Analysis of stablity and bifurcation for compressible fluid equations

2015 Fiscal Year Final Research Report

• Project/Area Number: 24340028
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Kyushu University
• Principal Investigator: KAGEI Yoshiyuki 九州大学, 数理(科)学研究科(研究院), 教授 (80243913)
• Co-Investigator(Kenkyū-buntansha): KAWASHIMA Shuichi 九州大学, 大学院数理研究院, 教授 (70144631) ; KOBAYASHI Takayuki 大阪大学, 基礎工学研究科, 教授 (50272133) ; NAKAMURA Tohru 熊本大学, 自然科学研究科, 准教授 (90432898)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: Navier-Stokes equation / compressible / asymptotic behavior / spectral anaylsis / stability / instability / bifurcation

Outline of Final Research Achievements

To establish mathematical theory for bifurcation and stability in the compressible Navier-Stokes equation, we studied the stability of stationary and time-periodic parallel flows. We proved that the asymptotic behavior of parallel flow is described by a linear heat equation when the space dimension n is greater than or equal to 3, and by a one-dimensional viscous Burgers equation when n=2. In the case of the Poiseuille flow, we derived a sufficient condition for the instability in terms of the Reynolds and Mach numbers. Furhtermore, we proved the bifurcation of a familiy of space-time-peirodic traveling waves when the Poiseuille flow is getting unstable. As a first step of the stability analysis of space-periodic patterns, we investigate the stablity of the motionless state on periodic infinte layer, and derived the asymptotic leading part of the perturbation by using the Bloch transformation.

Free Research Field

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