On Chorin’s Method for Stationary Solutions of the Oberbeck–Boussinesq Equation

Stability of stationary solutions of the Oberbeck–Boussinesq system (OB) and the corresponding artificial compressible system is considered. The latter system is obtained by adding the time derivative of the pressure with small parameter ε> 0 to the continuity equation of (OB), which was proposed by A. Chorin to find stationary solutions of (OB) numerically. Both systems have the same sets of stationary solutions and the system (OB) is obtained from the artificial compressible one as the limit ε→ 0 which is a singular limit. It is proved that if a stationary solution of the artificial compressible system is stable for sufficiently small ε> 0 , then it is also stable as a solution of (OB). The converse is proved provided that the velocity field of the stationary solution satisfies some smallness condition.