scholarly journals Global regularity of the two-dimensional magneto-micropolar fluid system with zero angular viscosity

2015 ◽  
Vol 35 (5) ◽  
pp. 2193-2207 ◽  
Author(s):  
Kazuo Yamazaki ◽  
Keyword(s):  
Micropolar Fluid ◽  
Global Regularity ◽  
Two Dimensional ◽  
Fluid System

Smoothness of Malliavin derivatives and dissipativity of solutions to two-dimensional micropolar fluid system

2017 ◽  
Vol 25 (3) ◽  
Author(s):  
Kazuo Yamazaki
Keyword(s):  
Micropolar Fluid ◽  
Kinematic Viscosity ◽  
Periodic Boundary Condition ◽  
Random Noise ◽  
Two Dimensional ◽  
Fluid System ◽  
Micropolar Fluids ◽  
Viscosity Coefficients ◽  
Nonsymmetric Stress ◽  
Nonsymmetric Stress Tensor

AbstractMicropolar fluids have microstructure and belong to a class of fluids with nonsymmetric stress tensor. We study the incompressible two-dimensional micropolar fluid system with periodic boundary condition forced by random noise that is white-in-time. In particular, we obtain a sufficient condition on the size of the angular viscosity coefficients in comparison to the vortex and kinematic viscosity coefficients so that the solution to this system is smooth in the Malliavin sense. In addition, we prove a result concerning an orthogonal projection onto a finite number of Fourier modes, taking advantage of the dissipative nature of the system.

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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