Stability of the density patches problem with vacuum for incompressible inhomogeneous viscous flows (Q6569710)

The article deals with the inhomogeneous incompressible Navier-Stokes equations in a bounded domain in two or three dimensions. The authors consider no-slip boundary conditions and the case where the initial density is only bounded, in particular allowing for vacuum as well as the density patch problem, where the domain splits into two disjoint parts with different constant densities at initial time. For such initial conditions, existence and uniqueness of strong solutions were established in a previous work, and the main result of the present article is to show stability of these solutions. In a first step, exponential decay of solutions is derived from higher-order energy estimates. After transformation to a Lagrangian frame, these decay properties lead to the asymptotic stability, which afterwards is transferred back to the solutions in an Eulerian frame. Here, the distance of two solutions is measured in terms of the relative kinetic energy and a negative Sobolev norm for the density.