Low Mach number limit of the full Navier-Stokes equations,

The low Mach number limit for classical solutions of the full Navier-Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are taken into account. In particular, we consider general initial data. The equations lead to a singular problem whose linearized is not uniformly well-posed. Yet, it is proved that the solutions exist and are uniformly bounded for a time interval which is independent of the Mach number Ma ∈ (0, 1], the Reynolds number Re ∈ [1,+∞] and the Péclet number Pe ∈ [1,+∞]. Based on uniform estimates in Sobolev spaces, and using a Theorem of G. Métivier and S. Schochet [30], we next prove that the penalized terms converge strongly to zero. This allows us to rigorously justify, at least in the whole space case, the well-known computations given in the introduction of the P.-L. Lions’ book [26].