Global well-posedness for 2D fractional inhomogeneous Navier-Stokes equations with rough density

This paper is concerned with the global well-posedness issue of the two-dimensional (2D) incompressible inhomogeneous Navier-Stokes equations with fractional dissipation and rough density. By establishing the Ltq(Lxp) -maximal regularity estimate for the generalized Stokes system and using the Lagrangian approach, we prove the global existence and uniqueness of regular solutions for the 2D fractional inhomogeneous Navier-Stokes equations with large velocity field, provided that the initial density is sufficiently close to the constant 1 in L2L and in the norm of some multiplier spaces. Moreover, we also consider the associated density patch problem, and show the global persistence of C1, 3 -regularity of the density patch boundary when the piecewise jump of density is small enough.