Some Serrin type blow‐up criteria for the three‐dimensional viscous compressible flows with large external potential force

Abstract

We provide a Serrin type blow-up criterion for the 3-D viscous compressible flows with large external potential force. For the Cauchy problem of the 3-D compressible Navier-Stokes system with potential force term, it can be proved that the strong solution exists globally if the velocity satisfies the Serrin's condition and the sup-norm of the density is bounded. Furthermore, in the case of isothermal flows with no vacuum, the Serrin's condition on the velocity can be removed from the claimed criterion.

Global regularity for the 3D compressible magnetohydrodynamics with general pressure

“…• We introduce some new type of estimates on functionals which are used for decoupling the velocity and magnetic field; • We obtain a blow-up criterion for (1.1)-(1.4) for general pressure term P (ρ) which is not restricted to the isothermal case; • We assure that the blow-up criterion dependsds only on density, which is an improvement for the results obtained from [Sue13a,Sue15] in which the magnetic field was present in the criterion; • We do not impose any extra compatibility condition on the initial data, which is required in the work [SWZ11,HLX11,Sue20c]. We give a brief description on the analysis applied in this work.…”

“…For the corresponding Navier-Stokes system, Xin [Xin98] proved that smooth solution will blow up in finite time in the whole space when the initial density has compact support, while Rozanova [Roz08] showed similar results for rapidly decreasing initial density. Fan-Jiang-Ou [FJO10], Sun-Wang-Zhang [SWZ11] and Suen [Sue20c] established some blow-up criteria for the classical solutions to 3D compressible flows, which were further extended by Lu-Du-Yao [LDY12] for MHD system. For the isothermal case when P (ρ) = Kρ for some K > 0, it was proved in [Sue13a,Sue15] that without vacuum in the initial density, when the density and magnetic field are essential bounded, the smooth solutions to (1.1)-(1.4) can be extended globally in time.…”