  The paper concerns with the global well-posedness issue of the 2D
incompressible inhomogeneous Navier-Stokes (INS) equations with fractional
dissipation and rough density. We first establish the $L^q_t(L^p)$-maximal
regularity estimate for the generalized Stokes system with fractional
dissipation, and then we employ it to obtain the global existence of solution
for the 2D fractional INS equations with large velocity field, provided that
the $L^2\cap L^\infty$-norm of density minus constant 1 is small enough.
Moreover, by additionally assuming that the density minus 1 is sufficiently
small in the norm of some multiplier spaces, we prove the uniqueness of the
constructed solution by using the Lagrangian coordinates approach. We also
consider the density patch problem for the 2D fractional INS equations, and
show the global persistence of $C^{1,\gamma}$-regularity of the density patch
boundary when the piecewise jump of density is small enough.
