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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-0004}, url = {http://global-sci.org/intro/article_detail/ata/13112.html} }