On continuation criteria for the full compressible Navier-Stokes equations in Lorentz spaces

In this paper, we derive several new sufficient conditions of non-breakdown of strong solutions for 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 $varepsilon$ such that the solution $(rho,u,theta)$ to full compressible Navier-Stokes equations can be extended beyond $t=T$ provided that one of the following two conditions holds (1) $rho in L^{infty}(0,T;L^{infty}(mathbb{R}^{3}))$, $uin L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))$ and $$| u|_{L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))}leq varepsilon, ~~text{with}~~ {2/p}+ {3/q}=1, q>3;$$ (2) $lambda<3mu,$ $rho in L^{infty}(0,T;L^{infty}(mathbb{R}^{3}))$, $thetain L^{p,infty}(0,T;L^{q,infty}(mathbb{R}^{3}))$ and $$|theta|_{L^{p,infty}(0,T; L^{q,infty}(mathbb{R}^{3}))}leq varepsilon, ~~text{with}~~ {2/p}+ {3/q}=2, q>3/2.$$ 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 to the full Navier-Stokes system. 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 vacuum in these systems could be allowed.