A note on the global existence of a two-dimensional chemotaxis-Navier–Stokes system

2017 ◽  
Vol 98 (7) ◽  
pp. 1224-1235 ◽  
Author(s):  
Jie Wu ◽  
Chunyan Wu
Keyword(s):  
Global Existence ◽  
Stokes System ◽  
Navier Stokes ◽  
Two Dimensional

scholarly journals GLOBAL EXISTENCE RESULTS FOR THE ANISOTROPIC BOUSSINESQ SYSTEM IN DIMENSION TWO

2011 ◽  
Vol 21 (03) ◽  
pp. 421-457 ◽  
Author(s):  
RAPHAËL DANCHIN ◽  
MARIUS PAICU
Keyword(s):  
Global Existence ◽  
Turbulent Flows ◽  
Vertical Direction ◽  
Boussinesq System ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Incompressible Euler Equation ◽  
Navier Stokes Equation ◽  
Transport Diffusion ◽  
Anisotropic Viscosity

Models with a vanishing anisotropic viscosity in the vertical direction are of relevance for the study of turbulent flows in geophysics. This motivates us to study the two-dimensional Boussinesq system with horizontal viscosity in only one equation. In this paper, we focus on the global existence issue for possibly large initial data. We first examine the case where the Navier–Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transport–diffusion equation with diffusion in the horizontal direction only. For both systems, we construct global weak solutions à la Leray and strong unique solutions for more regular data. Our results rest on the fact that the diffusion acts perpendicularly to the buoyancy force.

Stability of Navier–Stokes Equations

2006 ◽  
Author(s):  
Jean-Yves Chemin ◽  
Benoit Desjardins ◽  
Isabelle Gallagher ◽  
Emmanuel Grenier
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Well Posedness ◽  
Two Dimensional ◽  
Navier Stokes Equations ◽  
The Stability ◽  
Three Space

In this chapter we intend to investigate the stability of the Leray solutions constructed in the previous chapter. It is useful to start by analyzing the linearized version of the Navier–Stokes equations, so the first section of the chapter is devoted to the proof of the well-posedness of the time-dependent Stokes system. The study will be applied in Section 3.2 to the two-dimensional Navier–Stokes equations, and the more delicate case of three space dimensions will be dealt with in Sections 3.3–3.5.

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