2018年偏微分方程小型研讨会日程
会议地点:行健楼665 会议时间:12月7-8日
 
 
12月7日上午行健楼665
时间 内容
8:40-9:50 于品(清华大学)
On stable shock formations for 3-dimensional wave equations
10:00-11:10 何凌冰(清华大学)
Global stability of large solutions to Compressible Navier-Stokes equations in the whole space
12月7日下午行健楼665
2:30-3:40 王志强(复旦大学)
Observability and controllability of wave equations coupled by first order terms
4:00-5:10 金龙(清华大学)
Control and stabilization on hyperbolic surfaces
12月8日上午行健楼665
8:40-9:50 王成波(浙江大学)
具有适当曲率条件的黎曼流形上的非线性波动方程
10:00-11:10 王勇(中国科学院数学与系统科学研究院)
Global well-posedness for the Boltzmann equation with a class of large amplitude data
 
 
摘  要
 
On stable shock formations for 3-dimensional wave equations
于品(清华大学)
We will discuss a geometric perspective on shock formations for a class of quasilinear wave equations which admit global smooth solutions with small data. We exhibit a family of smooth initial data leading to breakdown of smoothness of the solution. The work combines the ideas from fluid mechanics and from general relativity. This is a joint work with Shuang Miao.
 
 
 
Global stability of large solutions to Compressible Navier-Stokes equations 
in the whole space
何凌冰(清华大学)
In this talk, we will focus on the global-in-time strong stability for CNS in the whole space. Assuming that the density is bounded in some Holder space, we first obtain that the solution will converge to its equilibrium with an explicit rate which as the same as that for the heat equation. Based on this new decay estimates, we prove the general global-in-time stability for CNS in the critical spaces. Finally, to show that our theory has wide application, we construct some large solution to CNS, which is beyond the close-to-equilibrium setting. The similar result can be generalized to the full NS system.
 
 
 
Observability and controllability of wave equations coupled by first order terms
王志强(复旦大学)
In this talk, we consider the observability and controllability of wave equations coupled by first order terms on a compact manifold.  We adopt the Dehman-Lebeau's approach to prove that: the weak observability inequality holds for such systems if and only if a class of ordinary differential equations related to the symbol of the first order terms along the Hamiltonian flow are exactly controllable.The observability constant and the observation time are also analyzed. By duality, we obtain a criteria for the  controllability of the dual control system in a finite co-dimensional space. Moreover, the criteria for exact controllability of such systems can be obtained under the further assumption of unique continuation property. Finally, some examples with special structures are given as applications where the unique continuation property indeed holds.
 
 
Control and stabilization on hyperbolic surfaces
金龙(清华大学)
In this talk, we discuss some recent results concerning the control and stabilization on a compact hyperbolic surface. In particular, we show that the Laplace eigenfunctions have uniform lower bounds on any nonempty open set; the linear Schrödinger equation is exactly controllable by any nonempty open set; and the energy of solutions to the linear damped wave equation with regular initial data decay exponentially for any smooth damping function. The new ingredient is the fractal uncertainty principle for porous sets by Bourgain–Dyatlov. This is partially based on joint work with SemyonDyatlov.
 
 
 
具有适当曲率条件的黎曼流形上的非线性波动方程
王成波(浙江大学)
本报告中,我们将探讨具有适当曲率条件的黎曼流形上的典型幂次型非线性波动方程具有小初值整体解的存在性问题。特别的,我们将对于一大类流形确定相应的临界指标。本报告基于与Chris Sogge与Yannick Sire最近的工作。
 
 
 
Global well-posedness for the Boltzmann equation 
with a class of large amplitude data
王勇(中国科学院数学与系统科学研究院)
In this talk, we will introduce some results on the global well-posedness of Boltzmann for a class of initial data with large amplitude oscillations. It is based on several joint works with RenjunDuan, Feiming Huang, Tong Yang and Zhu Zhang.