On the density patch problem for the 2-D inhomogeneous Navier-Stokes equations

In this paper, we first construct a class of global strong solutions for the 2-D inhomogeneous Navier-Stokes equations under very general assumption that the initial density is only bounded and the initial velocity is in $H^1(\mathbb{R}^2)$. With suitable assumptions on the initial density, which includes the case of density patch and vacuum bubbles, we prove that Lions' s weak solution is the same as the strong solution with the same initial data. In particular, this gives a complete resolution of the density patch problem proposed by Lions: {\it for the density patch data $\rho_0=1_{D}$ with a smooth bounded domain $D\subset\mathbb{R}^2$, the regularity of $D$ is preserved by the time evolution of Lions's weak solution.}