Global solutions to the 3-D incompressible inhomogeneous Navier-Stokes system

In this paper, we consider the global well-posedness of the 3-D incompressible inhomogeneous Navier-Stokes equations with initial data in the critical Besov spaces a(0) epsilon B-q,1(3/q) (R-3), u(0) = (u(0)(h), u(0)(3)) epsilon B-p,1(-1+3/p)(R-3) for p, q satisfying 1 < q <= p < 6 and 1/q - 1/p <= 1/3. More precisely, we prove that there exist two positive constants c(0), C-0 such that if (mu parallel to a(0)parallel to(3/q)(Bq,1) + parallel to u(0)(h)parallel to(-1+3/p)(Bp,1))exp(C-0 parallel to u(0)(3)parallel to(2)(-1+3/p)(Bp,1)/mu(2)) <= c(0)mu, then (1.3) has a unique global solution a epsilon (L) over tilde infinity(R+; B-q,1(3/q)(R-3)), u epsilon (L) over tilde infinity(R+; B-p,1(-1+3/p)(R-3)) boolean AND L-1(R+; B-p,1(1+3/p)(R-3)). In particular, this result implies the global well-posedness result in Abidi and Paicu (2007) [2] for the inhomogeneous Navier-Stokes system with small initial data. (c) 2012 Elsevier Inc. All rights reserved.