Global regularities of two-dimensional density patch for inhomogeneous incompressible viscous flow with general density
Abstract
Toward the open question proposed by P.-L. Lions in Lions96 concerning the propagation of regularities of density patch for viscous inhomogeneous flow, we first establish the global in time well-posedness of two-dimensional inhomogeneous incompressible Navier-Stokes system with initial density being of the form: η1 1_0+η2 1_0c, for any pair of positive constants (η1,η2), and for any bounded, simply connected Wk+2,p(2) domain 0. We then prove that the time evolved domain (t) also belongs to the class of Wk+2,p for any t>0. Thus in some sense, we have solved the aforementioned Lions' question %of density patch in Lions96 in the two-dimensional case. Compared with our previous paper LZ, here we remove the smallness condition on the jump, |η1-η2|, moreover, the techniques used in the present paper are completely different from those in LZ.
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