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

Raphaël Danchin (1) , Lingbing He (2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 Department of Mathematical Sciences,

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.

Mots clés: Compressible flows; Navier-Stokes; Critical regularity; Low Mach number; Besov spaces

Domaines: Equations aux dérivées partielles [math.AP]

Raphaël Danchin, Lingbing He. The incompressible limit in $L^p$ type critical spaces. Mathematische Annalen, 2016, 366 (3-4), pp.1365-1402. ⟨hal-01067758⟩