This article is devoted to the low Mach number limit of weak solutions to the compressible Navier–Stokes equations for polytropic fluids with periodic boundary conditions and ill‐prepared data. We derive formally the equation satisfied by the mean value of the velocity and the equations governing the dynamics of the nonlinear acoustic waves in dimension d= 2 or 3.

Abstract. The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on x. This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss the general low Mach number limit for standard compressible flows given in P.-L. Lions' book that means with constant viscosity coefficients.Mathematics Subject Classification. 35Q30.

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We perform the mathematical derivation of the compressible and incompressible Euler equations from the modulated nonlinear Klein-Gordon equation. Before the formation of singularities in the limit system, the nonrelativistic-semiclassical limit is shown to be the compressible Euler equations. If we further rescale the time variable, then in the semiclassical limit (the light speed kept fixed), the incompressible Euler equations are recovered. The proof involves the modulated energy introduced by Brenier (2000) [1]. RésuméOn obtient une dérivation mathématique des équations d'Euler compressibles et incompressibles à partir de l'équation de KleinGordon non linéaire modulée. Avant la formation de singularités pour le système limite, on démontre dans la limite non relativiste et semi-classique la convergence vers les équations d'Euler compressibles. Au moyen d'un changement d'échelle supplémentaire en temps, on démontre la limite semi-classique la vitesse de la lumière restant fixée, la limite vers les équations d'Euler incompressibles. La démonstration utilise l'énergie modulée introduite par Brenier (2000) [1].