Vol. 13, No. 1, 2020

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

Raphaël Danchin, Francesco Fanelli and Marius Paicu

Vol. 13 (2020), No. 1, 275–316
Abstract

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 subcritical regularity and that the initial density is a small perturbation (in the ${L}^{\infty }$ 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 (Comm. Pure Appl. Math. 55:11 (2002), 1365–1407) with a uniqueness statement, and is valid in any dimension $d\ge 2$ and for general pressure laws.

Keywords
compressible Navier–Stokes equations, bounded density, maximal regularity, tangential regularity, Lagrangian formulation
Mathematical Subject Classification 2010
Primary: 35Q35
Secondary: 35B65, 76N10, 35B30, 35A02
Milestones
Received: 24 April 2018
Accepted: 30 November 2018
Published: 6 January 2020
Authors
 Raphaël Danchin LAMA UMR 8050 Université Paris-Est Créteil Val-de-Marne Créteil France Francesco Fanelli Université Claude Bernard Lyon 1 CNRS UMR 5208 Institut Camille Jordan Villeurbanne France Marius Paicu Institut de Mathématiques de Bordeaux Université de Bordeaux Talence France