Global regularity problem of two‐dimensional magnetic Bénard fluid equations

2021 ◽  
Author(s):  
Liangliang Ma
Keyword(s):  
Global Regularity ◽  
Two Dimensional ◽  
Regularity Problem ◽  
Fluid Equations

Regularity results for the 2D critical Oldroyd-B model in the corotational case

2019 ◽  
Vol 150 (4) ◽  
pp. 1871-1913
Author(s):  
Zhuan Ye
Keyword(s):  
Stress Tensor ◽  
Critical Case ◽  
Global Regularity ◽  
A Priori ◽  
Regularity Result ◽  
Difficult Problem ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Time Singularities ◽  
Regularity Results

AbstractThis paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.

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