We consider a compressible viscous fluid effected by general form external force in . In part 1, an analysis of the linearized problem based on the weighted- method implies a condition on the external force for the existence and some regularities of the steady flow. In part 2, we study the stability of the steady flow with respect to the initial disturbance. What we proved is: if -norm of the initial disturbance is small enough, then the solution to the non-stationary problem exists uniquely and globally in time.
Yoshihiro SHIBATA. Koumei TANAKA. "On the steady flow of compressible viscous fluid and its stability with respect to initial disturbance." J. Math. Soc. Japan 55 (3) 797 - 826, July, 2003. https://doi.org/10.2969/jmsj/1191419003