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I will discuss the result on the non-uniqueness of solutions to the dual Minkowski problem. In particular we show that for the problem with constant right hand side, when $q>2n$ the solution is non-unique.

In this talk, we talk about the regularity of the free boundary in the Monge-Ampere obstacle problem. This is a joint work with Prof. Tang Lan and Prof. Wang Xu-Jia.

In this lecture, we present practical and provably secure (authenticated) key exchange protocol and password authenticated key exchange protocol, which are based on the learning with errors problems. These protocols are conceptually simple and have strong provable security properties. This type of new constructions were started in 2011-2012. These protocols are shown indeed practical. We will explain that all the existing LWE based key exchanges are variants of this fundamental design. In addition, we will explain some issues with key reuse and how to use the signal function invented for KE for authentication schemes.

In this talk, I will talk about our recent results for the well-posedness of the 3D incompressible Hall-magneto-hydrodynamic system (Hall-MHD). First, we provide an elementary proof of a global well-posedness result for small data with critical Sobolev regularity, in the spirit of Fujita-Kato’s theorem for the Navier-Stokes equations. Next, we present the long-time asymptotics of global (possibly large) solutions of the Hall-MHD system that are in the Fujita-Kato regularity class. A weak-strong uniqueness statement is also presented. Finally, we consider the so-called 2.5D flows for the Hall-MHD system (that is 3D flows independent of the vertical variable), and establish a global existence of strong solutions, assuming only that the initial magnetic field is small.

多孔介质广泛存在于自然与工程系统之中，其微观几何结构直接影响着整体宏观特性。虽然人们可以在孔隙尺度对多孔介质进行精准仿真，但是孔隙间的复杂结构，以及生产过程中不可避免的加工偏差，都对其宏观性质的估测带来了不确定性。为了解决上述问题，本报告将介绍一种基于闵可夫斯基泛函的不确定性量化框架，在对微观几何结构进行降维的前提下，通过广义多项式混沌或高斯过程，构建系统宏观特性的替代模型，从而降低总体计算成本。我们将通过浸透性和扩散系数这两个实例展示该量化框架的有效性。

We prove that the classical 3-D Navier-Stokes equations have a unique global Fujita-Kato solution provided that the $H^{-\frac12,0}$ norm of $\pa_3u_0$ is sufficiently small compared to some quantities of the initial data, which keep invariant under the natural scaling of N-S and dilating in the $x_3$ variable. This result provides some classes of large initial data which are large in Besov space $B^{-1}_{\infty,\infty}$ and can generate unique global solutions to 3-D Navier-Stokes system. In particular, we extend the previous results in a series of works by Chemin, Gallagher, Ping Zhang et al. for initial data with a slow variable to multi-scales slow variable initial data.

We show that any unique global solution (here we do not require any smallness condition beforehand) to 3-D axisymmetric Navier-Stokes equations in some scaling invariant spaces must eventually become a small solution. In particular, we show that the limits of $\|\omega^\theta(t)/r\|_{L^1}$ and $\|u^\theta(t)/\sqrt r\|_{L^2}$ are all $0$ as $t$ tends to infinity. And by using this, we can refine some decay estimates for the axisymmetric solutions.

tions The Poisson-Nernst-Planck (PNP) type of equations are one of the most extensively studied models for the transport of charged particles in many physical and biological problems. The solution to the PNP equahas many properties of physical importance, e.g., positivity, mass conservation, energy dissipation. It is desirable to design numerical methods that are able to preserve such properties at discrete level. In this talk, we will present two types of numerical schemes that can maintain physical properties. One is based on the so-called Slotboom variables; the other is based on the gradient flow structure of the PNP equations. Some numerical results are shown to demonstrate their performances. This is a joint work with Jie Ding, Chun Liu, Cheng Wang, Zhongming Wang, Xingye Yue, and many others.