A new blowup criterion for strong solutions to the three‐dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain

Abstract

A new blowup criterion for strong solutions to the three-dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain Communicated by Z. XinIn this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the threedimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in BMO norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO-type estimate for the LamK e system (2.6) and a variant of the Brezis-Waigner's inequality (2.3) play a critical role in the proof. which implies C 1 3 > 0. The compressible MHD system (1.1) is a combination of the compressible Navier-Stokes equations of fluid dynamics and Maxwells equations of electromagnetism. There have been a number of studies on MHD by physicists and mathematicians because of their

A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow

“…With Corollary 3, we are ready to prove the upper bound for ∇P L q with q > 3, which is important to derive the desired contradiction. Unlike the proof for lemma 3.7 of [2] where the upper bound for ∇ρ L q was obtained due to the given boundedness for ρ L ∞ t BM Ox in the contradiction arguments, here we have to derive the upper bound in terms of the pressure gradient. It turns out that the Brezis-Waigner's inequality (2.2) and the BM O estimate for the Lamé system play important parts.…”