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

Abstract

In this paper we prove the local existence and uniqueness of $C^{1+\gamma}$ solutions of the Boussinesq equations with initial data $v_0$, $\theta_0 \in C^{1+\gamma}$, $\omega_0, \Delta\theta_0\in L^q$ for $0 < \gamma < 1$ 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 $\theta$ controls the breakdown of $C^{1+\gamma}$ solutions of the Boussinesq equations.