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, depending on a small scaling parameter, whose linearized system is not uniformly well-posed. Yet, it is proved that solutions exist and they 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 & 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 P.-L. Lions' book [26].