GLOBAL WELL-POSEDNESS OF THE VISCOUS BOUSSINESQ EQUATIONS

By Thomas Y. Hou, Congming LI #. 2004

486 citations recorded by the source corpus.

We prove the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in $H^m$ with $m\ge 3$. It is known that when both the velocity and the density equations have finite positive viscosity, the Boussinesq system does not develop finite time singularities. We consider here the challenging case when viscosity enters only in the velocity equation, but there is no viscosity in the density equation. Using sharp and delicate energy estimates, we prove global existence and strong regularity of this viscous Boussinesq system for general initial data in $H^m$ with $m \ge 3$.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d07a09b40c1a4b3bd2a0539cea428bcb7840db4a

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In Applied Mathematics

In Computational Mechanics

In Mathematical Physics

Citations in the corpus

On the global well-posedness of the Boussinesq system with zero viscosity (2007, methodology): “For ν>0 and κ= 0 the problem of global well-posedness is well understood. In [5], Chae proved global well-posedness for initial data (v0,θ0) lying in Sobolev spaces Hs × Hs, with s>2 ( see also [14]). This result has been recently improved in [11] by taking the data in Hs × Hs,with s>0.Howe”
Continuous data assimilation for the 2D Bénard convection through velocity measurements alone (2014, unknown): “For recent results concerning the two-dimensional Boussinesq equations we refer the reader to [7], [11], [21], [23], [24], [29], and references therein.”
On the global well-posedness of a class of Boussinesq–Navier–Stokes systems (2009, background): “In [3,13], the authors independently proved that for large initial data (u, θ) ∈ H ×Hs with s > 2 the system is global well-posed.”
Global regularity of the 2D micropolar fluid flows with zero angular viscosity (2010, background): “Chae [5] and Hou and Li [15] independently established the global regularity of 2D Boussinesq equations (1.”
On the global well-posedness for Boussinesq system (2007, background): “This quantity is a stretching term in the three-dimensional incompressible Euler vorticity equation (for more details about this subject we can refer to [13]).”
On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity (2007, background): “More recently, D. Chae [4] and T. Y. Hou and C. Li [ 11 ] proved independently the global-in-time regularity when the initial data v0 and 0 belong to the Sobolev space Hs, with s > 2.”
Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion (2013, unknown): “In fact, in [18], the authors only required (u0, θ0) ∈ Hm(R2)×Hm−1(R2) with m ≥ 3.”
On the Global Well-posedness for the Boussinesq System with Horizontal Dissipation (2012, background): “Li [24] proved the global well-posedness independently for the two-dimensional Boussinesq system.”
Global regularity for the 2D Boussinesq equations with partial viscosity terms (2006, result): “After submission of this paper the author was informed of the preprint version of [8], where the authors obtained results similar to the part of Theorem 1.”
Global Regularity for Some Oldroyd‐B Type Models (2013, background): “A lot of works have been devoted recently to the two-dimensional Boussinesq system with partial dissipation [3], [10], [12], [6] for example.”
Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation (2011, methodology): “The global regularity for the case when ν1 = ν2 > 0 and η1 = η2 = 0 was proven by Chae [7] and by Hou and Li [19].”
Global regularity results for the 2D Boussinesq equations with vertical dissipation (2011, unknown): “The two cases (III) and (IV) were dealt with in [6] and in [19] and the global regularity was established for both cases.”
Global Well-Posedness for Euler–Boussinesq System with Critical Dissipation (2010, background): “In [7], Chae proved the global existence and uniqueness for initial data v0 0 ∈ H ×Hs, with s > 2, see also [20].”
Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations (2011, methodology): “1, are important in our energy framework, because the problem is set on a bounded domain, distinguishing itself from the Cauchy problem in [4,10].”
Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation (2015, background): “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].”