scholarly journals Global well-posedness for the Navier–Stokes–Boussinesq system with axisymmetric data

2010 ◽  
Vol 27 (5) ◽  
pp. 1227-1246 ◽  
Author(s):  
Taoufik Hmidi ◽  
Frédéric Rousset
Keyword(s):  
Boussinesq System ◽  
Navier Stokes ◽  
Well Posedness

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.

scholarly journals Local well-posedness of a dispersive Navier-Stokes system

2011 ◽  
Vol 60 (2) ◽  
pp. 517-576 ◽  
Author(s):  
C. David Levermore ◽  
Weiran Sun
Keyword(s):  
Stokes System ◽  
Navier Stokes ◽  
Well Posedness

Global Well-Posedness and Convergence Results for the 3D-Regularized Boussinesq System

2012 ◽  
Vol 64 (6) ◽  
pp. 1415-1435 ◽  
Author(s):  
Ridha Selmi
Keyword(s):  
Existence And Uniqueness ◽  
Analytical Study ◽  
Approximation Scheme ◽  
Boussinesq System ◽  
Well Posedness ◽  
Energy Methods ◽  
Unique Weak Solution ◽  
Frequency Space ◽  
Convergence Results ◽  
Fourier Theory

Abstract Analytical study of the regularization of the Boussinesq systemis performed in frequency space using Fourier theory. Existence and uniqueness of weak solutions with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray-Hopf solution of the Boussinesq system are established as the regularizing parameter α vanishes. The proofs are done in the frequency space and use energy methods, the Arselà-Ascoli compactness theorem and a Friedrichs-like approximation scheme.

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