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Low Mach Number Limit of the Full Navier-Stokes Equations

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Abstract

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].

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  1. MAB Université de Bordeaux I, 33405, Talence Cedex, France

    Thomas Alazard

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Communicated by A. Bressan

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Alazard, T. Low Mach Number Limit of the Full Navier-Stokes Equations. Arch. Rational Mech. Anal. 180, 1–73 (2006). https://doi.org/10.1007/s00205-005-0393-2

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