# International Workshop on Multidimensional Conservation Laws and Related Problems

## Introduction

The field of multidimensional conservation laws and related problems is experiencing fast and significant advances over the past years. This workshop is aimed to bring together researchers in the field to present recent advances and explore possible collaborations.

June 16-18, 2017

## Venue

601 Pao Yue-Kong Library

## Scientific Committee

• Shuxing Chen, Fudan University
• Yachun Li, Shanghai Jiao Tong University
• Weike Wang, Shanghai Jiao Tong University
• Ya-Guang Wang, Shanghai Jiao Tong University

## Organizers

• Myoungjean Bae, Pohang University of Science and Technology
• Shijin Deng, Shanghai Jiao Tong University
• Beixiang Fang, Shanghai Jiao Tong University
• Qilong Gu, Shanghai Jiao Tong University
• Chunjing Xie, Shanghai Jiao Tong University
• Feng Xie, Shanghai Jiao Tong University
• Xiongfeng Yang, Shanghai Jiao Tong University

## Speakers

• Hantaek Bae, UNIST
• Stefano Bianchini, SISSA
• Chao Chen, Fujian Normal University
• Jun Chen, Southern University of Science and Technology
• Ben Duan, Dalian University of Technology
• Mikhail Feldman, University of Wisconsin Madison
• Moonjin Kang, University of Texas at Austin
• Eun Heui Kim, California State University, Long Beach
• Jun Li, Nanjing University
• Tong Li,University of Iowa
• Tao Luo, City University of Hong Kong
• Tianwen Luo, Tsinghua University
• Ronghua Pan, Georgia Institute of Technology
• Aifang Qu, Shanghai Normal University
• Denis Serre, ENS Lyon
• Masahiro Suzuki, Nagoya Institute of Technology
• Chunpeng Wang, Jilin University
• Yi Wang, Chinese Academy of Sciences
• Shangkun Weng, Wuhan University
• Zhouping Xin, The Chinese University of Hong Kong
• Gang Xu, Jiangsu University
• Xiaozhou Yang, Wuhan Institute of Physics and Mathematics
• Huicheng Yin, Nanjing Normal University

## Schedule

Time Speaker Affiliation Title
Masahiro Suzuki Nagoya Institute of Technology Stability analysis and quasi-neutral limit for the Euler-Poisson equations
Aifang Qu Department of Mathematics, Shanghai Normal University Global unbounded weak solution of the Chaplygin gas
Mikhail Feldman University of Wisconsin-Madison Uniqueness for shock reflection problem
Denis Serre Ecole Normale Superieure de Lyon Expansion of a compressible gas in vacuum
Jun Chen Southern University of Science and Technology Stability of transonic flows past a wedge
Hantaek Bae Ulsan National Institute of Science and Technology, Korea Transport equation with nonlocal velocity
Ben Duan School of Mathematics, Dalian University of Technology Stability of steady solutions for the Euler-Poisson system
Jun Li Mathematics Department, Nanjing University Quasilinear wave equations in exterior domains

## Abstract

### Stability analysis and quasi-neutral limit for the Euler-Poisson equations

Masahiro Suzuki, Nagoya Institute of Technology

The purpose of this talk is to mathematically investigate the formation of a plasma sheath near the surface of materials immersed in a bulk plasma, and to obtain qualitative information of such a plasma sheath layer. Specifically, we study the asymptotic behavior and quasi-neutral limit of solutions to the Euler-Poisson equations in a half space or three-dimensional annular domain.

### Global unbounded weak solution of the Chaplygin gas

Aifang Qu, Department of Mathematics, Shanghai Normal University

Consider the existence of $L_{loc}^1$ solution of the Cauchy problem for the Euler system with state of Chaplygin type. The study of $L^1$ solution differs from that of the $L^\infty$ solution for this kind of linearly degenerate system significantly. In this talk, we give some discussion on it.

### Uniqueness for shock reflection problem

We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections, and existence of regular reflection solutions for potential flow equation. Then we will talk about recent results on uniqueness of regular reflection solutions for potential flow equation in a natural class of self-similar solutions. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, and prove uniqueness by a version of method of continuity. A property of solutions important for the proof of uniqueness is convexity of the free boundary. This talk is based on jont work with G.-Q. Chen and W. Xiang.

