TY - JOUR
T1 - Global  Regularity of 2-D Density Patches for Viscous Inhomogeneous Incompressible  Flow with  General Density: High Regularity Case
JO - Analysis in Theory and Applications
VL - 2
SP - 163
EP - 191
PY - 2019
DA - 2019/04
SN - 35
DO - http://doi.org/10.4208/ata.OA-0004
UR - https://global-sci.org/intro/article_detail/ata/13112.html
KW - Inhomogeneous incompressible Navier-Stokes equations, density patch, striated distributions, Littlewood-Paley theory.
AB - <p style="text-align: justify;">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 $\rho_0=\eta_1{1}_{\Omega_0}+\eta_2{1}_{\Omega_0^c}$, where $(\eta_1,\eta_2)$ is any pair of positive constants and $\Omega_0$ 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 $\rho(t)=\eta_1{1}_{\Omega(t)}+\eta_2{1}_{\Omega(t)^c}$, and the domain $\Omega(t)$ preserves the $W^{k+2,p}$-boundary regularity.</p>