Abstract : We prove uniqueness of the solutions ($u$, velocity and $\theta$, temperature) of the Boussinesq system in the whole space ${\mathbb{R}}^3$ in the critical functional spaces: continuous in time with values in $L^3$ for the velocity and $L^2$ in time with values in $L^{3/2}$ in space for the temperature. The proof relies on the property of maximal regularity for the heat equation.

Keywords : maximal regularity; uniqueness for the Boussinesq system; critical spaces