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 ${\rho }_{0}$ and velocity ${u}_{0}$ such that ${\rho }_{0}-\rho \in {Ḃ}_{p,1}^{3∕p}\left({ℝ}^{3}\right)$, ${u}_{0}\in {Ḃ}_{p,1}^{3∕p-1}\left({ℝ}^{3}\right)$, $p\in \left(\frac{6}{5},4\right)$ 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