Vol. 10, No. 2, 2017

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Optimal well-posedness for the inhomogeneous incompressible Navier–Stokes system with general viscosity

Cosmin Burtea

Vol. 10 (2017), No. 2, 439–479
Abstract

In this paper we obtain new well-posedness results concerning a linear inhomogeneous Stokes-like system. These results are used to establish local well-posedness in the critical spaces for initial density ρ0 and velocity u0 such that ρ0 ρ p,13p(3), u0 p,13p1(3), p (6 5,4) for the inhomogeneous incompressible Navier–Stokes system with variable viscosity. To the best of our knowledge, regarding the 3-dimensional case, this is the first result in a truly critical framework for which one does not assume any smallness condition on the density.

Keywords
inhomogeneous Navier–Stokes system, critical regularity, Lagrangian coordinates
Mathematical Subject Classification 2010
Primary: 35Q30, 76D05
Milestones
Received: 26 July 2016
Revised: 18 October 2016
Accepted: 28 November 2016
Published: 23 February 2017
Authors
Cosmin Burtea
Université Paris-Est Créteil
LAMA - CNRS UMR 8050
61 Avenue du Général de Gaulle
94010 Créteil
France