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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·)v=-p+θf,θ t +v·θ=0,divv=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:
35Q35PDEs in connection with fluid mechanics
35B40Asymptotic behavior of solutions of PDE
35A07Local existence and uniqueness theorems (PDE) (MSC2000)