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Nonlinear Analysis

Global regularities of 2-D density patch for viscous inhomogeneous incompressible flow with general density

Ping Zhang: AMSS, Chinese Academy of Sciences

Location:  Hill 525
Date & time: Friday, 10 February 2017 at 1:40PM - 1:41PM

Toward the open problem proposed by P.-L. Lions in [Mathematical topics in fluid mechanics. Vol.1. Incompressible models, 1998] concerning the propagation of regularitiesof density patch for viscous inhomogeneous incompressible flow, we first establish the global in time well-posedness oftwo-dimensional inhomogeneous incompressible Navier-Stokes system withinitial density being the summation of the characteristic function on a bounded, simply connected \[W^{k+2,p}(R^2)\] domain\[Om_0\] multiplied by a positive constant \[eta_1\] and the characteristic function on the completement of\[Om_0\] multiplied by a positive constant \[eta_2\] for any pair of positive constants \[(eta_1,eta_2).\] We then prove thatthe time evolved domain \[Om(t)\] also belongs to the class of\[W^{k+2,p}\] for any \[t>0.\] Thus in some sense, we have solved the aforementioned open problem of Lionsin the two-dimensional case.

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