Abstract : The aim of this mini-symposium is to bring together experts in the area of nonlinear PDEs from physics, such as Euler-type equations and Boltzmann equation, to present their recent research results in theoretical analysis and applications in physics. In this mini-symposium, people are expected to exchange new ideas, to discuss challenging issues, to explore new directions and topics, and to foster new collaborations and connections.
Organizer(s) : Renjun Duan, Xianpeng Hu, Tong Yang
[04052] Wave propagation and stabilization in the Boussinesq–Burgers system
Format : Talk at Waseda University
Author(s) :
Zhi-An Wang (The Hong Kong Polytechnic University )
Abstract : This talk will discuss the existence and stability of traveling wave solutions of the Boussinesq–
Burgers system describing the propagation of bores. Assuming the fluid is weakly dispersive, we
establish the existence of three different wave profiles by the geometric singular perturbation theory
alongside phase plane analysis. We further employ the method of weighted energy estimates to
prove the nonlinear asymptotic stability of the traveling wave solutions against small perturbations.
The technique of taking antiderivative is utilized to integrate perturbation functions because of the
conservative structure of the Boussinesq–Burgers system. Using a change of variable to deal with the
dispersion term, we perform numerical simulations for the Boussinesq–Burgers system to showcase
the generation and propagation of various wave profiles in both weak and strong dispersions. The
numerical simulations not only confirm our analytical results, but also illustrate that the Boussinesq–
Burgers system can generate numerous propagating wave profiles depending on the profiles of initial
data and the intensity of fluid dispersion, where in particular the propagation of bores can be generated
from the system in the case of strong dispersion.
[03930] Hypersonic similarity for steady potential flows over a two dimensional wedge
Format : Talk at Waseda University
Author(s) :
jie kuang (Academy of Mathematics and Systems Science)
wei xiang (city university of hong kong)
Yongqian Zhang (Fudan University)
Abstract : We will talk about our recent results on the hypersonic similarity for the potential flow over a two-dimensional wedge. The convergence is obtained in BV\cap L^1 spaces. Progress on related problems will be presented too.
[05017] Stability theory for the linear symmetric hyperbolic system with general relaxation
Format : Talk at Waseda University
Author(s) :
Yoshihiro Ueda (Kobe University)
Abstract : In this talk, we study the dissipative structure for the linear symmetric hyperbolic system with general relaxation. If the relaxation matrix of the system has symmetric properties, Shizuta and Kawashima(1985) introduced the suitable stability condition, and Umeda, Kawashima and Shizuta(1984) analyzed the dissipative structure. On the other hand, Ueda, Duan and Kawashima(2012,2018) focused on the system with non-symmetric relaxation and got partial results. Furthermore, they argued the new dissipative structure called the regularity-loss type. In this situation, this talk aims to extend the stability theory introduced by Shizuta and Kawashima(1985) and Umeda, Kawashima and Shizuta(1984) to our general system. Furthermore, we will consider the optimality of the dissipative structure. If we have time, I would like to discuss some physical models for its application and new dissipative structures.
[00454] Stability of Riemann shock wave via the method of a-contraction of shifts
Format : Talk at Waseda University
Author(s) :
Moon-Jin Kang (KAIST)
Abstract : I will present the so-called "a-contraction with shifts” method.
This method is quite useful in studying the stability of Navier-Stokes and Euler flows perturbed from Riemann solution containing a shock wave.
First, this method is energy based, and so allows us to seamlessly handle the composite wave of a viscous shock and rarefaction for its long-time behavior.
On the other hand, since the method can handle large perturbations of a viscous shock, and so provides the uniform stability of the shock w.r.t. the strength of viscosity, we can prove that the Riemann solution composed of a shock is stable and unique in the class of inviscid limits of solutions to the associated Navier-Stokes system.
[03959] Polynomial tail solutions for Boltzmann equation in the whole space
Format : Talk at Waseda University
Author(s) :
Renjun Duan (The Chinese University of Hong Kong)
Abstract : We are concerned with the Cauchy problem on the Boltzmann equation in the whole space. The goal is to construct global-in-time bounded mild solutions near Maxwellians with the perturbation admitting a polynomial tail in large velocities. The main difficulty to be overcome in case of the whole space is the polynomial time decay of solutions which is much slower than the exponential rate in contrast with the torus case.
