On the stability of stationary solutions of artificial compressible system

2018/04/27 Fri 15:30 - 17:30
寺本 有花

In this talk we consider the stability of stationary solutions of the incompressible Navier-Stokes system and the corresponding artificial compressible system. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero limit of the artificial Mach number $\epsilon$ which is a singular limit. It is shown that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small $\epsilon$. The result is applied to the Taylor problem. In general, the range of $\epsilon$ for which the above mentioned stability result holds shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of $\epsilon$ can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.