Unique global weak solutions, with energy equality and stability, exist for the 2D inhomogeneous Navier-Stokes equations when the initial density is bounded away from zero.
Navier-Stokes equation with variable d ensity and viscosity in critical space
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Inhomogeneous 2D Navier--Stokes equations: Existence, uniqueness, stability, continuity in time and energy conservation of weak solutions
Unique global weak solutions, with energy equality and stability, exist for the 2D inhomogeneous Navier-Stokes equations when the initial density is bounded away from zero.