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

Burtea, Cosmin

doi:10.2140/apde.2017.10.439

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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

DOI: 10.2140/apde.2017.10.439

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 $ρ_{0}$ and velocity $u_{0}$ such that $ρ_{0} - ρ \in Ḃ_{p, 1}^{3 ∕ p} (ℝ^{3})$ , $u_{0} \in Ḃ_{p, 1}^{3 ∕ p - 1} (ℝ^{3})$ , $p \in (\frac{6}{5}, 4)$ 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

Vol. 10, No. 2, 2017