召集人：段仁军（香港中文大学）、王勇（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

白祥, 中国科学院数学与系统科学研究院, 中国

Title: Well-posedness and asymptotic behavior for the Euler-alignment system 

Abstract: In this talk, we will present the global well-posedness theory for the Cauchy problem of the compressible Euler system with singular velocity alignment. Additionally, we show that asymptotic emergent behaviors for the system, providing time decay estimates with optimal decay rates. This is based on joint works with Qianyun Miao, Changhui Tan and Liutang Xue.

段仁军, 香港中文大学, 中国香港

Title: Boltzmann equation with mixed boundary in bounded domain

Abstract: We consider the Boltzmann equation in a bounded domain with a mixed-type boundary which is diffusively reflected in one portion and specularly reflected in the other portion with a further restriction that the diffusive part lies in between two parallel specular parts. Under such situation we construct the global small-amplitude solutions around global Maxwellians for the corresponding initial-boundary value problem. The proof involves the establishment of L2 hypocoercivity using the weak formulation and Korn’s inequality, and also involves the delicate application of the L2-L∞ bootstrap argument under such mixed-type boundary condition. Joint with Hongxu Chen.

樊心宇, 中国科学院数学与系统科学研究院, 中国

Title: Global smooth solutions to compressible Navier-Stokes equations under Dirichlet boundary conditions

Abstract: In this talk, we investigate the initial boundary value problems of compressible Navier-Stokes equations under non slip boundary conditions. We divide the whole domain into the inner and boundary parts along a free surface. The density is strictly positive in the boundary domain, while the vacuum is available in the inner domain.In order to establish the higher order estimates near the boundary, we apply the global tool introduced by Christoudoulou-Lindblad.

李维喜, 武汉大学, 中国

Title: Smoothing effect for the Boltzmann and Landau equations

Abstract: As specific degenerate elliptic operators, we may expect not only the usual smoothness but also the Gevrey regularity for subelliptic operators. However it is highly non-trivial to improve the Gevrey regularity to analyticity for degenerate elliptic operators. As a positive example and a classical subelliptic operator, the Kolmogorov operator indeed enjoy the analyticity regularity in the degenerate direction. Thus it is natural to ask the similar properties for the spatial inhomogeneous kinetic equations since these equations may be regarded as non-local and non-linear models of Kolmogorov operators. We will confirm this analytic regularization effect for the spatially inhomogeneous Boltzmann and Landau equations with hard potentials. Instead of the estimates on the usual derivatives, our argument relies crucially on a careful treatment on the derivatives with respect to a new time-average operator.  

刘双乾, 华中师范大学, 中国

Title: Shear flow problems for the Boltzmann equation

Abstract: I will report our recent study on the shear flow problem within the framework of the Boltzmann equation. Our focus includes the stability of shear flow for both the Maxwell molecular model and hard potentials. We will discuss the plane Couette flow and the heat transfer governed by the Boltzmann equation with diffusive reflection boundary conditions in a spatial interval. Additionally, the 3D kinetic Couette flow in the diffusive limit, as described by the Boltzmann equation, will be covered.

欧阳婧, 中国科学院数学与系统科学研究院, 中国

Title: The Global Well-posedness and Newtonian Limit for the Relativistic Boltzmann Equation in a Periodic Box 

Abstract: In this talk, we study the Newtonian limit for relativistic Boltzmann equation in a periodic box $\mathbb{T}^3$ . We first establish the global-in-time mild solutions of relativistic Boltzmann equation with uniform-in-c estimates and time decay rate. Then we rigorously justify the globalin-time Newtonian limits from the relativistic Boltzmann solutions to the solution of Newtonian Boltzmann equation in $L^1_pL^\infty_x$ . Moreover, if the initial data of Newtonian Boltzmann equation belong to $W^{1,\infty}(\mathbb{T}^3\times\mathbb{R}^3)$, based on a decomposition and $L^2-L^\infty$ argument, the global-in-time Newtonian limit is proved in $L^\infty_{x,p}$. The convergence rates of Newtonian limit are obtained both in $L^1_pL^\infty_x$ and $L^\infty_{x,p}$.

王勇, 中国科学院数学与系统科学研究院, 中国

Title: TBA 

Abstract: TBA

肖长国, 广西师范大学, 中国

Title: Hydrodynamic limit and Newtonian limit from the relativistic Boltzmann equation to the classical Euler equations

Abstract: The hydrodynamic limit and Newtonian limit are important in the relativistic kinetic theory. We justify rigorously the validity of the two independent limits from the special relativistic Boltzmann equation to the classical Euler equations without assuming any dependence between the Knudsen number $\varepsilon$ and the light speed $\mathfrak{c}$. The convergence rates are also obtained. This is achieved by Hilbert expansion of relativistic Boltzmann equation. New difficulties arise when tacking the uniform in $\mathfrak{c}$ and $\varepsilon$ estimates for the Hilbert expansion, which have been overcome by establishing some uniform-in-$\mathfrak{c}$ estimates for relativistic Boltzmann operators.  This talk is based on a joint work with Yong Wang.

