This paper is concerned with well-posedness of the natural convection in a viscous incompressible fluid. We first prove that the n-dimensional Boussinesq system is well-posed for initial data $(\vec{u}_0,\theta _0)$ either belonging to $({B}^{-1}_{\infty ,1}\cap {B^{-1,1}_{\infty ,\infty }})\times {B}^{-1}_{p,r}$ or to ${B^{-1,1}_{\infty ,\infty }}\times {B}^{-1,\varepsilon }_{p,\infty }$ with 1 ⩽ r ⩽ ∞, $\frac{n}{2}<p<\infty$, ɛ > 0 and then we prove that this system is well-posed for initial data belonging to $({B}^{-1}_{\infty ,1}\cap {B^{-1,1}_{\infty ,\infty }})\times ({B}^{-1}_{\frac{n}{2},1}\cap {B^{-1,1}_{\frac{n}{2},\infty }})$. well-posedness; $B^{s,\alpha }_{p,r}$; $\widetilde{L}^\rho _T(B^{s,\alpha }_{p,r})$.
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2012
American Institute of Physics
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