# Workshop on Recent Advances in PDEs (IX)

## Introduction

This workshop aims at exchange and collaboration between young researchers on fluid and kinetic equations. The topics include pointwise behavior of Vlasov-Poisson-Boltzmann system, mean field limit of synchronization models, and barotropic instability and linear inviscid damping for shear flow, etc.

## Date

September 20-21, 2019

## Venue

September 20, 2019: Large Conference Room(706), No. 6 Science Building, Minhang Campus, Shanghai Jiao Tong University
September 21, 2019: Room 306, No. 5 Science Building, Minhang Campus, Shanghai Jiao Tong University

• National Natural Science Foundation of China
• School of Mathematical Sciences, Shanghai Jiao Tong University
• Institute of Natural Sciences, Shanghai Jiao Tong University
• Key Laboratory of Scientific Engineering Computing, Ministry of Education

## Registration

No registration fee. Please Register Online

## Invited Speakers

• Zhen Lei, Fudan University
• Chaojiang Xu, Nanjing University of Aeronautics and Astronautics
• Mingying Zhong, Guangxi University
• Xiongtao Zhang, Huazhong University of Science and Technology
• Hao Zhu, Nankai University

## Schedule

Date: Sep 20. 2019
Venue: Large Conference Room(706), No. 6 Science Building

Time Speaker Title
15:30 - 16:30 Zhen Lei Liouville properties of the Navier-Stokes
16:30 - 17:30 Chaojiang Xu 千禧年七大数学难题之一：Navier-Stokes 方程的历史渊源

Date: Sep 21, 2019
Venue: Room 306, No. 5 Science Building

Time Speaker Title
08:50 - 09:00 Opening remarks
09:00 - 09:50 Mingying Zhong Green’s function and pointwise behavior of the Vlasov-Poisson-Boltzmann System (I)
10:00 - 10:50 Mingying Zhong Green’s function and pointwise behavior of the Vlasov-Poisson-Boltzmann System (II)
10:50 - 11:10 Group Photo & Tea Break
11:10 - 12:00 Xiongtao Zhang Mean-Field Limit for the Cucker-Smale Model (I)
12:00 - 14:30 Lunch
14:30 - 15:20 Xiongtao Zhang Mean-Field Limit for the Cucker-Smale Model (II)
15:20 - 15:40 Tea Break
15:40 - 16:30 Hao Zhu Barotropic instability and linear inviscid damping for shear ﬂows (I)
16:40 - 17:30 Hao Zhu Barotropic instability and linear inviscid damping for shear ﬂows (II)

## Program

### Liouville properties of the Navier-Stokes

Zhen Lei, Fudan University

Abstract: We establish a Liouville theorem for bounded ancient mild solutions to the axi-symmetric incompressible Navier-Stokes equations on $(-\infty, 0] \times (\mathbb{R}^2 \times \mathbb{T}^1)$. Connecting the compactness of $\mathbb{T}^1$ to the oscillation of the stream function is a crucial observation.

Chaojiang Xu, Nanjing University of Aeronautics and Astronautics

Abstract: 2000年五月Clay数学研究所在法兰西学院公布了七个“千年大奖问题”，每个问题的解决可以获得Clay研究所的一百万美元的奖金。本综合报告将面向数学专业的大学生和研究生，介绍这七大难题之一的Navier-Stikes方程的历史渊源。Navier-Stikes方程是流体力学的最基本的数学物理方程，我们将从阿基米德的浮力学理论开始追溯两千多年来这个方程的历史渊源，最后简单的介绍一些最新的研究进展。

### Green’s function and pointwise behavior of the Vlasov-Poisson-Boltzmann System

Mingying Zhong, Guangxi University

Abstract: In this talk, we study the pointwise space-time behaviors of the Green’s function and the global solution to the Vlasov-Poisson-Boltzmann (VPB) system in whole space. It is shown that due to the influence of electrostatic potential governed by the Poisson equation, the Green’s function admits only the macroscopic nonlinear diffusive waves, the singular kinetic waves, and the remainder term decaying exponentially in time but algebraically in space. These behaviors have essential difference from the Boltzmann equation, namely, the Huygen’s type sound wave propagation and the space-time exponential decay of remainder term for Boltzmann equation can not be observed for VPB system. Furthermore, we establish the pointwise space-time nonlinear diffusive behaviors of the global solution to the nonlinear VPB system in terms of the Green’s function. Some new strategies are introduced to deal with the difficulties caused by the electric fields.

### Mean-Field Limit for the Cucker-Smale Model

Xiongtao Zhang,Huazhong University of Science and Technology

Abstract:

In this talk, we introduce the rigorous mean field limit from Cucker-Smale mdoel (CS) to kinetic CS model (KCS) with regular and singular interaction respectively. For regular interaction, the global-intime stability of flocking states will be addressed. As a direct application of the uniform stability, we obtain the uniform-in-time mean-field limit in the Wasserstein metric. While for C-S model with singular interaction, uniform stability has not been obtained so far, and we will present a probabilistic approach for derivation of the KCS equation from the particle CS model with singular communication. More precisely, we introduce a suitable cutoff in the communication weight, and derive the estimates of the difference between particle trajectory and the kinetic trajectory for cutoff equations. Then we show the cutoff will tends to zero when particle number N tends to infinity and thus verify the mean field limit.

### Barotropic instability and linear inviscid damping for shear ﬂows

Hao Zhu, Nankai University

Abstract:

In this talk, we consider barotropic instability and linear damping around shear flows for incompressible fluids with Coriolis effects. For a class of shear flows, we develop a new method to find the sharp stability transition. We will apply this method to the Sinus flow. The addition of the Coriolis force is found to bring some fundamental changes. Moreover, we discuss the bifurcation of nontrivial traveling wave solutions and the proof of the linear damping by establishing the limiting absorption principle.