A Beale–Kato–Majda Criterion for Three Dimensional Compressible Viscous Heat-Conductive Flows

We prove a blow-up criterion in terms of the upper bound of ( ρ , ρ −1 , θ ) for a strong solution to three dimensional compressible viscous heat-conductive flows. The main ingredient of the proof is an a priori estimate for a quantity independently introduced in Haspot (Regularity of weak solutions of the compressible isentropic Navier–Stokes equation, arXiv:1001.1581, 2010 ) and Sun et al. (J Math Pure Appl 95:36–47, 2011 ), whose divergence can be viewed as the effective viscous flux.