# zbMATH — the first resource for mathematics

Local existence and blow-up criterion for the Boussinesq equations. (English) Zbl 0882.35096
Summary: We prove local existence and uniqueness of smooth solutions of the Boussinesq equations $v_t+(v\cdot\nabla) v=-\nabla p+\theta f,\quad \theta_t+ v\cdot\nabla\theta= 0,\quad \text{div }v= 0.$ We also obtain a blow-up criterion for these solutions. This shows that the maximum norm of the gradient of the passive scalar controls the breakdown of smooth solutions of the Boussinesq equations. As an application of this criterion, we prove global existence of smooth solutions in the case of zero external force.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B40 Asymptotic behavior of solutions to PDEs 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
Full Text:
##### References:
 [1] Ohkitani, Some mathematical aspects of 2D vortex dynamics. Proc. Miniconference of Partial Differential Equations and Applications, RIM-GARC Lecture Note Ser. 38 pp 35– (1997) · Zbl 0873.35071 [2] Majda, Vorticity and the mathematical theory of incompressible fluid flow (1986) · Zbl 0595.76021 [3] DOI: 10.1063/1.868044 · Zbl 0822.76087 · doi:10.1063/1.868044 [4] DOI: 10.1007/BF01212349 · Zbl 0573.76029 · doi:10.1007/BF01212349 [5] DOI: 10.1002/cpa.3160410704 · Zbl 0671.35066 · doi:10.1002/cpa.3160410704
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.