Abstract : Boundary effect and stability analysis in fluid mechanics involves many mathematical problems, including boundary layer, free boundary, stability and instability of the hydrodynamic equations under the high Reynolds number, etc. The research on these problems is not only mathematically important and challenging, but also provides specific views for explaining certain physical phenomena and mechanical laws. In order to enhance the exchange on the latest research results and facilitate the cooperations on hydrodynamic stability theory, we would like to organize a mini-symposium named “Recent progress on mathematical theory of boundary layer” .
Organizer(s) : Zhifei Zhang, Guilong Gui, Yong Lu, Chao Wang
[03046] Eckhaus instability of the compressible Taylor vortex
Format : Talk at Waseda University
Author(s) :
Yoshiyuki Kagei (Tokyo Institute of Technology)
Abstract : This talk is concerned with the bifurcation and stability of the compressible Taylor vortex. It is shown that Taylor vortices bifurcate near the criticality for the incompressible problem when the Mach number is sufficiently small. The localized stability of the compressible Taylor vortices is considered and it is shown that the Eckhaus instability of compressible Taylor vortices occurs as in the case of the incompressible ones.
[03064] Stability of shear flows in inviscid and viscous fluids
Format : Talk at Waseda University
Author(s) :
Weiren Zhao (New York University Abu Dhabi)
Abstract : In this talk, I will present some recent progress in the asymptotic stability of shear flows in both inviscid and viscous fluids. The inviscid damping and enhanced dissipation phenomenon will be discussed in both linear and nonlinear models.
[03247] Tollmien-Schlichting waves in the subsonic regime
Format : Talk at Waseda University
Author(s) :
Di Wu (South China University of Technology)
Nader Masmoudi (New York University Abu Dhabi)
Yuxi Wang (Sichuan University)
Zhifei Zhang (Peking University)
Abstract : The Tollmien-Schlichting (T-S) waves play a key role during the early stage of the boundary layer transition. In a breakthrough work, Grenier, Guo and Nguyen gave a first rigorous construction of the T-S waves of temporal mode for the incompressible fluid. In this paper, we construct the T-S waves of both temporal mode and spatial mode to the linearized compressible Navier-Stokes system around the boundary layer flow in the whole subsonic regime. The proof is based on a new iteration scheme via solving two quasi-compressible systems related to the incompressible part and compressible part respectively. For the incompressible part, the key ingredient is to solve an Orr-Sommerfeld type equation, which is based on a new Airy-Airy-Rayleigh iteration instead of Rayleigh-Airy iteration introduced by Grenier, Guo and Nguyen.
[04253] Global Existence of Weak Solutions for Compressible Navier--Stokes--Fourier Equations with the Truncated Virial Pressure Law
Format : Talk at Waseda University
Author(s) :
Fei Wang (Shanghai Jiao Tong University)
Didier Bresch (Univ. Savoie Mont Blanc)
Pierre-Emmanuel Jabin (Pennsylvania State University)
Abstract : This paper concerns the existence of global weak solutions {\it \`a la Leray} for compressible Navier--Stokes--Fourier system with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.
[03513] Nonlinear Stability of the Taylor-Couette flow
Format : Talk at Waseda University
Author(s) :
Te Li (National University of Singapore)
Abstract : Hydrodynamic stability at high Reynolds number is a central topic in fluid mechanics. It is closely related to turbulence. Whereas the laminar velocity profile is linearly stable for all Reynolds number, Reynolds experiment reveals that laminar flows could be unstable and transit to turbulence at high Reynolds number. This phenomenon is described as subcritical transition. And the mechanism behind is not well understood yet. To investigate the dynamical nonlinear stability, we develop a systematic approach to establish sharp resolvent estimates for the linearized operator around the 2D Taylor-Couette flow. One of the main difficulties is that the linearized operator is non-self-adjoint. Based on the resolvent estimates, we first show the sharp enhanced-dissipation decay rate of the solution for the linearized system. We also derive the space-time estimates for the nonlinear part using the resolvent estimates. Combining all above, we obtain the nonlinear transition threshold for the Taylor-Couette flow. This talk is based on a series of joint works by X. An-T. He-L.
[04401] On the solvability of the linearized Triple-Deck system
Format : Online Talk on Zoom
Author(s) :
Yasunori Maekawa (Kyoto University)
David Gerard-Varet (Universite Paris Cite et IMJ-PRG)
Sameer Iyer (University of California)
Abstract : We establish the solvability of the linearized Triple-Deck system in Gevrey $3/2$ regularity in the tangential variable, under the concavity assumptions on the background flow. This talk is based on the joint work with David Gerard-Varet (Universite Paris Cite et IMJ-PRG) and Sameer Iyer (University of California).
[03016] On dynamic stability for steady Prandtl solutions
Format : Online Talk on Zoom
Author(s) :
Yue Wang (Capital Normal University)
Abstract : In this talk, I will first review some properties of steady Prandtl solutions. Then I will introduce our recent work on dynamic stability of steady Prandtl solutions( a joint work with Yan Guo and Zhifei Zhang) and others’ related results.
