Stability of the density patches problem with vacuum for incompressible inhomogeneous viscous flows

We consider the inhomogeneous incompressible Navier-Stokes system in a smooth twoor three-dimensional bounded domain, in the case where the initial density is only bounded. Existence and uniqueness for such initial data was shown recently in Danchin and Mucha [Comm. Pure Appl. Math. 72 (2019)], but the stability issue was left open. After having shown that the solutions constructed therein have exponential decay, a result of independent interest, we prove the stability with respect to initial data, first in Lagrangian coordinates, and then in the Eulerian frame. We actually obtain stability in the energy space for the velocity and in a Sobolev space with negative regularity for the density. Let us underline that, as opposed to prior works, our stability estimates are valid even in the case of a vacuum. In particular, our result applies to the classical density patches problem, where the density is a characteristic function.