Finite Time Blow-Up Behaviors of Strong Solutions to the Compressible Navier-Stokes Equations

In this paper, we are interested in the finite time blow-up behaviors of strong solutions to the compressible Navier-Stokes equations with vacuum as far field density. Via the blow-up criterion obtained by [H. Wen and C. Zhu, Adv. Math., 2013], we are able to obtain a lower bound of the blow-up rate for \(||\nabla u||_{L^2}\) ${||\mathrm{\nabla }u||}_{L^2}$ and \(||P||_{L^2}\) ${||P||}_{L^2}$ provided that \(\gamma >3\) $\gamma >3$ and the viscous coefficients satisfy the additional condition \(\frac{29}{3}\mu >\lambda \) $\frac{29}{3}\mu >\lambda$ .