Abstract : Kinetic theory has been expanding its frontier and emerged as promising in various fields of engineering and science. At the same time, it has been a source of unsolved mathematical problems at fundamental levels, which are still actively studied. This mini-symposium aims at bringing in international experts on mathematical analysis, modeling, and computation of kinetic theory and related topics, in order to present the field’s state-of-the-art results and foster future academic exchanges and collaborations among researchers from different sub-fields. We propose three sessions which include 12 speakers from different generations of the field and 2 leading experts Tai-Ping Liu and Shih-Hsien Yu as chairpersons who can enhance the communication of the groups.
[01451] Boundary singularity of a mono-speed Lorentz model for molecules with the infinite-range potential
Format : Talk at Waseda University
Author(s) :
Shigeru Takata (Kyoto University)
Masanari Hattori (Kyoto University)
Hayato Iida (Kyoto University)
Abstract : Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect
of the grazing collision for the infinite-range potential on the velocity distribution function
(VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the
steady one-dimensional boundary-value problem, it is confirmed that the grazing collision
suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However,
as the price for that regularization, the collision integral becomes no longer finite in the
direction of the molecular velocity parallel to the boundary. Consequently, the gradient of
the macroscopic quantity diverges, even stronger than the case of the finite-range potential.
A conjecture about the diverging rate in approaching the boundary is made as well for a wide
range of the infinite-range potentials, accompanied by numerical evidences.
[01849] On the Existence and Regularity for the Stationary Linearized Boltzmann Equation in a Small Domain
Format : Talk at Waseda University
Author(s) :
I-Kun Chen (National Taiwan University)
Ping-Han Chuang (National Taiwan University)
Jhe-Kuan Su (National Taiwan University)
Chun-Hsiung Hsia (National Taiwan University)
Daisuke Kawagoe (Kyoto University)
Abstract : We consider the incoming boundary value problem for the
stationary linearized Boltzmann equation in a bounded domain with C^2
boundary of positive Gaussian curvature. We prove the existence of
H^1 of solutions under assumptions that the boundary data is good
enough and the domain is small enough. A counter example is provided
to demonstrate the role of the geometry.
[05464] Regularity estimates for the non-cutoff soft potential Boltzmann equation with typical rough and slowly decaying data
Format : Talk at Waseda University
Author(s) :
Lingbing He (Tsinghua University)
Jie Ji (Peking University)
Abstract : or the non-cutoff soft potential Boltzmann equation, if the Boltzmann collision operator is strictly elliptic in the $v$ variable, it is conjectured that the solution to the equation will become infinitely smooth instantly for both spatial and velocity variables for any positive time, even if the initial data has only polynomial decay in high velocity regimes. This conjecture is significant because it is closely connected to the regularity problem of weak solutions, especially for the smoothing property of so-called ``regular point''. In this work, we show that the conjecture may not hold for the general weak solution due to the degenerate and non-local properties of the collision operator. We demonstrate this in three steps: (i) constructing so-called ``typical rough and slowly decaying data''; (ii) proving that such data induces only finite smoothing effect for weak solutions in Sobolev spaces; and (iii) proving that this finite smoothing property induces local properties for any positive time, including that the Leibniz rule does not hold for high derivatives of the collision operator (even in the weak sense) and that there is a discontinuity in the $x$ variable for the average of weak solutions on certain domains.
[00646] Vanishing angular singularity limit to the hard-sphere Boltzmann equation
Format : Talk at Waseda University
Author(s) :
Jin Woo Jang (Pohang University of Science and Technology)
Bernhard Kepka (University of Bonn)
Alessia Nota (Università degli Studi dell'Aquila)
Juan J. L. Velázquez (University of Bonn)
Abstract : In this talk we consider Boltzmann's collision kernel for inverse power law interactions $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $ d=3 $. We introduce the proof of the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near $\theta\simeq 0$ in the limit $ s\to \infty $. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.
[04112] On BGK-type models with velocity-dependent collision frequency
Format : Talk at Waseda University
Author(s) :
Doheon Kim (Hanyang University)
Seok-Bae Yun (Sungkyunkwan University)
Abstract : In the original Bhatnagar-Gross-Krook (BGK) model, the collision term in the Boltzmann equation is replaced by a simpler expression, so that the model is easier-to-handle and satisfies the conservation laws and the H-Theorem. This model contains the collision frequency as a parameter independent of the particle velocity. In this talk, I will introduce variants of the BGK model which contain velocity-dependent collision frequency, thereby mimicing the behavior of the Boltzmann equation more closely.
[02844] Green's function for solving IBVP of evolutionary PDEs
Format : Talk at Waseda University
Author(s) :
Hung-Wen Kuo (National Cheng Kung University)
Abstract : We propose a new method to solve the initial-boundary value problem for hyperbolic-dissipative PDEs based on the spirit of LY algorithm. Utilizing the idea of Laplace wave train and the notions of Rayleigh surface wave operators, we are able to obtain the complete representations of the Green's functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions.
[00644] H\"OLDER REGULARITY OF THE BOLTZMANN EQUATION PAST AN OBSTACLE
Format : Talk at Waseda University
Author(s) :
donghyun lee (POSTECH)
chanwoo Kim (University of Wisconsin, madison)
Abstract : Regularity and singularity of the Boltzmann equation with various shape of domains is a challenging research theme in the Boltzmann theory. In this talk, we discuss about H\"oler regularity of the Boltzmann equation outiside of convex object under specular reflection boundary condition.
[03811] Solution to the Boltzmann equation without cutoff in $(L^1\cap L^p)_k$
Format : Talk at Waseda University
Author(s) :
Shota Sakamoto (Kyushu University)
Renjun Duan (Chinese University of Hong Kong)
Yoshihiro Ueda (Kobe University)
Abstract : We consider a Cauchy problem of the Boltzmann equation without angular cutoff near the global Maxwellian on the whole space. In this case, the control of the $L^1$ norm on the Fourier side is not sufficient for global existence due to low-frequency terms. Therefore, we employ the $L^p$ norm estimates with respect to the frequency to control such parts. This $L^1\cap L^p$ strategy will close a priori estimates when combined with a time-weighted energy method.
Abstract : In this talk, we consider the Boltzmann equation with angular-cutoff for very soft potential case. We prove a regularization mechanism that transfers the microscopic velocity regularity to macroscopic space regularity in the fractional sense. A precise pointwise estimate of the fractional derivative of collision kernel, and a connection between velocity derivative and space derivative in the fractional sense are exploited to overcome the high singularity for very
soft potential case.
[03301] Dynamical behaviors in stochastic kinetic flocking models
Format : Talk at Waseda University
Author(s) :
Xiongtao Zhang (Wuhan University)
Abstract : We will introduce some recent works on the stochastic flocking models. We are interested in the case when the noise is multiplicative and the flocking interaction vanishes at the far field. We will show rigorous proof of mean-field limit (weak or strong) and the emergence of flocking (conditional or unconditional) under various assumptions.
[04712] Kinetic study of a gas undergoing resonant collisions
Format : Online Talk on Zoom
Author(s) :
Francesco Salvarani (DVRC &University of Pavia)
Laurent Boudin (Sorbonne Université)
Thomas Borsoni (Sorbonne Université)
Julien Mathiaud (CEA & Université de Bordeaux)
Alex Rossi (Friedrich-Alexander-Universität Erlangen-Nürnberg)
Abstract : We study a kinetic model for a gas undergoing résonant collision. After proving the main properties of the model, we study the compactness of the corresponding linearized Boltzmann operator.
