## 2020年流体力学数学理论研讨会

# 会议日程

11月6日（周五），地点：北京师范大学后主楼1124

14：00-15：00

王勇Hilbert expansion of the Boltzmann equation with specular boundary condition in half-space

15：00-16: 00

王益 Vanishing dissipation limit of planar wave pattern to the multi-dimensional compressible Navier-Stokes

16：00-17: 00

黄飞敏 Stability of rarefaction wave for stochastic Burgers equation

11月7日（周六），地点：腾讯会议ID：874 110 003会议密码：123456

9: 00-10:00 (北京时间)20: 00-21:00 (美国东部时间)

陈明 Center manifolds without a phase space in equations from elasticity and hydrodynamics

10:15-11:15（北京时间）21:15-22:15（美国东部时间）

王德华 Euler equations, transonic flows and isometric embeddings

# 报告及摘要

Hilbert expansion of the Boltzmann equation with specular boundary condition in half-space

王勇

（中科院数学与系统科学研究院）

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. Based on a systematic derivation and study of the viscous layer equations and the $L^2$ to $L^\infty$ framework, we establish the validity of the Hilbert expansion for the Boltzmann equation with specular reflection boundary conditions, which leads to derivations of compressible Euler equations and acoustic equations. The talk is based on a joint work with Yan Guo and Feimin Huang.

Vanishing dissipation limit of planar wave pattern to the multi-dimensional compressible Navier-Stokes equations

王益

（中科院数学与系统科学研究院）

The talk is concerned with our recent results on the vanishing dissipation limit of planar rarefaction wave to both 2D/3D 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.

Stability of rarefaction wave for stochastic Burgers equation

黄飞敏

（中科院数学与系统科学研究院）

The large time behavior of strong solutions to the stochastic Burgers equation is considered in this paper. It is first shown that the unique global strong solution to the one dimensional stochastic Burgers equation time-asymptotically tend to a rarefaction wave, that is, the rarefaction wave is non-linearly stable under white noise perturbation for stochastic Burgers equation. A time-convergence rate is also obtained. Moreover, an important inequality (denoted by Area Inequality) is derived. This inequality plays essential role in the estimates, and may have applications in the related problems, in particular for the time-decay rate of solutions of both the stochastic and deterministic PDEs. As an application, the stability of planar rarefaction wave is shown stable for a two dimensional viscous conservation law with stochastic force.

Center manifolds without a phase space in equations from elasticity and hydrodynamics

陈明

（匹兹堡大学）

Elliptic PDEs on tubular domains arise in many applied settings. Inspired by a recent work of Faye-Scheel, we present a new center manifold reduction method which is “without a phase space” in the sense that we never explicitly reformulate our PDE as an evolution equation. Under suitable hypotheses, the resulting center manifold captures all sufficiently small bounded solutions. Compared with classical methods, we find our reduction theorem to be more directly related to the original physical problems and particularly convenient for calculations. Moreover our analysis is casted in Holder spaces, which is often desirable for elliptic problems. We then apply this machinery to two examples: one from nonlinear elasticity and another from hydrodynamics. This is a joint work with Samuel Walsh and Miles Wheeler.

Euler equations, transonic flows and isometric embeddings

王德华

（匹兹堡大学）

In this talk, we will discuss the Euler equations of gas dynamics and applications in transonic flows and isometric embeddings in geometry. First the basic theory of Euler equations will be reviewed. Then we will present the results on the transonic flows past an obstacle including the global existence of weak solutions. Finally we will present the fluid dynamic formulation of the isometric embedding problem in geometry and discuss the global solutions based on various approaches.