摘要
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 densityρ0=η11Ω0+η21Ω0^c,where(η1,η2)is any pair of positive constants andΩ0 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 functionρ(t)=η11Ω(t)+η21Ω(t)^c,and the domainΩ(t)preserves the W^k+2,p-boundary regularity.
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 ρ0=η11?0+η21?0c,where（η1,η2）is any pair of positive constants and ?0 is a bounded,simply connected domain with Wk+2,p（R2）boundary regularity.We prove that for any positive time t,the density function ρ（t）=η11（?（t））+η21?（t）c,and the domain ?（t） preserves the Wk+2,p-boundary regularity.
基金
MCM for the hospitality and the financial support
supported by SFB 1060
Universitat Bonn during the last part of the work
partially supported by NSF of China under Grants Nos.11371347 and 11688101
innovation grant from National Center for Mathematics and Interdisciplinary Sciences
