11 月 20 日 08：00-19：00 

腾讯会议 ID: 810 798 667 密码: 211120


11 月 21 日 08：00-19：00 

腾讯会议 ID: 215 753 229 密码: 211121

11 月 22 日 08：00-19：00 

腾讯会议 ID: 978 862 380 密码: 211122


报告题目与摘要  
不可压缩欧拉方程的涡解  
曹道民（中国科学院数学与系统科学研究院&广州大学）
报告人将报告近年来在不可压欧拉方程定常涡解方面的研究，特别地要介绍在二维全空间中不可压欧拉方程行波涡对解（traveling vortex pair)、涡线对 (vortex sheet）的存在性和带边界二维区域上不可压欧拉方程定常涡线解的存在性。报告人介绍的结果主要来源于和赖善发、秦国林、詹伟城、邹昌君合作的

论文。


Residues of destructed resonant torus
程崇庆（南京大学）

In this talk, we introduce a method to handle the problem of complete  degeneracy in the study of lower dimensional invariant tori surviving from destructed  resonant torus.


The Bernoulli-type free boundary problem and its application
杜力力（四川大学）
In this talk, we will introduce the mathematical theory on the Bernoulli’s type free  boundary problem, including the existence and uniqueness of the solution to the free  boundary problem and the regularity of the free boundary.Moreover, as an important  application of this theory, we will introduce the recent existence results on the steady  incompressible inviscid impinging jets with gravity.


Continued gravitational collapse for gaseous star and pressureless  

Euler-Poisson system
 黄飞敏（中国科学院数学与系统科学研究院）
The gravitational collapse of an isolated self-gravitating gaseous star for $\gamma$-law pressure $p(\rho)=\rho^\gamma$ ($1<\gamma<\frac43$) in the mass-supcritical case is investigated. In this talk, all spherically symmetric solutions  of the pressureless Euler-Poisson system are classified. Precisely speaking, for fixed  radius $r$, there exists a unique critical velocity $v^*(r)>0$ depending on the mean  density in the ball $B(0,r)$ for the pressureless Euler-Poisson system such that if the  initial velocity $\chi_1(r)\geq v^*(r)$ (Escape case), then the dust runs away from  the gravitational force forever along an escape trajectory, and if the initial velocity  $\chi_1(r)< v^*(r)$ (Collapse case), then the dust collapses at the origin in a finite  time $t^*(r)$ even it expands initially, i.e., $\chi_1(r)>0$. Moreover, it is proved that  there exist a class of spherically symmetric solutions of gaseous star, which  

formulate a continued gravitational collapse in finite time, based on the background  of the pressureless solutions if $\chi_1(r)< v^*(r)$ for all $r\in[0,1]$. It is noted that  $\chi_1(r)$ could be positive, that is, the star might expand initially, but finally  collapse. The talk is based on a joint work with Yue Yao.


Low Mach number limit of Navier-Stokes equations with large  
temperature variations in bounded domains
琚强昌（北京应用物理与计算数学研究所）

The low Mach number limit of full compressible Navier-Stokes equations with  large temperature variations is verified rigorously in a three-dimensional boundeddomain.Weighted uniform estimates of the solutions are derived delicately in a time interval which is independent of the Mach number,in particular,for the high-order derivatives, when the initial data are well prepared only in the sense of L2-norm.The effects of large temperature variations and solid boundaries create some essential difficulties in showing the uniform estimates.This is a recent joint work with Prof. Ou, Yaobin from Renmin University.


Analysis of steady solutions for the incompressible Euler system in  
an infinitely long nozzle
李从明（上海交通大学）
Stagnation point in flows is an interesting phenomenon in fluid mechanics. It  induces many challenging problems in analysis. This talk presents a recent joint work  with Yingshu Lv and Chunjing Xie. We first derive a Liouville type theorem for  Poiseuille flows in the class of incompressible steady inviscid flows in an infinitely  long strip, where the flows can have stagnation points. With the aid of this Liouville type theorem, we show the uniqueness of solutions with positive horizontal velocity  for steady Euler system in a general nozzle when the flows tend to the horizontal  

velocity of Poiseuille flows at the upstream. Finally, this kind of flows are proved to  exist in a large class of nozzles.


On the Free Boundary Problem of 3-D Full compressible Euler  
Equations Coupled With a Nonlinear Poisson Equation
 罗涛（香港城市大学）
For the problem of full compressible Euler Equations with variable entropy  coupled with a nonlinear Poisson equation in three spatial dimensions with a  general free boundary not restricting to a graph, we identify a stability condition  for the electric potential of which the outer normal derivative is positive on the free  surface besides the Taylor sign condition for the pressure to obtain a priori  estimates on the Sobolev norms of the fluid variables and bounds for geometric  quantities of free surface. This talk is based on a joint work with K. Trivisa and H.  Zeng.


