On the energy inequality for weak solutions to the three-dimensional compressible magnetohydrodynamic equations

2018 ◽  
Vol 59 (1) ◽  
pp. 011507
Author(s):  
Xin Liu
Keyword(s):  
Weak Solutions ◽  
Energy Inequality ◽  
Three Dimensional ◽  
Magnetohydrodynamic Equations

scholarly journals A New Regularity Criterion for the Three-Dimensional Incompressible Magnetohydrodynamic Equations in the Besov Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
TianLi LI ◽  
Wen Wang ◽  
Lei Liu
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Besov Spaces ◽  
Pressure Field ◽  
Three Dimensional ◽  
Regularity Criterion ◽  
Mhd Equations ◽  
Regularity Criteria ◽  
Magnetohydrodynamic Equations ◽  
Incompressible Magnetohydrodynamic Equations

Regularity criteria of the weak solutions to the three-dimensional (3D) incompressible magnetohydrodynamic (MHD) equations are discussed. Our results imply that the scalar pressure field π plays an important role in the regularity problem of MHD equations. We derive that the weak solution u , b is regular on 0 , T , which is provided for the scalar pressure field π in the Besov spaces.

The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier–Stokes equations

2007 ◽  
Vol 463 (2082) ◽  
pp. 1491-1508 ◽  
Author(s):  
P.E Kloeden ◽  
J Valero
Keyword(s):  
Weak Solutions ◽  
Weak Topology ◽  
Stokes Equations ◽  
Energy Inequality ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Connected Subset ◽  
Weakly Compact ◽  
Navier Stokes Equations ◽  
Strong Connectedness

The attainability set of the weak solutions of the three-dimensional Navier–Stokes equations which satisfy an energy inequality is shown to be a weakly compact and weakly connected subset of the space H , i.e. the Kneser property holds in the weak topology for such weak solutions. The proof of weak connectedness uses the strong connectedness of the attainability set of the weak solutions of the globally modified Navier–Stokes equations, which is first proved. The weak connectedness of the weak global attractor of the three-dimensional Navier–Stokes equations is also established.

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