Abstract:

The current paper is principally motivated by establishing the global wellposedness to the three-dimensional Boussinesq system with zero diffusivity in thesetting of axisymmetric flows without swirling with v0 ∈ H12(R3) ∩ B˙ 30,1(R3) anddensity ρ0 ∈ L2(R3)∩B˙ 30,1(R3). This respectively enhances the two results recentlyaccomplished in Danchin and Paicu (2008) and Hmidi and Rousset (2010). Ourformalism is inspired, in particular for the first part from Abidi (2008) concerningthe axisymmetric Navier–Stokes equations once v0 ∈ H12(R3) and external forcef ∈ L2 loc(R+; Hβ(R3)), with β > 1 4 . This latter regularity on f which is thedensity in our context is helpless to achieve the global estimates for Boussinesqsystem. This technical defect forces us to deal once again with a similar proof tothat of Abidi (2008) but with f ∈ Lβ loc(R+; L2(R3)) for some β > 4. Second, weexplore the gained regularity on the density by considering it as an external forcein order to apply the study already obtained to the Boussinesq system.

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