scholarly journals Regularity Criteria for Hyperbolic Navier-Stokes and Related System

2012 ◽  
Vol 2012 ◽  
pp. 1-7
Author(s):  
Jishan Fan ◽  
Tohru Ozawa
Keyword(s):  
Besov Space ◽  
Strong Solutions ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
Navier Stokes ◽  
Related System

We prove a regularity criterion for strong solutions to the hyperbolic Navier-Stokes and related equations in Besov space.

scholarly journals A Geometrical Regularity Criterion in Terms of Velocity Profiles for the Three-Dimensional Navier–Stokes Equations

2019 ◽  
Vol 72 (4) ◽  
pp. 545-562 ◽  
Author(s):  
C V Tran ◽  
X Yu
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Regularity Criterion ◽  
Velocity Profiles ◽  
Regularity Criteria ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Large Velocity ◽  
Whole Space ◽  
Velocity Region

Summary In this article, we present a new kind of regularity criteria for the global well-posedness problem of the three-dimensional Navier–Stokes equations in the whole space. The novelty of the new results is that they involve only the profiles of the magnitude of the velocity. One particular consequence of our theorem is as follows. If for every fixed $t\in (0,T)$, the ‘large velocity’ region $\Omega:=\{(x,t)\mid |u(x,t)|>C(q)\left|\mkern-2mu\left|{u}\right|\mkern-2mu\right|_{L^{3q-6}}\}$, for some $C(q)$ appropriately defined, shrinks fast enough as $q\nearrow \infty$, then the solution remains regular beyond $T$. We examine and discuss velocity profiles satisfying our criterion. It remains to be seen whether these profiles are typical of general Navier–Stokes flows.

Global regularity criterion for the dissipative systems modelling electrohydrodynamics involving the middle eigenvalue of the strain tensor

2021 ◽  
pp. 1-14
Author(s):  
Fan Wu
Keyword(s):  
Global Regularity ◽  
Stokes Equations ◽  
Strain Tensor ◽  
Three Dimensional ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
Dissipative Systems ◽  
Navier Stokes ◽  
Incompressible Viscous Fluids ◽  
Systems Modelling

In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].

scholarly journals Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Jishan Fan ◽  
Tohru Ozawa
Keyword(s):  
Liquid Crystal ◽  
Nematic Liquid Crystal ◽  
Stokes System ◽  
Type System ◽  
Parabolic Type ◽  
Strong Solutions ◽  
Regularity Criteria ◽  
Navier Stokes ◽  
Liquid Crystal Flow ◽  
Local Strong Solutions

We study a system describing the evolution of a nematic liquid crystal flow. The system couples a forced Navier-Stokes system describing the flow with a parabolic-type system describing the evolution of the nematic crystal director fields (Q-tensors). We prove some regularity criteria for the local strong solutions. However, we do not provide estimates on the rates of increase of high norms.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

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