The incompressible limit in $$L^p$$ type critical spaces

This paper aims at justifying the low Mach number convergence to the incompressible Navier–Stokes equations for viscous compressible flows in the ill-prepared data case. The fluid domain is either the whole space, or the torus. A number of works have been dedicated to this classical issue, all of them being, to our knowledge, related to \(L^2\) spaces and to energy type arguments. In the present paper, we investigate the low Mach number convergence in the \(L^p\) type critical regularity framework. More precisely, in the barotropic case, the divergence-free part of the initial velocity field just has to be bounded in the critical Besov space \(\dot{B}^{d/p-1}_{p,r}\cap \dot{B}^{-1}_{\infty ,1}\) for some suitable \((p,r)\in 2,4]\times 1,+\infty ].\) We still require \(L^2\) type bounds on the low frequencies of the potential part of the velocity and on the density, though, an assumption which seems to be unavoidable in the ill-prepared data framework, because of acoustic waves. In the last part of the paper, our results are extended to the full Navier–Stokes system for heat conducting fluids.