Abstract
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 ε such that the solution (ρ, u, θ) to the full compressible Navier-Stokes equations can be extended beyond t = T provided that one of the following two conditions holds: 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*: Third, without the condition on ρ in (0.1) and (0.3), the results also hold for the 3D non-homogeneous incompressible Navier-Stokes equations. The appearance of a vacuum in these systems could be allowed.
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