### 2020南航偏微分方程青年学者会议

College of Science, Nanjing University of Aeronautics and Astronautics

2020南航偏微分方程青年学者会议

1. 会议内容及日程安排

11031日（星期六）上午报到，注册，开幕式，学术报告；下午学术报告；

2111日（星期天）上午学术报告；下午学术报告。

1. 会议线上会议平台：腾讯会议

11031日（星期六）：

ID: 925 845 181, 密码：666888

2111日（星期日）：

ID: 363 797 853, 密码：666888.

 2020年10月31日（星期六） 9:45-09:50 会议报到、注册（进入腾讯线上会议） 09:50-10:00 开幕式（mg游戏平台王春武院长会议致辞） 会议报告 时间 报告人 主题 主持人 10:00-11:00 王益（中科院数学与系统科学研究院） Vanishing dissipation limit of planar rarefaction wave and planar contact discontinuity to the multi-dimensional compressible Navier-Stokes equations 徐超江 11:00-12:00 刘双乾（华中师范大学） The uniform shear flow for the Boltzmann equation 王春朋 12:00-14:00 午歇 14:00-15:00 徐润章（哈尔滨工程大学） Global well-posedness of coupled parabolic systems 章志飞 15:00-16:00 赵立丰（中国科学技术大学） Asymptotic behaviours for energy-subcrtical damped Klein-Gordon equations 李军 2020年11月1日（星期日） 09:00-10:00 徐桂香（北京师范大学） Recent developments for the (generalized) derivative Schrodinger equations 袁海荣 10:00-11:00 肖清华（中科院武汉物理与数学研究所） Global Hilbert expansion for the relativistic Vlasov-Maxwell-Boltzmann system 李维喜 11:00-14:00 午歇 14:00-15:00 訾瑞昭（华中师范大学） Suppression of blow-up in Patlak-Keller-Segel-Navier-Stokes system via the Couette flow 王泽军 15:00-16:00 李玉祥（东南大学） Chemotaxis fluid system with nonlinear diffusion 徐江

Chemotaxis fluid system with nonlinear diffusion

Chemotaxis fluid systems decribe the directional movement of bateria in the fluiud. In this talk, we first review the main results of chemotaxis fluid system with nonlinear diffusion, then I will talk about a recent result of Tao Weirun and me. We consider a chemotaxis-Stokes system with slow p-Laplacian diffusion under homogeneous boundary conditions of Neumann type. It is proved that global bounded weak solutions exist whenever p>23/11.

The uniform shear flow for the Boltzmann equation

The uniform shear flow for the rarefied gas is governed by the time-dependent spatially homogeneous Boltzmann equation with a linear shear force. The main feature of such flow is that the temperature may increase in time due to the shearing motion that induces viscous heat and the system becomes far from equilibrium. For Maxwell molecules, we establish the unique existence, regularity, shear-rate-dependent structure and non-negativity of self-similar profiles for any small shear rate. The non-negativity is justified through the large time asymptotic stability even in spatially inhomogeneous perturbation framework, and the exponential rates of convergence are also obtained with the size proportional to the second order shear rate. The analysis supports the numerical result that the self-similar profile admits an algebraic high-velocity tail that is the key difficulty to overcome in the proof.

Vanishing dissipation limit of planar rarefaction wave and planar contact discontinuity to the multi-dimensional compressible Navier-Stokes equations

The talk is concerned with our recent results on the vanishing viscosities limit of planar rarefaction wave to both 2D compressible isentropic Navier-Stokes equations and 3D full compressible Navier-Stokes equations and the vanishing dissipation limit of planar contact discontinuity of 3D full compressible Navier-Stokes equations, which, in particular, impies the positive answer to the uniqueness of planar contact discontinuity for 3D compressible Euler equations in the class of zero dissipation limit of compressible Navier-Stokes equations.

Global Hilbert expansion for the relativistic Vlasov-Maxwell-Boltzmann system

In this talk, we discuss the validity of Hilbert expansion for the relativistic Vlasov-Maxwell-Boltzmann system. The global-in-time validity of its Hilbert expansion is established and the limiting relativistic Euler-Maxwell system is                            derived as the mean free path goes to zero. Our method is based on an extended $L^2-L^{\infty}$ framework and the Glassey-Strauss Representation of the electromagnetic field.

Recent developments for the (generalized) derivative Schrodinger equations

We will survey some recent developments about the stability of the solitary waves for the (generalized) derivative Schrodinger equation. We firstly show the existence/nonexistence of traveling waves for NLS with derivative (DNLS) by the structure analysis, ODE argument and the variational argument. Secondly, we consider the orbital stability of the solitary waves for the derivative Schrodinger equation in the energy space. Last, we will show the orbital instability of the solitary waves for the generalized derivative Schrodinger equation in the degenerate case. They are joint works with Professor Changxing Miao, Yifei Wu and Xingdong Tang. We will also point out some open problem about gDNLS.

Global well-posedness of coupled parabolic systems

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.

Asymptotic behaviours for energy-subcrtical damped Klein-Gordon equations

In this talk, we will investigate the long time behaviour of damped Klein-Gordon equation in the energy subcritical case. In fact, we will establish the soliton  resolution for this equation which means that the global bounded solution evolves into the super-position of moving solitons. Moreover, we will give some further description and classification results about the 2-solitary waves. The talk is based on joint works with Ze Li, Raphael Cote, Yvan Martel and Xu Yuan.

Suppression of blow-up in Patlak-Keller-Segel-Navier-Stokes system via the Couette flow

In this talk, we consider the two dimensional Patlak-Keller-Segel-Navier-Stokes system near the Couette flow $(Ay,0)$ in $\mathbb{T}\times \mathbb{R}$. It is shown that if $A$ is large enough, the solution to the system stays globally regular. Both the parabolical-parabolic case and the the parabolic-elliptic case are investigated. In particular, for the parabolic-parabolic case, an extra smallness assumption on the initial chemical gradient $\|(\nb c_\mathrm{in})_\neq\|_{L^2}$ is needed to control the mixing destabilizing effect. This is a joint work with Zeng Lan and Zhang Zhifei.