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

Abstract

In this paper we obtain new well-possedness results concerning a linear inhomogenous 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 ${\rho }_0-\rho \in {\dot{B}}_{p,1}^{\frac{3}{p}}({\mathbb{R}}^3)$, $u_0\in {\dot{B}}_{p,1}^{\frac{3}{p}-1}({\mathbb{R}}^3)$, $p\in \left(\frac{6}{5},4\right)$, for the inhomogeneous incompressible Navier-Stokes system with variable viscosity. To the best of our knowledge, regarding the $3D$ case, this is the first result in a truly critical framework for which one does not assume any smallness condition on the density.

Cited by in corpus (1)

• The decay and stability of solutions for the 3D density-dependent incompressible Boussinesq system