega)= /int_/Omega /frac12|/nabla u|^2+ /mu /int_{/partial/Omega} f(x,u(x)) d/sigma,$$

among all open set $/Omega$ within the unit ball of $/mathbb R^3$ , with a fixed volume, and $u/in H^1(/Omega,/mathbb S^2)$.  Here $f:/mathbb R^3/times /mathbb R /to/mathbb R$ is a suitable nonnegative function, which is given.

While the existence of minimizers remains open in the full generality, there has been some partial progress when $/Omega$ is assumed to be convex. In this talk, I will discuss some results for $/Omega$ that are not necessarily convex. This is a joint work with my student Qinfeng Li.

Speaker: Xu Zhang

Title: An invitation to control theory of stochastic distributed parameter systems.

Abstract: Control theory for ODE systems is now relatively mature. There exist a huge list of works on control theory for (deterministic) distributed parameter systems though it is still quite active; while the same can be said for control theory for stochastic systems in finite dimensions. In this talk, I will give a short introduction to control theory of stochastic distributed parameter systems (governed by stochastic differential equations in infinite dimensions, typically by stochastic PDEs), which is, in my opinion, almost at its very beginning stage. I will mainly explain the new phenomenon and difficulties in the study of controllability and optimal control problems for these sort of equations. In particular, I will show by some examples that both the formulation of stochastic control problems and the tools to solve them may differ considerably from their deterministic / finite-dimensional counterparts. Interestingly enough, one has to develop new mathematical tools to solve some problems in this field. Meanwhile, in some sense, the stochastic distributed parameter control system is the most general control system in the framework of classical physics, and therefore the study of this field may provide some useful hints for that of quantum control systems.

Speaker: Feng Zhou

Title : Isolated singularities for some nonlinear elliptic equations.

Abstract: We will talk about some results on the isolated singularities for some nonlinear elliptic equations including the nonlinear Choquard equations and the equations involving the Hardy-Leray potentials. We prove the nonexistence and existence of isolated singular positive solutions for these equations. We present some suitable distributional identities of the solution and we obtain the qualitative properties for the minimal singular solutions. This is based on joint works with H.Y. Chen.

 Speaker: ZhiFei Zhang

Title: Some mathematical progress on the hydrodynamics stability

Abstract: In this talk, I will present some recent results on the linear inviscid damping of shear flows, the metastability of Kolmogorov flow and spectral analysis of the Oseen vortices operator.

Speaker： Ping Zhang

Title: On the critical one component regularity for 3-D

 Navier-Stokes system

Abstarct:Given an initial data $v_0$ with vorticity ~$/Om_0=/na/times v_0$ in~$L^{/frac 3 2}$ (which

implies that~$v_0$ belongs to the Sobolev   space~$H^{/frac12}$),  we prove that the solution~$v$ given by the classical Fujita-Kato theorem  blows up in a finite time~$T^/star$   only if, for any $p$ in~$ ]4,6[$ and any unit vector~$e$ in~$/R^3,$ there holds $/int_0^{T^/star}/|v(t)/cdote/|_{/dH^{/f12+/f2p}}^p/,dt=/infty.$ We remark that all these quantities  are scaling invariant under the scaling transformation of Navier-Stokes system.

Speaker: Dong Ye

Title :Liouville Theorem for stable at infinity solution to Lane-Emden system 

 Abstract: Consider the classical Lane-Emden system. We prove that 

for any $(p, q)$ under the Sobolev hyperbola, the only nonnegative 

solution stable outside a compact set is zero. To handle the case 

$p < 1$, we will make use of $m$-biharmonic equation.