A Blow-up criterion for the 3-D compressible Navier-Stokes equations

In this paper, the blow-up criterion for the strong solutions to the compressible Navier-Stokes equations in ℝ³ is studied. We obtain that if sup_o≤t≤T ∥ρ(t)∥_L∞ + ∥∇ρ(t)∥_{LTp(Lq(ℝ3))} + ∫_o^T ∥∇u(t)∥_Ḃ∞,∞0 dt <; ∞, for 2/p + 3/q ≤ 2, 3 <; q <; ∞, then the strong solutions (ρ u) can be continued beyond T.