本年（2026年）度のセミナーの記録
- 4 月 10 日
- 飯田 祥樹 氏 （京都大学） Yoshiki Iida (Kyoto University)
- 4 月 17 日
- 出口 直人 氏 （京都大学） Naoto Deguchi (Kyoto University)
- 4 月 27 日
- Ulisse Stefanelli 氏 （University of Vienna）
● 2026 年 4 月 10 日 (Fri) 16:00 〜 17:00
- 講演者
- 飯田 祥樹 氏 （京都大学 大学院理学研究科）
Yoshiki Iida (Kyoto University)
- 講演題目
- 滑らかでない境界を持つ層状領域上の線形化 primitive 方程式
Linearized primitive equations on non-smooth layers
- 講演要旨
-
本講演では，地球上の海洋の挙動を記述する，3 次元層状領域上の primitive 方程式を考察する．
特に本方程式の線形化問題に焦点を当て，海底の形状を記述する側の境界が必ずしも滑らかでない場合を考える．
例えば，境界の正則性が $C^1$-級や Lipschitz 連続しかないとき，対応するレゾルベント方程式の解の正則性は期待できない．
そこで，対応する線形化作用素（静水圧 Stokes 作用素）の定義を Shen (2012), Geng–Shen (2025) に基づいて与え，これが解析半群を生成することを示す．
特に，鍵となる圧力項の評価を重点的に紹介する．
● 2026 年 4 月 17 日 (Fri) 16:00 〜 17:00
- 講演者
- 出口 直人 氏 （京都大学 数理解析研究所）
Naoto Deguchi (Kyoto University)
- 講演題目
- Stability analysis of exterior compressible viscous flows induced by time-bounded external forces
- 講演要旨
-
Consider the boundary value problem for the compressible Navier-Stokes equation in a three-dimensional exterior domain, subject to a time-bounded external force which decays at spatial infinity.
We obtain the existence result of the time-bounded solution when the external force is sufficiently small.
Furthermore, under the smallness assumption for initial perturbations, we prove the global existence and time-decay estimates for solutions to the initial-boundary value problem for the perturbation equation around the time-bounded solution.
The analysis performed in the L^2 based homogeneous Besov space on the exterior domain to control both the slow spacial decay of the time-bounded solution and the hyperbolic effect arising from the equation of the mass conservation law.
The L^\infty-in-time estimate in the Besov space plays a crucial role in the proofs.
● 2026 年 4 月 27 日 (Mon) 16:00 〜 17:00
- 講演者
- Ulisse Stefanelli 氏 （University of Vienna）
- 講演題目
- A free boundary problem in accretive growth
- 講演要旨
-
Accretive growth, in which material is added at the boundary of a system, is a central phenomenon in biology, natural processes, and engineering.
Mathematically, it can be described by a stationary Hamilton-Jacobi equation governing the motion of the boundary, coupled with a PDE for an activation field (such as nutrients, temperature, or stress) defined on the evolving domain.
This results in a free boundary problem with a highly nonlinear, coupled structure.
In this talk, I will present an existence analysis for such a problem.
I will begin with the growth subproblem, where I establish sharp regularity properties of the sublevel sets of the solution.
In particular, I will show that the growing domains satisfy a uniform Poincaré inequality.
This provides the analytical framework needed to prove an existence result for the fully coupled free boundary system.
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