A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations

Let (p, u) be a strong or smooth solution of the nonhomogeneous incompressible Navier-Stokes equations in (0, T*) x Omega, where T* is a finite positive time and Omega is a bounded domain in R-3 with smooth boundary or the whole space R-3. We show that if (p, u) blows up at T*, then integral(T*)(0) vertical bar u(t)vertical bar(s)(LTW)(Omega dt = infinity for any (r, s) with 2/s + 3/r = 1 and 3 < r <= infinity. As immediate applications, we obtain a regularity theorem and a global existence theorem for strong solutions.