Stability of Bifurcating Stationary Solutions of the Artificial Compressible System

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Abstract

The artificial compressible system gives a compressible approximation of the incompressible Navier–Stokes system. The latter system is obtained from the former one in the zero limit of the artificial Mach number ϵ which is a singular limit. The sets of stationary solutions of both systems coincide with each other. It is known 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, then it is also stable as a solution of the artificial compressible one for sufficiently small ϵ. In general, the range of ϵ 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 ϵ can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.

Original languageEnglish
Pages (from-to)1213-1228
Number of pages16
JournalJournal of Mathematical Fluid Mechanics
Volume20
Issue number3
DOIs
Publication statusPublished - Sep 1 2018
Externally publishedYes

Keywords

  • Artificial compressible system
  • Bifurcation
  • Incompressible Navier–Stokes system
  • Singular perturbation
  • Stability

ASJC Scopus subject areas

  • Mathematical Physics
  • Condensed Matter Physics
  • Computational Mathematics
  • Applied Mathematics

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