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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.
In this paper, inspired by the study of the energy flux in local energy inequality of the 3D incompressible Navier-Stokes equations, we improve almost all the blow up criteria involving temperature to allow the temperature in its scaling invariant sp
In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, o
In this paper, we derive regular criteria via pressure or gradient of the velocity in Lorentz spaces to the 3D Navier-Stokes equations. It is shown that a Leray-Hopf weak solution is regular on $(0,T]$ provided that either the norm $|Pi|_{L^{p,infty}
The energy equalities of compressible Navier-Stokes equations with general pressure law and degenerate viscosities are studied. By using a unified approach, we give sufficient conditions on the regularity of weak solutions for these equalities to hol
We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in $mathbb{R}^n.$ We focus on the so-called critical Besov regularity framework. In this setting, it is natural