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

Abstract

Abstract 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 L t q ( L x p ) -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 L 2 ∩ L ∞ and in the norm of some multiplier spaces. Moreover, we also consider the associated density patch problem, and show the global persistence of C 1 , γ -regularity of the density patch boundary when the piecewise jump of density is small enough.