On the blow up criterion for the 2-D compressible Navier-Stokes equations

A Beale–Kato–Majda blow-up criterion for the 3-D compressible Navier–Stokes equations

“…Fan, S. Jiang and Y. Ou [12] also obtained a similar result for the compressible heat-conductive flows. On the other hand, for the 2D compressible Navier-Stokes equations in T 2 , B. Desjardins [11] proved more regularity of weak solution under the assumption that the density is upper bounded; Very recently, L. Jiang and Y. Wang [17], Y. Sun and Z. Zhang [26] obtained a blow-up criterion in terms of the upper bound of the density for the strong solution.…”

Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids

“…Comparable results for the compressible Navier-Stokes system are in short supply and subject to various restrictions on the geometry of domains and/or viscosity coefficients (see e.g. [2], [13], [16]). This is mostly due to the fact that the viscosity provides only a partial smoothing effect on some but not all quantities in question.…”

A blow-up criterion of strong solutions to the 2D compressible Navier-Stokes equations

“…Note that the L 1 (0, T, L ∞ (Ω)) bound for ∇u immediately implies the upper bound for ρ. It is strongly expected that the upper bound of ρ is enough to control the blow-up of strong solutions to the 2D compressible Navier-Stokes equations, as it is shown in [8,18] for the periodic case. In this paper, we show this is really true for a bounded smooth domain in R 2 , despite of the presence of vacuum for the initial density.…”

“…Recently, based on the approach of [26], Jiang and Wang [18] gave a blow-up criterion for the strong solution in T 2 (see also [11] for a similar result). More precisely, they proved…”