肖峰, 湖南师范大学, 中国

Title: Hilbert expansion of the Boltzmann equation in the incompressible Euler level in a channel

Abstract: We mainly talk about the hydrodynamic limit from the Boltzmann equation with specular reflection boundary condition to the incompressible Euler in a channel. Based on the multi-scaled Hilbert expansion, the equations with boundary conditions and compatibility conditions for interior solutions, viscous and Knudsen boundary layers are derived under different scaling, respectively. Then some uniform estimates for the interior solutions, viscous and Knudsen boundary layers are established. With the help of these uniform estimates and the L^2 − L^\infty framework, the solutions to the Boltzmann equation are constructed by the truncated Hilbert expansion with multiscales, and hence the hydrodynamic limit in the incompressible Euler level is justified. This talk is based on recent work joint with Feimin Huang, Weiqiang Wang and Yong Wang.

肖清华, 中国科学院精密测量科学与技术创新研究院, 中国

Title: The Hilbert expansion for some kinetic equations via weighted energy method

Abstract: Our talk is concerned with global Hilbert expansion for the relativistic   Vlasov-Maxwell-Landau (RVML) , the Vlasov-Maxwell-Landau (VML), and the non-cutoff Vlasov-Maxwell-Boltzmann (VMB) systems in the entire space. Based on weighted energy methods, we prove that the unique classical solution of the RVML, the VML,  or the non-cutoff VMB system converges globally over time to the smooth solution to  the corresponding hydrodynamic system as the Knudsen number approaches zero.

熊杭, 湘潭大学, 中国

Title: Global well-posedness of Vlasov-Poisson-Boltzmann system with  neutral initial data and small relative entropy

Abstract: The dynamics of dilute plasma particles such as electrons and ions can be modeled by the fundamental two species Vlasov-Poisson-Boltzmann system which describes mutual interactions of plasma particles through collisions in the self-induced electric field. In this paper, we are concerned with global well-posed theory of mild solutions to this system. We established the global existence and uniqueness of mild solutions to the two species Vlasov-Poisson-Boltzmann system in the torus for a class of initial data with bounded time-velocity-weighted $L^{\infty}$ norm under nearly neutral condition and some smallness condition on $L^1_xL^\infty_v$ norm with time-velocity-weight as well as  defect mass, energy and entropy so that the initial data allow large amplitude oscillations.

熊林杰, 湖南大学, 中国

Title: The Vlasov-Poisson-Boltzmann System with a Repulsive Potential Near Vacuum

Abstract: The problem of very soft potential cases $\gamma \in (-3,-2]$ for the Vlasov–Poisson-Boltzmann system near vacuum has remained unresolved in Guo's work [Comm. Math. Phys.218(2001), no.2, 293–313.] for over two decades, limited by the decay of spatial density. We provide a positive answer to this problem with the assistance of an external potential in the present work. Specifically, by employing optimal decay estimates derived by A. Velozo Ruiz and R. Velozo Ruiz [Comm. Math. Phys. 405 (2024), no. 3, Paper No. 80, 45 pp.] using a commuting vector field approach, we construct global classical solutions for the Vlasov–Poisson-Boltzmann system with the repulsive potential $\Phi(x) = \frac{-|x|^2}{2} $ for the whole range of cutoff soft potentials $\gamma \in (-3,0)$.

杨东成, 华南理工大学, 中国

Title: Global stability of viscous contact waves for the Vlasov-Poisson-Landau system

Abstract: In this talk, we are concerned with the stability of contact waves for the two species Vlasov-Poisson-Landau system with slab symmetry for the physical Coulomb interaction. Our results are twofold. First, we construct the unique global-in-time solution near a local Maxwellian for the Vlasov-Poisson-Landau system with suitably small initial data, where the fluid components of the local Maxwellian are called viscous contact waves in the sense of hydrodynamics. Second, we prove the large time asymptotics of global solutions towards such local Maxwellian, and establish the time decay rates of the disparity between two species and the electric field. The proof is based on the two different sets of decompositions of the solutions and the intricate weighted energy method.  

 杨雄锋, 上海交通大学, 中国

Title: The long wavelength approximation limit of the Vlasov-Poisson (VP) system in torus

Abstract: 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. It is a joint work with Dr. Zhao Lixian. 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. It is a joint work with Dr. Zhao Lixian.

钟明溁, 广西大学, 中国

Title: Diffusion Limit with Optimal Convergence Rate of Classical Solutions to the Vlasov-Maxwell-Boltzmann System

Abstract: We study the diffusion limit of the classical solution to the Vlasov-Maxwell-Boltzmann (VMB) system with initial data neara global Maxwellian. By introducing a new decomposition of the solution to identify the essential components for generating the initial layer, we prove the convergence and establish the opitmal convergence rate of the classical solution to the VMB system to the solution of the  Navier-Stokes-Maxwell  system based on the spectral analysis.