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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 \(\epsilon \) 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 \(\epsilon \). In general, the range of \(\epsilon \) 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.

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Acknowledgements

I am deeply grateful to Professor Yoshiyuki Kagei for his constant encouragement. This work was partly supported by JSPS KAKENHI Grant Number 17J04702.

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  1. Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan

    Yuka Teramoto

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  1. Yuka Teramoto
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Correspondence to Yuka Teramoto.

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Communicated by T. Nishida.

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Teramoto, Y. Stability of Bifurcating Stationary Solutions of the Artificial Compressible System. J. Math. Fluid Mech. 20, 1213–1228 (2018). https://doi.org/10.1007/s00021-018-0364-7

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  • DOI: https://doi.org/10.1007/s00021-018-0364-7

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