A well-posedness result for viscous compressible fluids with only bounded density

Raphaël Danchin (1, 2) , Francesco Fanelli (3, 4, 5) , Marius Paicu (6, 7)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
3 UCBL - Université Claude Bernard Lyon 1
4 ICJ - Institut Camille Jordan
5 EDPA - Équations aux dérivées partielles, analyse
6 IMB - Institut de Mathématiques de Bordeaux
7 UB - Université de Bordeaux

We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in the L ∞ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension d ≥ 2 and for general pressure laws.

Mots clés: Lagrangian formulation; maximal regularity; tangential regularity; bounded density; compressible Navier-Stokes equations; Lagrangian formulation

Domaines: Equations aux dérivées partielles [math.AP]

Raphaël Danchin, Francesco Fanelli, Marius Paicu. A well-posedness result for viscous compressible fluids with only bounded density. Analysis & PDE, 2020, 13 (1), ⟨10.2140/apde.2020.13.275⟩. ⟨hal-01778175⟩