# American Institute of Mathematical Sciences

March  2019, 18(2): 751-794. doi: 10.3934/cpaa.2019037

## Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities

 Department of Mathematics, Texas A & M University, College Station, Texas, USA

Received  February 2018 Revised  July 2018 Published  October 2018

Fund Project: This work is part of the doctoral dissertation of the author under the supervision of Professor Zhouping Xin at the Institute of Mathematical Sciences of the Chinese University of Hong Kong, Hong Kong. The author would like to express great gratitude to Prof. Xin for his kindly support and professional guidance.

This work is devoted to studying the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such a scenario, the boundary no longer provides friction and therefore the perturbation of the angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the stability of the uniformly rotating solutions with a small angular velocity. In particular, the initial perturbation which preserves the angular momentum will be stable in the sense that the global strong solution to the Navier-Stokes equations exists and the perturbation is uniformly bounded and small in time.

Citation: Xin Liu. Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities. Communications on Pure & Applied Analysis, 2019, 18 (2) : 751-794. doi: 10.3934/cpaa.2019037
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