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

TL;DR

This paper establishes local well-posedness for the inhomogeneous incompressible Navier-Stokes equations with variable viscosity in critical spaces, without smallness assumptions on the initial density, advancing the mathematical understanding of fluid dynamics.

Contribution

It provides the first critical framework result for 3D inhomogeneous Navier-Stokes with variable viscosity without density smallness constraints.

Findings

Well-posedness in critical spaces for initial data in Besov spaces

First 3D result without smallness assumption on density

New well-posedness results for linear inhomogeneous Stokes-like systems

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.