We prove that there exists a strong solution to the Dirichlet boundary value problem for the steady Navier-Stokes equations of a compressible heat-conductive fluid with large external forces in a bounded domain Ω⊂Rd (. d=. 2, 3), provided that the Mach number is appropriately small. At the same time, the low Mach number limit is rigorously verified. The basic idea in the proof is to split the equations into two parts, one of which is similar to the steady incompressible Navier-Stokes equations with large forces, while another part corresponds to the steady compressible heat-conductive Navier-Stokes equations with small forces. The existence is then established by dealing with these two parts separately, establishing uniform in the Mach number a priori estimates and exploiting the known results on the steady incompressible Navier-Stokes equations.
Dou, C., Jiang, F., Jiang, S., & Yang, Y. F. (2015). Existence of strong solutions to the steady Navier-Stokes equations for a compressible heat-conductive fluid with large forces. Journal Des Mathematiques Pures et Appliquees, 103(5), 1163–1197. https://doi.org/10.1016/j.matpur.2014.10.009