A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity

Abstract

We are concerned with an initial boundary value problem for the compressible magnetohydrodynamic equations with viscosity depending on the density. It is shown that for the initial density away from vacuum, the strong solution to the problem exists globally if the gradient of velocity satisfies ∇u L 2 (0,T ;L ∞ ) < ∞. Our method relies upon the delicate energy estimates and elliptic estimates.

Blow‐up criterion for compressible Navier–Stokes equations with degenerate viscosities

“…Comparing with the blow‐up results established in Li et al, 23 the main work in this paper is getting the estimate of $$$ {\left\Vert \nabla \rho /\rho \right\Vert}_{L&#x0005E;{\infty}\left(0,T;{L}&#x0005E;6\right)} $$$. Besides the high degeneracy in momentum equations, viscosity coefficients vanish as the density allows vacuum which prevents us from using a similar method proposed by previous works 21–23 . From the observation of the usual energy estimate and the mathematical entropy, as well as careful computation, we can see that the key step in proving Theorem 1.2 is to derive the $$$ {L}&#x0005E;{\infty}\left(0,T;{L}&#x0005E;2\right) $$$‐estimate on $$$ \nabla \rho /\rho $$$ which is essentially equivalent to $$$ \nabla \rho $$$ when the initial density has a uniformly positive lower bound.…”

“…When the viscosity coefficients depend on the density, Cai and Sun 21 established several blowup criteria to the local solution. Zhong 22 proved the local solution is the global one if the gradient of velocity satisfies ||∇u|| L 2 (0,T;L ∞ ) < ∞. However, all the results mentioned above on the blow-up of the solutions of compressible flows are for viscosities with a uniformly positive lower bound, that is, both 𝜇(𝜌) ≥ 𝜇 and 𝜆(𝜌) ≥ 𝜆, where 𝜆, 𝜇 are positive constants.…”

“…When the viscosity coefficients depend on the density, Cai and Sun 21 established several blowup criteria to the local solution. Zhong 22 proved the local solution is the global one if the gradient of velocity satisfies $$$ {\left\Vert \nabla u\right\Vert}_{L&#x0005E;2\left(0,T;{L}&#x0005E;{\infty}\right)}&lt;\infty $$$. However, all the results mentioned above on the blow‐up of the solutions of compressible flows are for viscosities with a uniformly positive lower bound, that is, both $$$ \mu \left(\rho \right)\ge \underset{\_}{\mu } $$$ and $$$ \lambda \left(\rho \right)\ge \underset{\_}{\lambda } $$$, where $$$ \underset{\_}{\lambda },\underset{\_}{\mu } $$$ are positive constants.…”

“…Besides the high degeneracy in momentum equations, viscosity coefficients vanish as the density allows vacuum which prevents us from using a similar method proposed by previous works. [21][22][23] From the observation of the usual energy estimate and the mathematical entropy, as well as careful computation, we can see that the key step in proving Theorem 1.2 is to derive the L ∞ (0, T; L 2 )-estimate on ∇𝜌∕𝜌 which is essentially equivalent to ∇𝜌 when the initial density has a uniformly positive lower bound. When inf 𝜌 0 = 0, some of the new difficulties arise due to the appearance of vacuum.…”

On the blow‐up phenomena of the compressible Navier‐Stokes‐Korteweg system with degenerate viscosity

“…Recently, Jiu et al 32 proved the blow up phenomena of the classical solutions to the Navier‐Stokes equations with density‐dependent viscosity. Inspired by work from the blow‐up phenomena for CNS system, many mathematicians also began to study the blow‐up problem of the full compressible magnetohydrodynamic (MHD) system 40–43 …”