召集人：黄飞敏（中国科学院数学与系统科学研究院）

时   间：2024.07.14—2024.07.20 

会 议 日 程

 2024年7月15日星期一

2024年7月16日星期二

2024年7月17日星期三

2024年7月18日星期四

报 告 信 息

曹文涛, 首都师范大学, 中国

Title: Compressible Euler equations and isometric embedding

Abstract: In this talk I will discuss the relations and recent progress of isometric embedding from differential geometry and compress Euler equations.

方北香, 上海交通大学, 中国

Title: On Transonic Shocks in a Finite Nozzle with Prescribed Pressure at the Exit

Abstract: In this talk, we are concerned with the inviscid steady Euler flow pattern with a single shock front in a finite nozzle, which enters the nozzle with a supersonic state and leave with a subsonic one. In order to determine the location of the shock front, it is suggested to impose a pressure condition at the exit by R. Courant and K.O. Friedrichs in their monograph “Supersonic Flow and Shock Waves”. This talk will introduce recent advances on this problem based on a joint work with Prof. Zhouping Xin, and the talk will be divided into two parts. In the first part, we shall report the proof for the existence of transonic shocks in an almost flat finite nozzle with prescribed pressure at the exit. The nozzle is almost flat in the sense that its boundary is a general perturbation of a flat one. The key difficulty is the information of the approximate location of the shock front. We are going to overcome this difficulty by designing a free boundary problem for the linearized Euler system which will yield useful information on the initial approximation location of the shock front. With these information, we can then further apply a nonlinear iteration scheme to determine the whole flow patten in the nozzle, including the location of the shock front. In the second part, we shall introduce our further related advances on this topic as well as open problems to be investigated.

耿世锋, 湘潭大学, 中国

Title: Asymptotic behavior of solutions for the compressible Euler equations with time-dependent damping

Abstract: In this talk, we study the compressible Euler equations with time-dependent damping $-\frac{1}{(1+t)^{\lambda}}\rho u$. We first artfully construct the asymptotic profile, a special linear wave equation with time-dependent damping in the critical case $\lambda=1$, Then we rigorously prove that the solutions time-asymptotically converge to the linear wave equation with critical time-depending damping. We propose a time asymptotic expansion around the self-similar solution of the generalized porous media equation (GPME) and rigorously justify this expansion as $\lambda \in (\frac17,1)$. In other word, instead of the self-similar solution of GPME, the expansion is the best asymptotic profile of the solution to the compressible Euler equations with time-dependent damping.

琚强昌, 北京应用物理与计算数学研究所, 中国

Title: Low Mach number limit of global strong solutions to the full compressible Navier-Stokes equations around the plane Couette flow 

Abstract: We study the global existence and low Mach number limit of strong solutions to the 2-D full compressible Navier-Stokes equations around the plane Couette flow in a horizontally periodic layer with non-slip and isothermal boundary conditions. It is shown that the plane Couette flow is asymptotically stable for sufffciently small initial perturbations, provided that the Reynolds number, Mach number and temperature difference between the top and the lower walls are small. For the case that both the top and the lower walls maintain the same temperature, we further prove that such global strong solutions converge to a steady solution of the incompressible Navier-Stokes equations as the Mach number goes to zero.

浦赟, 中国科学院数学与系统科学研究院, 中国

Title: Global solutions for steady supersonic Euler flows past a cylindrically symmetric cone 

Abstract: In this talk, we try to establish the global existence of entropy solutions for three-dimensional cylindrically symmetric supersonic Euler flows past a cone with a relatively large opening angle. In this problem, the uniform incoming flow and the generating curve of the cone are prescribed, and we are required to determine the surrounding flow field. A modified Glimm scheme is developed to solve this problem and an improved energy estimate is applied.

苏佩, Université Paris-Saclay, 法国

Title: Regularity issue for the system describing elastic structure interacting with the Navier-Stokes equations 

Abstract: We are interested in the interaction of a viscous incompressible fluid with an elastic structure, where the structure is located on a part of the fluid boundary. It reacts to the surface forces induced by the fluid and deforms the reference domain to the moving domain. The fluid equations are coupled with the structure via the kinematic condition and the action-reaction principle on the interface.

We study the 2D visco-elastic shell interacts with 3D Navier-Stokes equations. Especially in a general reference geometry (the shell deforms along the normal direction of the flexible boundary), we prove a counterpart of the classical Ladyzhenskaya-Prodi-Serrin condition yielding conditional regularity and uniqueness of a solution. This requires additionally the deformation of the shell is Lipschitz continuous.

This is based on joint work with D. Breit (Clausthal), P. Mensah (Clausthal) and S. Schwarzacher (Uppsala).

唐桂荣, 中国科学院数学与系统科学研究院, 中国

Title: Global Well-posedness of Pressureless Euler-Poisson System in one dimension

Abstract: Global well-posedness of pressureless Euler–Poisson System is of great significance in describing the evolution of Universe. In this talk, we study the global existence and uniqueness results for the pressureless Euler-Possion system with the $\rho_0$ being a Radon measure, \rho_0(\mathbb{R})<+\infty$ and $u_0 \in L^{\infty}(\rho_0)$. Global weak solutions can be constructed explicitly using the initial data by analyzing minimizers of one functional. The uniqueness result can be derived by Oleinik entropy condition and the weak convergence of energy. At last, we derive the generalized Rankine-Hugoniot condition of the $\delta$-shock and give an example to explain the necessity of weak convergence of energy.