[02793] Analytic regularization effect for the spatially inhomogeneous Boltzmann equation
Format : Talk at Waseda University
Author(s) :
Wei-Xi LI (Wuhan University)
Abstract : We verify in this work the spatially inhomogeneous Boltzmann equation with strong angular singularity will admit the analytic smoothing effect, just like its diffusive models such as the Landau and Fokker-Planck equations. To overcome the degeneracy in the spatial variable, a family of well-chosen vector fields with time-dependent coefficients will play a crucial role in the proof
[04010] Dispersive limit of kinetic models for collisional plasma
Format : Talk at Waseda University
Author(s) :
Zhu Zhang (The Hong Kong Polytechnic University)
Tong Yang (The Hong Kong Polytechnic University)
Abstract : The motion of charged particles can be described by the Vlasov-Poisson-Boltzmann (VPB) system. Compared to the classical Boltzmann equation for dilute gases, solutions to VPB are expected to have dispersive behavior because of the dispersion mechanism on the dynamics of plasma in different scales of physical interest. By a formal spectrum analysis, we can observe asymptotic relations between VPB and some limiting dispersive equations in a suitable regime. Then we justify that the propagation of non-linear ions-acoustic waves are governed by the KdV equation. This is a joint work with T. Yang.
[04067] Wave propogation and stabilization in the Boussinesq-Burgers system
Format : Talk at Waseda University
Author(s) :
Xianpeng Hu (City University of Hong Kong)
Anita Yang (The Chinese University of Hong Kong)
Abstract : In this talk, we will present the existence and stability of traveling wave solutions of the Boussinesq–Burgers system describing the propagation of bores. Assuming the fluid is weakly dispersive, we establish the existence of three different wave profiles by the geometric singular perturbation theory alongside phase plane analysis. We further employ the method of weighted energy estimates to prove the nonlinear asymptotic stability of the traveling wave solutions against small perturbations. The technique of taking antiderivative is utilized to integrate perturbation functions because of the conservative structure of the Boussinesq–Burgers system. Using a change of variable to deal with the dispersion term, we perform numerical simulations for the Boussinesq–Burgers system to showcase the generation and propagation of various wave profiles in both weak and strong dispersions. The numerical simulations not only confirm our analytical results, but also illustrate that the Boussinesq–Burgers system can generate numerous propagating wave profiles depending on the profiles of initial data and the intensity of fluid dispersion, where in particular the propagation of bores can be generated from the system in the case of strong dispersion. The talk is based on a recent joint work with Prof. Zhian Wang and Prof. Kun Zhao.
[04667] Vacuum free boundary problems in ideal compressible MHD
Format : Talk at Waseda University
Author(s) :
Tao Wang (Wuhan University)
Abstract : We present the joint works with Professor Yuri Trakhinin on the local well-posedness of vacuum free boundary problems in ideal compressible magnetohydrodynamics (MHD) with or without surface tension.
[01253] Well-posedness of some free boundary problems in compressible fluids
Format : Talk at Waseda University
Author(s) :
Wenbin Zhao (Peking University)
Abstract : In this talk, we will discuss some free boundary problems in compressible fluids. We derive the evolution equation of the free surface and identify the stability condition of the problem. This method gives a unified approach to treat both incompressible and compressible fluids.
[04043] Long time instability of compressible symmetric shear flows
Format : Talk at Waseda University
Author(s) :
Xianpeng Hu (City University of Hong Kong)
Andrew Yang (City University of Hong Kong)
Abstract : It is well-known that at high Reynolds numbers, the linearized Navier-Stokes equations around the inviscid stable shear profile admit growing mode solutions due to the destabilizing effect of small viscosities. This phenomenon, which is related to Tollmien-Schlichting instability, has been rigoriously justified by Grenier-Guo-Nguyen [Adv. Math. 292 (2016); Duke J. Math. 165 (2016)] on incompressible Navier-Stokes equations. In this work, we aim to construct the Tollmien-Schlichting waves for the compressible Navier-Stokes equations over symmetric shear flows in a channel. We will also discuss the effect of temperature fields on the stability of these shear flows.
