GLOBAL EXISTENCE RESULTS FOR THE ANISOTROPIC BOUSSINESQ SYSTEM IN DIMENSION TWO

By Raphaël Danchin, Marius Paicu. 2008

200 citations recorded by the source corpus.

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 a 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.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/21bd2150b38035a4cfc0e570cce4f7149fc23897

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In Applied Mathematics

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3. Changxing Miao
4. Edriss S. Titi
5. Zhen Lei
6. Darryl D. Holm
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8. Qi S. Zhang

In Computational Mechanics

1. Xiaoxin Zheng
2. Changxing Miao
3. Edriss S. Titi
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5. Darryl D. Holm
6. Evelyn Lunasin
7. John D. Gibbon

In Mathematical Physics

1. Jinkai Li
2. Xiaoxin Zheng
3. Changxing Miao
4. Edriss S. Titi
5. Zhen Lei
6. Qi S. Zhang
7. Vlad Vicol

Citations in the corpus

• Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation (2015, result): “Some generalizations of the results in [3, 14], toward weakening the assumption on the initial data, were established in Hmidi– Keraani [13] and Danchin–Paicu [10]; in particular, no smoothness assumptions on the initial data are required in [10].”
• Nonlinear maximum principles for dissipative linear nonlocal operators and applications (2011, unknown): “[4, 18, 27] and references therein.”
• Global regularity results for the 2D Boussinesq equations with vertical dissipation (2011, unknown): “[13] and [26] examined the two cases (V) and (VI) and obtained the global existence of suitably regular solutions.”
• On the Global Well-posedness for the Boussinesq System with Horizontal Dissipation (2012, background): “In the sprite of [7, 17], we prove linear losing a priori estimates for the general anisotropic system with convection.”