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A new blowup criterion for strong solutions to the three‐dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain
Author(s) -
Chen Yingshan,
Hou Xiaofeng,
Zhu Limei
Publication year - 2017
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4407
Subject(s) - bounded function , mathematics , magnetohydrodynamic drive , compressibility , norm (philosophy) , domain (mathematical analysis) , dirichlet problem , mathematical analysis , magnetohydrodynamics , dirichlet distribution , magnetic field , physics , mechanics , quantum mechanics , political science , law , boundary value problem
In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three‐dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in B M O norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO‐type estimate for the Lam é system [Disp. Item 2.6. 2.6 LU:=μΔU+(μ+λ)∇divU=F,inΩ,U(x)=0,on∂Ω,U(x)=0,asx→∞. ...] and a variant of the Brezis‐Waigner's inequality [Disp. Item 2.3. 2.3 ∥f∥L∞(Ω)⩽C1+∥f∥BMO(Ω)ln(e+∥∇f∥Lp(Ω)). ...] play a critical role in the proof. Copyright © 2017 John Wiley & Sons, Ltd.