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

$(\vec{u}_0,\theta _0)$
(u0,θ0) either belonging to
$({B}^{-1}_{\infty ,1}\cap {B^{-1,1}_{\infty ,\infty }})\times {B}^{-1}_{p,r}$
(B,11B,1,1)×Bp,r1
or to
${B^{-1,1}_{\infty ,\infty }}\times {B}^{-1,\varepsilon }_{p,\infty }$
B,1,1×Bp,1,ɛ
with 1 ⩽ r ⩽ ∞,
$\frac{n}{2}<p<\infty$
n2<p<
, ɛ > 0 and then we prove that this system is well-posed for initial data belonging to
$({B}^{-1}_{\infty ,1}\cap {B^{-1,1}_{\infty ,\infty }})\times ({B}^{-1}_{\frac{n}{2},1}\cap {B^{-1,1}_{\frac{n}{2},\infty }})$
(B,11B,1,1)×(Bn2,11Bn2,1,1)
. well-posedness;
$B^{s,\alpha }_{p,r}$
Bp,rs,α
;
$\widetilde{L}^\rho _T(B^{s,\alpha }_{p,r})$
L̃Tρ(Bp,rs,α)
.

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