Global Regularity of 2-D Density Patches for Viscous Inhomogeneous Incompressible Flow with General Density: High Regularity Case (Analysis in Theory and Applications, 2019)
This paper is a continuation work of [26] and studies the propagation of the high-order boundary regularities of the two-dimensional density patch for viscous inhomogeneous incompressible flow. We assume the initial density $ρ_0=η_1{1}_{Ω_0}+η_2{1}_{Ω_0^c}$, where $(η_1,η_2)$ is any pair of positive constants and $Ω_0$ is a bounded, simply connected domain with $W^{k+2,p}(R^2)$ boundary regularity. We prove that for any positive time $t$, the density function $ρ(t)=η_1{1}_{Ω(t)}+η_2{1}_{Ω(t)^c}$, and the domain $Ω(t)$ preserves the $W^{k+2,p}$-boundary regularity.
Keywords: Inhomogeneous incompressible Navier-Stokes equations; density patch; striated distributions; Littlewood-Paley theory.