Statistical properties of 2D Navier-Stokes equations with time  

periodic forcing and degenerate stochastic forcing
吕克宁（Brigham Young University）
 We consider the incompressible 2D Navier-Stokes equations with periodic boundary conditions driven by a deterministic time periodic forcing and a degenerate stochastic forcing. We show that the system possesses a unique ergodic periodic invariant measure which is exponentially mixing under a Wasserstein metric. We also prove the weak law of large numbers for the continuous time inhomogeneous solution process. In addition, we obtain the weak law of large numbers and central limit  theorem by restricting the inhomogeneous solution process to periodic times. The  

results are independent of the strength of the noise and hold true for any value of  viscosity with a lower bound $\nu_1$ characterized by the Grashof number  $G_1$ associated with the deterministic forcing. In the laminar case, there is a larger  lower bound $\nu_2$ of the viscosity characterized by the Grashof number  $G_2$ associated with both the deterministic and random forcing. We prove that in this  laminar case, the system has trivial dynamics for any viscosity larger than  $\nu_2$ by demonstrating the existence of a unique globally exponentially stable  random periodic solution that supports the unique periodic invariant measure.


Smooth sonic-supersonic flows in critical nozzles

王春朋（吉林大学）
This talk concerns the global existence of smooth sonic-supersonic potential flows  in a two-dimensional expanding nozzle with the critical geometry at the inlet. The flow  is governed by a quasilinear non-strictly hyperbolic equation with degeneracy at the  inlet. An interesting phenomena is that the existence of such sonic-supersonic flows  depends on the height of the inlet.


1-D Navier-Stokes equation with BV data: well-posedness and  
wave propagation
 王海涛（上海交通大学）
It is established recently by Liu-Yu a constructive existence theory of weak solution to isentropic Navier-Stokes equation with initial data of small total variation.  The key ingredient in their work is the pointwise structures of heat kernel with BV  coefficient. In this talk, we will first review their result. Then, by refining the heat  kernel estimates, we prove the regularity and uniqueness of the weak solution.  Moreover, if the initial perturbation in BV class is localized in space, we can describe  the wave propagation precisely. This talk is based on joint works with Shih-Hsien Yu  and Xiongtao Zhang.


Thermal convection in superposed free flow and porous media:  
analysis and numerics
王晓明（南方科技大学）
We report on a few recent results related to thermal convection in a fluid layer  overlying a saturated porous media based on the Navier-Stokes-Darcy-Boussinesq  (NSDB) model with appropriate interface boundary conditions. The existence of global  in time weak solution for the NSDB system together with a weak-strong uniqueness  result are presented first.The stability of the pure conduction state at small Rayleigh  number is introduced next. The loss of stability of the pure conduction state as the  Rayleigh number crosses a threshold value is studied via a hybrid approach that combines analysis with numerical computation. In particular, we discover that the  transition between shallow and deep convection associated with the variation of the  ratio of free-flow to porous media depth is accompanied by the change of the most  unstable mode from the lowest possible horizontal wave number to higher wave  numbers, which could occur with variation of the height ratio as well as the Darcy  number and the ratio of thermal diffusivity among others.Numerical methods that  decouples the heat, the free flow, the porous media flow while maintaining energy  

stability are presented as well.


Derivation of a field-road model
王学锋（香港中文大学（深圳））
 Of concern is the scenario of a road running through a large field, in which a  diffusive species has different diffusion rates on the road and in the field, respectively.  8 years ago, Beresticky and collaborators proposed a field-road model, which consists  of the KPP equation for the areal density function in the field, and a reaction-diffusion  equation for the linear density function on the road (the x-axis), with the two density  functions coupled by a Robin boundary on the x-axis. We wonder if this model can be  derived from a more basic model. Indeed, we do so by first assuming the width of the  road is positive, and then proposing a full model on the whole plane, with reasonable  transmission conditions on the edges of the road, we then send the width of the road  to 0. The resulting limiting model covers the field-road of Beresticky et al, and more.  This is a joint work of Haomin Huang, Siyu Liu and Yantao Wang.


A free boundary problem modeling tumor growth  
with Gibbs-Thomson relation
 吴俊徳（苏州大学）
In this talk, we discuss well-posedness and asymptotic behavior of a FBP tumor  model with Gibbs-Thomson relation. The tumor model consists of a diffusion equation  for the nutrient concentration and an elliptic equation for the pressure, and the  movement of the tumor boundary satisfies the kinematic condition.


Global well-posedness of coupled parabolic systems

徐润章（哈尔滨工程大学）
The initial boundary value problem of a class of reaction-diffusion systems  (coupled parabolic systems) with nonlinear coupled source terms is considered in  order to classify the initial data for the global existence, finite time blowup and  
longtime decay of the solution. The whole study is conducted by considering three  cases according to initial energy: low initial energy case, critical initial energy case  and high initial energy case. For the low initial energy case and critical initial energy  case the sufficient initial conditions of global existence, long time decay and finite time blowup are given to show a sharp-like condition. And for the high initial  energy case the possibility of both global existence and finite time blowup is proved first, and then some sufficient initial conditions of finite time blowup and global  

existence are obtained respectively.


