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Global regularity for the 3D MHD system with damping

Zujin ZhangXian Yang — 2016

Colloquium Mathematicae

We study the Cauchy problem for the 3D MHD system with damping terms ε | u | α - 1 u and δ | b | β - 1 b (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

An improved regularity criteria for the MHD system based on two components of the solution

Zujin ZhangYali Zhang — 2021

Applications of Mathematics

As observed by Yamazaki, the third component b 3 of the magnetic field can be estimated by the corresponding component u 3 of the velocity field in L λ ( 2 λ 6 ) norm. This leads him to establish regularity criterion involving u 3 , j 3 or u 3 , ω 3 . Noticing that λ can be greater than 6 in this paper, we can improve previous results.

Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations

Zujin ZhangChenxuan Tong — 2022

Applications of Mathematics

We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that | ω r ( x , t ) | + | ω z ( r , t ) | C r 10 , 0 < r 1 2 . By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing ω r , ω z and ω θ / r on different hollow cylinders, we are able to improve it and obtain | ω r ( x , t ) | + | ω z ( r , t ) | C | ln r | r 17 / 2 , 0 < r 1 2 .

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