The mild solution of the inhomogeneous fractional incompressible Navier-Stokes equation on <inline-formula><tex-math id="M1">$ \mathbb{R}^n $</tex-math></inline-formula>

In this paper, we study the existence of local and global mild solutions for a nonhomogeneous fractional incompressible Navier-Stokes system when initial values belong to subcritical and critical Lebesgue spaces. With the help of the pointwise estimates of the fractional heat kernel, the corresponding bilinear operator estimates have been established for subcritical case. However, it is unbounded in the critical Banach space $ C(0,+\infty; L^{\frac{n}{2\alpha-1}}( \mathbb{R}^n)) $. To get rid of this difficulty, we consider a subspace of $ C(0,+\infty; L^{\frac{n}{2\alpha-1}}( \mathbb{R}^n)) $ which enables us to obtain the related bilinear operator estimates. Based on these bilinear operator estimates, the existence of local and global mild solutions of nonhomogeneous fractional incompressible Navier-Stokes equations are established.