A Blow-Up Criterion for the Nonhomogeneous Incompressible Navier--Stokes Equations

Let (ρ , u) be a strong or smooth solution of the nonhomogeneous incompressible Navier--Stokes equations in (0, T^*) × Ω, where T^* is a finite positive time and Ω is a bounded domain in R³ with smooth boundary or the whole space R³. We show that if (ρ , u) blows up at T^*, then ∫₀T^* |u (t) |Lwʳ (Ω )ˢ \, dt = ∞ for any $(r,s)$ with 2s + 3r =1 and 3 < r ≤ ∞. As immediate applications, we obtain a regularity theorem and a global existence theorem for strong solutions.