Well-posedness of 3-D inhomogeneous Navier–Stokes equations with highly oscillatory initial velocity field

Abstract

Without smallness assumption on the variation of the initial density function, we first prove the local well-posedness of 3-D incompressible inhomogeneous Navier–Stokes equations with initial data (a0,u0) in the critical Besov spaces Bλ,13λ(R3)×B˙p,13p−1(R3) for λ, p given by Theorem 1.1. Then we prove this system is globally well-posed provided that ‖u0‖B˙p,13p−1 is sufficiently small. In particular, this result implies the global well-posedness of 3-D inhomogeneous Navier–Stokes equations with highly oscillatory initial velocity field and any initial density function with a positive lower bound.