Mathematical Analysis of space-time nonuniform dynamics of equations for viscous compressible fluids

2020 Fiscal Year Final Research Report

• Project/Area Number: 16H03947
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Tokyo Institute of Technology (2019-2020); Kyushu University (2016-2018)
• Principal Investigator: KAGEI Yoshiyuki 東京工業大学, 理学院, 教授 (80243913)
• Co-Investigator(Kenkyū-buntansha): 水町 徹 広島大学, 先進理工系科学研究科(理), 教授 (60315827) ; 川島 秀一 早稲田大学, 理工学術院, 教授(任期付) (70144631) ; 前川 泰則 京都大学, 理学研究科, 教授 (70507954) ; 中村 徹 熊本大学, 大学院先端科学研究部(工), 准教授 (90432898) ; 小川 知之 明治大学, 総合数理学部, 専任教授 (80211811)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 函数論方程式 / 圧縮性Navier-Stokes方程式 / 安定性 / 分岐 / 漸近挙動 / スペクトル解析

Outline of Final Research Achievements

The spectra of linearized operators around spatio-temporal periodic states of the compressible Navier-Stokes system were investigated in detail to obtain a precise description of the large time behavior of solutions around such periodic states. The structure of the spectrum of the linearized operator of the artificial compressible system was studied around the bifurcation point of stationary solutions and it was proved that if the artificial Mach number is sufficiently small, then the spectrum is decomposed into two parts, one is given by a perturbation of the spectrum for the incompressible system and the other one arises from the compressible aspect of the equations. This analysis was extended to the case of the linearized operator at the Couette flow of the compressible Navier-Stokes equations.

Free Research Field

数学解析

Academic Significance and Societal Importance of the Research Achievements

圧縮性Navier-Stokes方程式はさまざまな興味深い流れのパターンを記述するが，方程式のもつ双曲型の側面から生じる技術的困難のため，時空非一様な解のまわりのダイナミクスの数学解析はいまだ発展途上にあり，数理構造の解明とそのための数学解析手法の開発が望まれている．本研究では特に圧縮性Navier-Stokes方程式の時間的および空間的な周期構造をもつ流れのパターンの安定性解析について有効な解析手法を与え，解の時間無限大における漸近挙動の様相を明らかにした．