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

Abstract

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 $\ddot{I}{}_0=1_D$ 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.}

23 pages