Mini Workshop on nonlinear PDEs related to fluid dynamics
Date and Time: July 27 (Sat), 2024; 13:00--17:20
Venue: Room 213, Main Bidg. Ookayama Campus, Tokyo Institute of Technology
https://www.titech.ac.jp/english/0/maps/ookayama
Date and Time: July 27 (Sat), 2024; 13:00--17:20
Venue: Room 213, Main Bidg. Ookayama Campus, Tokyo Institute of Technology
https://www.titech.ac.jp/english/0/maps/ookayama
Speakers:
Chun-Hsiung Hsia (National Taiwan University)
Masahiro Suzuki (Nagoya Institute of Technology)
Ryo Takada (The University of Tokyo)
Xin Zhang (Tongji University)
Program
13:00--13:50 Chun-Hsiung Hsia (National Taiwan University)
On the geometric effect of the regularity of the linearized Boltzmann equation
14:10--15:00 Ryo Takada (The University of Tokyo)
Existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term
15:20--16:10 Xin Zhang (Tongji University)
On steady solutions of the Hall-MHD system in Besov spaces
16:30--17:20 Masahiro Suzuki (Nagoya Institute of Technology)
Ionized Gas in an Annular Region
Abstract
Chun-Hsiung Hsia (National Taiwan University)
Title: On the geometric effect of the regularity of the linearized Boltzmann equation
Abstract: This is a joint work with I-Kun Chen, Daisuke Kawagoe and Jhe-Kuan Su. We study the incoming boundary value problem for the stationary linearized Boltzmann equation in bounded convex domains. The geometry of the domain has a dramatic effect on the space of solutions. We prove the existence of solutions in W1,p spaces for 1 ≤ p < 2 for small domains. In contrast, if we further assume the positivity of the Gaussian curvature on the boundary, we prove the existence of solutions in W1,p spaces for 1 ≤ p < 3 provided that the diameter of the domain is small enough. In both cases, we provide counterexamples in the hard sphere model; a bounded convex domain with a flat boundary for p = 2, and a small ball for p = 3.
Ryo Takada (The University of Tokyo)
Title: Existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term
Abstract: In this talk, we consider the existence of solutions to the fractional semilinear heat equation with a singular inhomogeneous term. We show the temporal decay estimates for the fractional heat semigroup in several Zygmund spaces, and establish the critical integral estimates for the inhomogeneous term and the nonlinear term by making use of the real interpolation method in weak Zygmund spaces. Then, we prove the local-in-time existence of solutions to the inhomogeneous fractional semilinear heat equation in uniformly local weak Zygmund spaces including the inhomogeneous functions with optimal singularities. This talk is based on the joint work with Professor Kazuhiro Ishige (The University of Tokyo) and Professor Tatsuki Kawakami (Ryukoku University).
Xin Zhang (Tongji University)
Title: On steady solutions of the Hall-MHD system in Besov spaces
Abstract: In this talk, we present some recent results on the well-posedness and ill-posedness issues for the incompressible stationary Hall-magnetohydrodynamic (Hall-MHD) system in $\mathbb{R}^3.$ Inspired by the stationary incompressible Navier-Stokes problem, we first show the existence and uniqueness of solutions provided with the forces in $\dot B^{3/p-3}_{p,r}(\mathbb{R}^3)$ for $1\leq p <3$ and $r=1$. Moreover, this result can be extended to any $1\leq r\leq \infty$ whenever $p=2,$ without any additional assumption on the physical parameters. On the other hand, we also give some ill-posedness results for Hall-MHD system in \emph{critical} function spaces $\dot B^{3/p-1}_{p,r}(\mathbb{R}^3)$ ($p\geq 3$). This talk is based on a joint work with Jin Tan (Cergy Paris Universit\'{e}) and Hiroyuki Tsurumi (Tokushima University).
Masahiro Suzuki (Nagoya Institute of Technology)
Title: Ionized Gas in an Annular Region
Abstract: We consider a plasma that is created by a high voltage difference $\lambda$, which is known as a Townsend discharge. We consider it to be confined to the region $\Omega$ between two concentric spheres, two concentric cylinders, or more generally between two star-shaped surfaces. We first show that if the plasma is initially relatively dilute, then either it may remain dilute for all time or it may not, depending on a certain parameter $\kappa(\lambda,\Omega)$. Secondly, we prove that there is a connected one-parameter family of steady states. This family connects the non-ionized gas to a plasma, either with a sparking voltage $\lambda^*$ or with very high ionization, at least in the cylindrical or spherical cases. This talk is based on a joint work with Professor W. A. Strauss (Brown University).
Contact: Yoshiyuki Kagei (Tokyo Tech) kagei@math.titech.ac.jp
Sponsor: JSPS KAKENHI Grant-in-Aid for Scientific Research (A) JP24H00185