Special Session 63: Singular limit problems arising from nonlinear PDEs

Local exact Lagrangian controllability of the 1D compressible Navier--Stokes equations

Kai K Koike Tokyo Institute of Technology Japan Co-Author(s):    Franck Sueur, Gast\`{o}n Vergara-Hermosilla

Abstract: We consider barotropic compressible Navier--Stokes equations in the interval $[0,\pi]$ with homogeneous Dirichlet boundary conditions. Our result is the following: given two sufficiently close subintervals $I$ and $J$ of $(0,1)$, we construct a smooth external force $f$ in the momentum equation supported in $(1,\pi)$ such that the flow map moves $I$ exactly onto $J$ in a given time $T>0$. The essential point in the proof is to find two external forces $f_1$ and $f_2$ that have ``independent`` stretching effect on $I$. Such forces are constructed using the linearized adjoint system and the independence is proved using a unique continuation property which we prove based on Fourier analytic techniques.

COUPLING AND PROPAGATION OF SINGULARITIES IN THE INITIAL LAYER FOR BOLTZMANN EQUATION

Hung-Wen Kuo National Cheng Kung University Taiwan Co-Author(s):    Tai-Ping Liu and Shih-Hsien Yu

Abstract: In the study of the relationship between the Boltzmann equation in kinetic theory and fluid dynamics, it is essential to understand the singular layers. In the classical paper by Harold Grad, he identified three basic layers: boundary, shock, and initial layers. In this talk, I will present joint work with Tai-Ping Liu and Shih-Hsien Yu on the formation and propagation of singularities in the initial layer of the solutions to the Boltzmann equation in both the space-time $(x, t)$ and microscopic velocity $\boldsymbol{\xi}$ domains. Singularities transport in the space-time $(x, t)$ domain and interact with the nonlinear collision operator in the microscopic velocity ${\boldsymbol \xi}$ domain, creating rich singular coupling in the $(x, t, {\boldsymbol \xi})$ domain. Our work identifies the essential features of the collision operators using generalized Carleman-Hilbert coordinates, ensuring that the Green`s function approach is sufficient to reveal the explicit structure of the singularities in the initial layers.

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.

3D hard sphere Boltzmann equation: explicit structure and the transition process from polynomial tail to Gaussian tail

Haitao Wang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Yu-Chu Lin, Kung-Chien Wu

Abstract: We study the Boltzmann equation with hard sphere in a near-equilibrium setting. The initial data is compactly supported in the space variable and has a polynomial tail in the microscopic velocity. We show that the solution can be decomposed into a particle-like part (polynomial tail) and a fluid-like part (Gaussian tail). The particle-like part decays exponentially in both space and time, while the fluid-like part dominates the long time behavior and exhibits rich wave motion. The nonlinear wave interactions in the fluid-like part are precisely characterized. Furthermore, the transition process from the polynomial to the Gaussian tail is quantitatively revealed.

GLOBAL SOLUTION OF 3-D KELLER-SEGAL MODEL WITH COUETTE FLOW IN WHOLE SPACE

Weike WANG Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: we introduce both parabolic-elliptic Keller-Segel model and parabolic-parabolic Keller-Segel model in the background of a Couette flow with spatial variables in $R^3$. It is proved that for both parabolic-elliptic and parabolic-parabolic cases, a Couette flow with a suffciently large amplitude prevents the blow-up of solutions. This result is totally different from either the classical Keller-Segel model or the case with a large shear flow and the periodic spatial variable $x$; for those two cases, the solution may blow up. Here, we apply Green`s function method to capture the suppression of blow-up and prove the global existence of the solutions.

Stability analysis of boundary layers and inviscid limits of MHD equations

Feng Xie Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will discuss the inviscid limit of solutions to initial boundary value problems of viscous hydrodynamics equations with high Reynolds numbers. Due to the mismatch of boundary conditions between the viscous hydrodynamics equations and the limit system (ideal hydrodynamics equations), the boundary layer correction functions usually are introduced as studying the process of inviscid limit of solutions. We will discuss both the stability of boundary layers and the justification of Prandtl boundary layer expansions. First, we will review the progress of analysis of boundary layer equations and inviscid limits; Then, we will introduce the results obtained in recent years in the study of inviscid limits and the stability analysis of boundary layers on MHD.

On the Sobolev stability threshold for 3D Navier-Stokes equations with rotation near the Couette flow

Xiaojing Xu Beijing Normal University Peoples Rep of China Co-Author(s):    Wenting Huang, Ying Sun

Abstract: In this talk, we introduce the dynamic stability of periodic, plane Couette flow in the three-dimensional Navier-Stokes equations with rotation at high Reynolds number $\mathbf{Re}$. Our aim is to determine the stability threshold index on $\mathbf{Re}$, we demonstrate that if initial data satisfies $\left\|u_{\mathrm{in}}\right\|_{H^{\sigma}}\frac{9}{2}$ and some $\delta=\delta(\sigma)>0$ depending only on $\sigma$, then the solution to the 3D Navier-Stokes equations with rotation is global in time without transitioning away from Couette flow. In this sense, Coriolis force contributes as a factor enhancing fluid stability by improving its threshold from $\frac{3}{2}$ to 1. This is a jointed work with Wenting Huang, and Ying Sun.

Convergence rate for the incompressible limit of the Navier-Stokes equations

Xin Xu Ocean University of China Peoples Rep of China Co-Author(s):    Xin Xu

Abstract: \begin{abstract} We study the incompressible limit of the Navier-Stokes equations in a bounded domain with slip boundary conditions. Based on the previous convergence results, we further give the convergence rates of the solutions. \end{abstract}

The limit from Vlasov-Poisson system to KdV/ZK equations

Xiongfeng Yang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Zhao Lixian

Abstract: This talk presents the long wavelength approximation limit of the Vlasov-Poisson (VP) system in torus. We derive formally two-directional wave packets as the solutions of Korteweg-de Vries (KdV) equations from 1-D VP system, the two distinct wave packets as the solutions of Zakharov-Kuznetsov (ZK) equations from 3-D VP system with magnetic field, and the two-way waves as the solutions to the corresponding Kadomtsev-Petviashvili equations from 2-D VP system without magnetic field. A rigorous justification of this long-wave limit is established by the relative entropy method.

BV solutions to the Navier-Stokes equation

Xiongtao Zhang Wuhan University Peoples Rep of China Co-Author(s):    Haitao Wang and Shih-Hsien Yu

Abstract: In this talk, I will introduce the well-posedness and the large time behaviors of the BV solutions to the compressible Navier-Stokes equation. The method is based on the Green`s function of the heat kernel and NS equations.