### Expansion of a compressible gas in vacuum

Denis Serre, Ecole Normale Superieure de Lyon

Tai-Ping Liu [2] introduced the notion of physical solution’’ of the isentropic Euler system when the gas is surrounded by vacuum. This notion can be interpreted by saying that the front is driven by a force resulting from a H"older singularity of the sound speed. We address the question of when this acceleration appears or when the front just move at constant velocity (ballistic motion).

We know from [1,4] that smooth isentropic flows with a ballistic front exist globally in time, for suitable initial data. In even space dimension, these solutions may persist for all $t\in{\mathbb R}$~; we say that they are {\em eternal}. We derive a sufficient condition in terms of the initial data, under which the boundary singularity must appear. As a consequence, we show that, in contrast to the even-dimensional case, eternal flows with a ballistic front don’t exist in odd space dimension. Our argument is related to that of Milnor [3] in his proof of the hairy ball Theorem.

In one space dimension, we give a refined definition of physical solutions. We show that for a shock-free flow, their asymptotics as both ends $t\rightarrow\pm\infty$ are intimately related to each other.

See my paper [5]

### References

[1] M. Grassin. Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J., 47(1998), pp 1397-1432

[2] Tai-Ping Liu. Compressible flow with damping and vacuum. Japan J. Indust. Appl. Math., 13(1996), pp 25-32

[3] J. Milnor. Analytic proofs of the “hairy ball theorem’’ and the Brouwer fixed point theorem. The American Mathematical Monthly, 85 (1978), pp 521-524

[4] D. Serre. Solutions classiques globales des equations d’Euler pour un fluide parfait compressible. Annales de l’Institut Fourier 47 (1997), pp 139-153

[5] D. Serre. Expansion of a compressible gas in vacuum. Bulletin of the Institute of Mathematics, Academia Sinica, Taiwan, 10 (2015), pp 695-716.

### Stability of transonic flows past a wedge

Jun Chen, Southern University of Science and Technology

We will talk about the stability of transonic flows past a 2-D wedge governed by the full Euler equations. Given a piecewise constant transonic flow past a straight wedge, if the incoming flow and the wedge are perturbed, there exists a unique subsonic solution in the downstream together with a perturbed shock in between. Corner singularity and decay of the subsonic flow at far field are handled through elliptic estimates. The analysis discloses the relation between the shock polar and the regularity and asymptotic behavior of the downstream subsonic flow.

### Transport equation with nonlocal velocity

Hantaek Bae, Ulsan National Institute of Science and Technology, Korea

We consider 1D equations with nonlocal velocity field

\begin{eqnarray}\label{transport equation} \theta_t+u\theta_x-\delta u_{x} \theta+\Lambda^{\gamma}\theta=0 \end{eqnarray}

where $u=\mathcal{N}(\theta)$ is given by one of the form

1. $u=\mathcal{H}\theta$;

2. $u= (1-\partial_{xx} )^{-\alpha}\theta$.

In this talk, we address the existence of weak solutions of transport equation. When $0<\gamma<1$, we take initial data having finite energy. When $\gamma=1$, we take initial data having infinite energy involving Muckenhoupt weights.

### Stability of steady solutions for the Euler-Poisson system

Ben Duan, School of Mathematics, Dalian University of Technology

In this talk, we will discuss the Euler-Poisson flows in nozzles. Our motivation and four types of flow patterns will be introduced, including subsonic flows, supersonic flows, transonic shocks and smooth transonic flows. We may focus on the case of subsonic flows, the unique existence of multi-dimensional irrotational flow and 2-dimensional Euler-Poisson flow will be reported.

### Quasilinear wave equations in exterior domains

Jun Li, Mathematics Department, Nanjing University

In the theory of compressible aerodynamics, a basic problem is to consider long time stability of motions of compressible gases in 3-D exterior domains. It can be reformulated as quasilinear wave equations in exterior domains with Neumann boundary conditions. In this talk, I will introduce the related backgrounds both in fluid dynamics and in quasilinear wave equations. In addition, when the equations fulfill null form, I will explain the ideas to prove the global well-posedness of this kind of initial-boundary value problem.