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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)
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[3] DOI: 10.1063/1.868044 · Zbl 0822.76087 · doi:10.1063/1.868044
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