On the global well-posedness for Boussinesq system

Abidi, Hammadi; Hmidi, Taoufik

doi:10.1016/j.jde.2006.10.008

Cited by 192 publications

(190 citation statements)

References 10 publications

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“…Danchin and Paicu very recently explored the global regularity issue for the cases when there is either horizontal dissipation (ν 1 > 0 and ν 2 = κ 1 = κ 2 = 0) or horizontal thermal diffusion (κ 1 > 0 and ν 1 = ν 2 = κ 2 = 0) and obtained global solutions at several regularity levels (see [11]). Other interesting recent results on the 2D Boussinesq equations can be found in [1,9,10,12,13,17,20,22]. The global regularity problem for the Boussinesq equations with vertical dissipation and thermal diffusion, namely (1.1), was first studied by Adhikari et al in [2].…”

Section: Introductionmentioning

confidence: 98%

Global regularity results for the 2D Boussinesq equations with vertical dissipation

Adhikari

Cao

2011

Journal of Differential Equations

126

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This paper furthers the study of Adhikari et al. (2010) [2] on the global regularity issue concerning the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. It is shown here that the vertical velocity v of any classical solution in the Lebesgue space L q with 2 q < ∞ is bounded by C 1 q for C 1 independent of q. This bound significantly improves the previous exponential bound. In addition, we prove that, if v satisfies T 0 sup q 2 v(·,t) 2 L q q dt < ∞, then the associated solution of the 2D Boussinesq equations preserve its smoothness on [0, T ]. In particular, v L q C 2 √ q implies global regularity.

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Section: Introductionmentioning

confidence: 98%

Global regularity results for the 2D Boussinesq equations with vertical dissipation

Adhikari

Cao

2011

Journal of Differential Equations

126

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“…Le système (1) est un modèle simplifié courant pour l'évolution de fluides géophysiques (voir par exemple [27] ou [29]). Le champ de vecteurs u et le scalaire Π désignent alors respectivement la vitesse et la pression du fluide considéré et θ, une quantité scalaire (1) transportée par le fluide. Pour avoir un modèle plus réaliste, il conviendrait de rajouter des termes de forces extérieures aux deux équations.…”

Section: Introductionunclassified

“…Le cas de données régulières dans des espaces de Sobolev a été résolu (indé-pendamment) par T. Hou et C. Li dans [20], et par D. Chae dans [5]. Très récemment, H. Abidi et T. Hmidi ont établi dans [1] l'existence globale et unicité sans condition de petitesse pour des données initiales (θ 0 , u 0 ) appartenant à un très gros sous-espace de (L 2 (R 2 ))…”

Section: Introductionunclassified

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Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux

Danchin¹,

Paicu²

2008

Bul. Soc. Math. France

131

135

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Résumé. -Dans cet article, on étudie le système de Boussinesq décrivant le phéno-mène de convection dans un fluide incompressible et visqueux. Ce système est composé des équations de Navier-Stokes incompressibles avec un terme de force verticale dont l'amplitude est transportée sans dissipation par le flot du champ de vitesses.On montre que les résultats classiques pour le système de Navier-Stokes standard demeurent vrais pour le système de Boussinesq bien qu'il n'y ait pas d'amortissement sur le terme de force.Plus précisément, on établit l'existence de solutions faibles globales d'énergie finie en n'importe quelle dimension et l'existence de solutions fortes uniques globales en dimension N ≥ 3 pour de petites données initiales. Dans le cas particulier de la dimension deux, les solutions d'énergie finie sont uniques pour n'importe quelle donnée initiale dans L 2 (R 2 ). DANCHIN (R.) & PAICU (M.)Abstract (The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity)We are concerned with the so-called Boussinesq equations with partial viscosity. These equations consist of the ordinary incompressible Navier-Stokes equations with a forcing term which is transported with no dissipation by the velocity field. Such equations are simplified models for geophysics (in which case the forcing term is proportional either to the temperature, or to the salinity or to the density).In the present paper, we show that the standard theorems for incompressible NavierStokes equations may be extended to Boussinesq system despite the fact that there is no dissipation or decay at large time for the forcing term.More precisely, we state the global existence of finite energy weak solutions in any dimension, and global well-posedness in dimension N ≥ 3 for small data. In the twodimensional case, the finite energy global solutions are shown to be unique for any data in L 2 (R 2 ).

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“…In [15] Miao and Xue established the global regularity of the 2D Boussinesq equations with fractional dissipation and thermal diffusion whose total fractional power is greater than or equal to 1. Some other interesting recent results on the 2D Boussinesq equations can be found in [1,5,6,8,9].…”

Section: Introductionmentioning

confidence: 91%

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

Adhikari

Cao

2010

Journal of Differential Equations

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This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the L r -norm of the vertical velocity v for any 1 < r < ∞ is globally bounded and that the L ∞ -norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace (− ) δ for δ > 0 would guarantee the global regularity of classical solutions.

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On the global well-posedness for Boussinesq system

Abstract: International audienc

Cited by 192 publications

References 10 publications

Global regularity results for the 2D Boussinesq equations with vertical dissipation

Global regularity results for the 2D Boussinesq equations with vertical dissipation

Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

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