Global Well-Posedness for Inhomogeneous Navier–Stokes Equations in Endpoint Critical Besov Spaces

Abstract

We study global well-posedness for the inhomogeneous Navier–Stokes equations on $${\mathbb {R}}^n$$ , $$n\ge 2$$ , with initial velocity in endpoint critical Besov spaces $$B^{-1+n/q}_{q,\infty }({\mathbb {R}}^n)$$ , $$n\le q<2n$$ , and merely bounded initial density with a positive lower bound. First, we consider a multiplication property of $$L^\infty $$ -functions in some Bessel potential and Besov spaces. Based on it and on maximal $$L^\infty _\gamma $$ -regularity of the Stokes operator in little Nicolskii spaces, we show solvability for the momentum equations with fixed bounded density. Finally, proof for existence of a solution to the inhomogeneous Navier–Stokes equations is done via an iterative scheme when $$B^{-1+n/q}_{q,\infty }$$ -norm of initial velocity and relative variation of initial density are small, while uniqueness of a solution is proved via a Lagrangian approach when initial velocity belongs to $$B^{-1+n/q}_{q,\infty }({\mathbb {R}}^n)\cap B^{-1+n/q}_{r,\infty }({\mathbb {R}}^n)$$ for slightly larger $$r>q$$ .