Global regularity of density patch for the 3D inhomogeneous Navier–Stokes equations

Abstract

This paper is concerning with the regularity propagation of density patches for the 3D inhomogeneous incompressible Navier–Stokes equations. By careful time-weighted energy estimates, Stokes estimates and the singular integral operators, we prove that the 3D density patches preserve the $$C^{k,\gamma }(k=1,2)$$ regularity for the initial interface given by $$\rho _0(x)=\eta _11_{\Omega _0}+\eta _21_{\Omega ^c_0}$$. In particular, we do not need the smallness assumption on $$|\eta _1-\eta _2|$$.