Local existence and blow-up criterion of Hölder continuous solutions of the Boussinesq equations

In this paper we prove the local existence and uniqueness of C 1+γ solutions of the Boussinesq equations with initial data υ 0 , θ 0 ∈ C 1+γ , ω 0 , ∇θ 0 ∈ L q for 0 < γ < 1 and 1 < q < 2. We also obtain a blow-up criterion for this local solutions. More precisely we show that the gradient of the passive scalar θ controls the breakdown of C 1+γ solutions of the Boussinesq equations.