On the regularity criteria for the three-dimensional compressible Navier-Stokes system in Lorentz spaces

In this paper, we are concerned with the continuation criteria to the 3D isentropic compressible Navier-Stokes equations in Lorentz spaces. We prove that there exists a positive constant epsilon$$ \varepsilon $$ such that no blowup occurs at time T$$ T $$ in this system provided that the supernorm of the density is bounded and the space-time Lorentz spaces norm ||rho u||Lp,infinity(0,T;Lq,infinity)$$ {\left\Vert \sqrt{\rho }u\right\Vert}_{L circumflex {p,\infty}\left(0,T;{L} circumflex {q,\infty}\right)} $$ with 2/p+3/q=1$$ 2/p+3/q equal to 1 $$ (3 <= q<infinity)$$ \left(3\le q<\infty \right) $$ is small. As a direct application, we established some Serrin's blow-up criteria in the Lorentz spaces.