王丹丽, 中国科学院数学与系统科学研究院, 中国

Title: Well posedness and limit theorems for a class of stochastic dyadic models 

Abstract: We consider stochastic inviscid dyadic models with energy-preserving noise. It is shown that the models admit weak solutions which are unique in law. Under a certain scaling limit of the noise, the stochastic models converge weakly to a deterministic viscous dyadic model, for which we provide explicit convergence rates in terms of the parameters of noise. A central limit theorem underlying such scaling limit is also established. In case that the stochastic dyadic model is viscous, we show the phenomenon of dissipation enhancement for suitably chosen noise. This is based on joint work with Prof.Dejun Luo (SIAM J. Math. Anal., 2023).

王术, 北京工业大学, 中国

Title: Quasi-neutral limit problem of Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics

Abstract: In this talk we discuss quasi-neutral limit problem of Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics and review the known main progress on rigorously establishing quasi-neutrality theory from the view of mathematics. Quasi-neutrality is one basic assumption in physics such as semiconductors and plasma, which is firstly proposed by W.Van Roosbroech in Bell System Tech. J., 1950. Firstly, we introduce some multi-scaling structure stability theory on quasi-neutral limit and boundary layer, initial layer and mixed layer problem for drift-diffusion models in semiconductors. Then, we give some recent progresses on well-posedness theory and quasi-neutral limit problem on Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics.

王腾, 北京工业大学, 中国

Title: Asymptotic behavior of solution for the compressible Navier-Stokes equations

Abstract: In this talk, we first show the nonlinear stability result of the planar viscous shock up to a time-dependent shift for the three-dimensional compressible Navier-Stokes equations under the generic perturbations, in particular, without zero mass conditions. Next, we are concerned with the vanishing dissipation limiting problem of one-dimensional non-isentropic Navier-Stokes equations with shock data.

王天怡, 武汉理工大学, 中国

Title: On the asymptotic behavior of Steady Euler equations

Abstract: In this talk, we will start with the results on low Mach number limit which reveal how the external forces affect the convergence rates of the flows at far fields. And， the far field convergence rates of both incompressible and compressible flows at far fields as the boundary of the nozzle goes to flat even when the forces do not admit convergence rates at far fields. Then, for the unbounded domain, the maximum principle is applied to estimate the potential function, by choosing the proper compared functions. The convergence rates of velocity at the far field are obtained by the weighted Schauder estimates. Furthermore, we construct the examples to show the optimality of our convergence rates, and show the expansion of the incompressible airfoil flow at infinity, which indicates the higher convergence rates. We also will mention the recent work on the far-field convergence rate and stability of steady Euler flows with large vorticity and characteristic discontinuities in arbitrary infinitely long nozzles.This talk is based on the joint works with Mingjie Li, Lei Ma, Wei Xiang, Chunjin Xie, Jiaojiao Zhang.

王亚光, 上海交通大学, 中国

Title: Study of boundary layers in geophysical flow 

Abstract: In this talk, we shall review the mathematical analysis of boundary layers in  geophysical flow in the large Reynolds number and beta-plane parameter limit, including the behavior of boundary layers, the well-posedness of boundary layer equations, and the justification of the boundary layer expansions.

吴晓春, 中南大学, 中国

Title: Convergence rates for compressible Euler equations with damping

Abstract: In this talk, we will show the large time behavior of the compressible Euler equations for polystroic gas with damping. More specifically, for the time-dependent damping, we will show that the compressible Euler equations with damping time-asymptotically converges to the generalized porous media equation. For damping with the constant coefficient, we justify a time-asymptotic expansion around the diffusion wave is a better asymptotic profile.

杨钊, 中国科学院数学与系统科学研究院, 中国

Title: An intro to the spectral theory of nonlinear waves 

Abstract: Played a pivotal role in applied mathematics, waves traveling at constant speed are present in all kinds of PDE models. Their stability is a vital part of mathematical modeling because it distinguishes solutions expected to be observable in nature from others that are merely mathematical artifacts. In the first half of this short course, I will discuss various aspects related mainly to the spectral and linearized stability of “front” and “back” solutions, concerning their essential and point spectrum, Evans function, and Green kernels. In the second half of the course, I will turn to the spectrum of periodic waves and discuss briefly about Floquet theory and its applications.  At the end, if time permits, I will present some recent results related to the course materials.

袁谦, 中国科学院数学与系统科学研究院, 中国

Title: The incompressible limit of viscous vortex sheets with large data

Abstract: In this talk, we shall discuss the incompressible limit of the classical solutions around vortex sheets to the Navier-Stokes equations for ill-prepared initial data. The results are valid for all time and in addition, there are not any smallness assumptions on the background vortex sheets. This work is joint with Wenbin Zhao.

郑好, 中国科学院数学与系统科学研究院, 中国

Title: Introduction on quantum hydrodynamic models and some recent results

Abstract: Quantum hydrodynamic (QHD) models study hydrodynamic phenomena where quantum effects must be taken into account, such as superfluidity, Bose-Einstein condensation, quantum plasmas, or semiconductor devices. In this talk, I will give a briefly introduction on the physical background of several QHD models, as well as some related mathematical problems and their literatures. Last, I will introduce some recent results about the 1-dimensional collisional Quantum Hydrodynamics (QHD) system and its relaxation-time limit, which is a work in collaborating with Paolo Antonelli and Pierangelo Marcati.