Suppression of explosion by mixing
 徐霄乾（昆山杜克大学）
In the study of incompressible fluid, one fundamental phenomenon that arises  in a wide variety of application is dissipation enhancement by so-called mixing flow.  In this talk, I will give a brief introduction to the idea of mixing flow and the role it  plays in the field of advection-diffusion-reaction equation. More specifically, I will  explain why the presence of fluid can enhance the dissipation and prevent the  singularity formation for some types of evolution equations, even with degeneracy.


Analysis on Tollmien-Schlichting wave in the  
Prandtl-Hartmann Regime
 杨彤（香港城市大学）
In this talk, we will present the instability induced by the Tollmien-Schlichting wave  governed by the MHD system in the Prandtl-Hartmann regime. The interaction of the  inviscid mode and viscous mode that leads to the instability is analyzed by the  introduction of a new decomposition of the Orr-Sommerfeld operator on the velocity  and magnetic fields. The critical Gevrey index for the instability is justified by constructing the growing mode in the essential frequency and it is shown to be the same  as the incompressible Navier-Stokes equations in the Prandtl regime. This result  justifies rigorously the physical understanding that the transverse magnetic field to the  boundary in the Prandtl-Hartmann regime has no extra stabilizing effect on the  Tollmien-Schlichting wave. This is a joint work with Chengjie Liu and Zhu Zhang.


On the global existence or blowup of smooth large data solutions to the  
second order quasilinear wave equations
尹会成（南京大学）
In this talk, for the second order quasilinear wave equations with the short pulse initial data (a class of large initial data which are firstly introduced by D.Christodoulou),  we shall discuss the sufficient and necessary conditions for the global existence or  blowup of smooth solutions. These works are joint with Prof.Xin Zhouping and  Prof.Ding Bingbing.


Mosquito suppression models consisting of two sub-equations  
switching each other
庾建设（广州大学）
The release of Wolbachia-infected mosquitoes in 2016 and 2017 enabled nearelimination of the sole dengue vector Aedes albopictus on Shazai and Dadaosha islands  in Guangzhou. Mathematical analysis may offer guidance in designing effective mass  release strategies for the area-wide application of this Wolbachia incompatible and  sterile insect technique in the future. The two most crucial questions in designing  release strategies are how often and in what amount should Wolbachia-infected  mosquitoes be released in order to guarantee the success of population suppression. In  this talk, I will introduce our recent works on answering the two questions which have  
been published in the following three papers.  
• J. Differ. Equations, 2020, 269(7): 6193-6215.
• J. Differ. Equations, 2020, 269(12): 10395-10415.  
• SIAM J. Appl. Math., 2021, 81(2): 718-740.
By treating the released mosquitoes as a given function, we proposed mosquitosuppression models consisting of two sub-equations switching each other. An almost  complete characterization of interactive dynamics of wild and released mosquitoes are offered, including the global asymptotic stability of zero solution and the exactnumber of periodic solutions of these models. It is well known that to obtain existence  and also uniqueness conditions for periodic solutions is mathematically challenging for  many dynamical systems and there are few such results existed. I hope the methods and  techniques used in these three papers can be usefully applied to other model analysis as  well.


The blow up solutions to Boussinesq equations on $R^3$ with  
dispersive temperature
张立群（中国科学院数学与系统科学研究院）
The three-dimensional incompressible Boussinesq system is one of the important  equations in fluid dynamics. The system describes the motion of temperaturedependent incompressible flows. And the temperature naturally has diffusion. Very  recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,\alpha}$ finite time blow up solutions for Euler systems with finite energy. Inspired by their works, we  constructed $C^{1,\alpha}$ finite time blow-up solution for Boussinesq equations  where temperature has diffusion and finite energy. The main difficulty is that the  Laplace operator of temperature equation is not coercive under Sobolev weighted  norm which is introduced by Elgindi. We introduced a new time scaling formulation  and new weighted Sobolev norms, under which we obtain the coercivity estimate. The  new norm is well-coupled with the original norm, which enable us to finish the proof. This is a jointed work with Gao Chen and Zhang Xianliang.


Linear inviscid damping and enhanced dissipation for monotone  
shear flows
章志飞（北京大学）
In this talk, we will introduce a compactness method to establish the linear  inviscid damping and enhanced dissipation estimates for the 2-D linearized NavierStokes system around a class of monotone shear flows in a finite channel. These  estimates should be crucial for nonlinear stability.


Gradient estimates for harmonic maps into singular spaces

 朱熹平（中山大学）

In his seminal work in geometric analysis, S. T. Yau proved the famous gradient  estimate for harmonic functions on smooth manifolds. Later in 1980, S. Y. Cheng  extended the Yau's gradient estimate to harmonic maps between smooth manifolds. In  this talk we will discuss the question how to obtain Yau's gradient estimates for  harmonic maps from smooth manifolds to singular metric spaces.