Methods and Applications of Analysis

Volume 18 (2011)

Number 4

On the Boussinesq system: regularity criteria and singular limits

Pages: 391 – 416

DOI: http://dx.doi.org/10.4310/MAA.2011.v18.n4.a3

Authors

Luigi C. Berselli (Università degli Studi di Pisa, Dipartimento di Matematica Applicata “U. Dini”, Pisa, Italy)

Stefano Spirito (Università dell’Aquila, Dipartimento di Matematica, Coppito, Italy)

Abstract

We consider the 3D Boussinesq system and we prove several criteria, not involving the density, for the continuation of smooth solutions. We give particular emphasis to the results in bounded domains, under various boundary conditions. The results we prove are partially known and we collect them in a unified framework, mainly with the perspective of understanding the stabilization/ smoothing required by numerical methods (especially by large scales methods). In the final section, we also consider the vanishing viscosity/diffusivity limits, proving (locally-in-time) sharp singular limits for smooth solutions of the Cauchy problem.

Keywords

Boussinesq equations, regularity criteria, vanishing viscosity/diffusivity limits

2010 Mathematics Subject Classification

Primary 35Q35. Secondary 35B30, 35B65, 76B03.

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