ON CONTINUATION CRITERIA FOR THE FULL COMPRESSIBLE NAVIER-STOKES EQUATIONS IN LORENTZ SPACES

In this paper, we derive several new sufficient conditions of the non-breakdown of strong solutions for both the 3D heat-conducting compressible Navier-Stokes system and nonhomogeneous incompressible Navier-Stokes equations. First, it is shown that there exists a positive constant " such that the solution (rho, u, theta) to the full compressible Navier-Stokes equations can be extended beyond t = T provided that one of the following two conditions holds: [GRAPHICS] . To the best of our knowledge, this is the first continuation theorem allowing the time direction to be in Lorentz spaces for the compressible fluid. Second, we establish some blow-up criteria in anisotropic Lebesgue spaces for the finite blow-up time T* : (1) assuming that the pair (p, (q) over right arrow) satisfies 2/p + 1/q(1) + 1/q(2) + 1/q(3) = 1 (1 < q(i) < infinity) and (1.17), then [GRAPHICS] . Third, without the condition on rho in (0.1) and (0.3), the results also hold for the 3D nonhomogeneous incompressible Navier-Stokes equations. The appearance of a vacuum in these systems could be allowed.