Strong Convergence of Low Mach Number Limit for the Compressible Navier–Stokes Equations in the Scaling Critical Spaces

Abstract We consider the Cauchy problem for the compressible Navier–Stokes equations in whole space and low Mach number limit problem. In this paper, we show that the incompressible part of the velocity strongly converges to the solution of the incompressible Navier–Stokes equations as the Mach number goes to 0 in the scaling critical space. We also show that the density and the compressible part of the velocity vanish. Moreover, we derive the diverging of the time derivative of the compressible part of the velocity as Mach number goes to 0. The proofs are based on the $$L^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> -Maximal regularity for the heat equations and the Strichartz estimates for the wave equations.