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

Abstract

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 $\epsilon$ 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) $\rho \in L^{\infty }(0,T;L^{\infty }({\mathbb{R}}^3))$, $u\in L^{p,\infty }(0,T;L^{q,\infty }({\mathbb{R}}^3))$ and $$\Vert u{\Vert }_{L^{p,\infty }(0,T;L^{q,\infty }({\mathbb{R}}^3))}\le \epsilon ,\text{with}2/p+3/q=1,q>3;$$ (2) $\lambda <3\mu ,$ $\rho \in L^{\infty }(0,T;L^{\infty }({\mathbb{R}}^3))$, $\theta \in L^{p,\infty }(0,T;L^{q,\infty }({\mathbb{R}}^3))$ and $$\Vert \theta {\Vert }_{L^{p,\infty }(0,T;L^{q,\infty }({\mathbb{R}}^3))}\le \epsilon ,\text{with}2/p+3/q=2,q>3/2.$$ 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.

22 pages

References in corpus (1)

• $\epsilon$-regularity criteria in Lorentz spaces to the 3D Navier-Stokes equations