Strong solutions to the Cauchy problem of two-dimensional incompressible fluid models of Korteweg type

This paper studies the local existence of strong solutions to the Cauchy problem of the incompressible fluid models of Korteweg type with vacuum as far field density. The corresponding 3D problem has been solved by Tan and Wang (2010) [21]. Notice that the technique used by Tan and Wang fails treating the 2D case, because the Lp-norm (p>2) of the velocity u cannot be controlled in terms only of ρu and ∇u here. In the present paper, we will use the framework of weighted approximation estimates introduced by Liang (2015) [14] for Navier-Stokes equations to obtain the local existence of strong solutions provided the initial density does not decay very slowly at infinity, with the compact support case included.