Special Session 114: New developments in Analysis of Mathematical Fluid Dynamics

Stability of Stationary Solutions to the Nonisentropic Euler--Poisson System in a Perturbed Half Space

Mingjie LI Minzu University of China Peoples Rep of China Co-Author(s):    Masahiro Suzuki

Abstract: The main concern of the talk is to mathematically investigate the formation of a plasma sheath near the surface of nonplanar walls. We study the existence and asymptotic stability of stationary solutions for the nonisentropic Euler-Poisson equations in a domain of which boundary is drawn by a graph, by employing a space weighted energy method. Moreover, the convergence rate of the solution toward the stationary solution is obtained, provided that the initial perturbation belongs to the weighted Sobolev space. Because the domain is the perturbed half space, we first show the time-global solvability of the nonisentropic Euler-Poisson equations, then construct stationary solutions by using the time-global solutions. This is a joint work with Professor Masahiro Suzuki.

Asymptotic stability for n-dimensional magnetohydrodynamic equations

Jitao Liu Beijing University of Technology Peoples Rep of China Co-Author(s):

Abstract: This talk is concerned with the stability theory of n-dimensional incompressible and compressible magnetohydrodynamic ({\it MHD for short}) equations with only kinematic viscosity or magnetic diffusion in the periodic domain {{\mathbb{T}}^n}. I will present some new results on the asymptotic stability and sharp decay estimates of this system when the magnetic field close to an equilibrium state satisfying the Diophantine condition. In the present works, by exploiting and effectively utilizing the structure of perturbation system, a new dissipative mechanism is found out and applied so that we can sharply improve the spaces of existing works, where the decay estimates and asymptotic stability of solutions are taking place. Some key ideas of our method will be discussed. This talk is based on joint works with Quansen Jiu and Yaowei Xie.

Vanishing viscosity limits for the free boundary problem of compressible flows

Yu Mei Northwestern Polytechnical University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we present some results of vanishing viscosity limits for the free boundary problem of compressible isentropic flows. For the free boundary compressible Navier-Stokes equations of Newtonian fluids with or without surface tension, we established the uniform regularities of solutions in Sobolev conormal and Lipschitz spaces, and justified the vanishing viscosity and surface tension limits by a strong convergence argument. On the other hand, for the free boundary compressible viscoelastic equations of neo-Hookean fluids with or without surface tension, we obtained the uniform Sobolev regularities of solutions and proved the vanishing viscosity limits in Sobolev spaces, which indicates the stabilizing effect of elasticity.

Stability and large-time behavior of nD tropical climate model with zero thermal dissipation

Dongjuan Niu Capital Normal University Peoples Rep of China Co-Author(s):    Wu, Huiru; Tang, Houzhi

Abstract: In this talk, we are concerned with stability problem and large- time behavior of 2D and 3D tropical climate model with zero thermal dissipation under small initial data. Specifically, the first one focuses on the global well-posedness and large-time behavior of solutions of the system with the dissipation and part damping in the whole space. The second one investigates the stability problem and the optimal decay rate of 3D tropical climate model with zero thermal dissipation under small initial data. The third one works on the stability near Couette flow to the system with the dissipation and damping in the strip domain.

Isothermal Limit of Entropy Solutions of the Euler Equations for Isentropic Gas Dynamics

Tianyi Wang Wuhan University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we want to present the isothermal limit of entropy solutions in $L^\infty$, containing the vacuum states, of the Euler equations for isentropic gas dynamics. First, We want to start with the explicit asymptotic analysis of the Riemann solutions containing the vacuum states. Then, we want to show the entropy solutions in $L^\infty$ of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent $\gamma \rightarrow 1$. This is achieved by combining careful entropy analysis and refined kinetic formulation with compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent $\gamma\to 1$. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to $\gamma>1$. This is the joint work with Gui-Qiang G. Chen, and Fei-Min Huang.

Time-periodical solution to compressible Euler Equation

Huimin Yu Shandong Normal University Peoples Rep of China Co-Author(s):    Peng Qu and Xiaomin Zhang

Abstract: In this talk, we will discuss some recent work on the time-periodical solutions of compressible Euler equations with damping or geometric effects. We will focus on the solutions triggered by time periodic boundary conditions. Firstly, the steady states are analyzed. Then, the existence, stability and uniqueness of the supersonic/subsonic temporal periodic solutions around the steady states are given. Finally, the time periodic transonic shock will be discussed.

Incompressible limit of viscous vortex sheets with large data

Qian Yuan Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Qian Yuan, Wenbin Zhao

Abstract: In this talk, we shall discuss the incompressible limit of vortex sheets for the Navier-Stokes equations with 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, as well as some components of the initial perturbations.