This paper is concerned with well-posedness of the Boussinesq system. We prove that the $n$ ($n\ge2$) dimensional Boussinesq system is well-psoed for small initial data $(\vec{ u}_0,\theta_0)$ ($\nabla\cdot\vec{ u}_0=0$) either in $({ B}^{ -1}_{ \infty,1}\cap{ B^{ -1,1}_{ \infty,\infty}})\times{ B}^{ -1}_{ p,r}$ or in ${ B^{ -1,1}_{ \infty,\infty}}\times{ B}^{ -1,\epsilon}_{ p,\infty}$ if $r\in[1,\infty]$, $\epsilon>0$ and $p\in(\frac{ n}{ 2},\infty)$, where $B^{ s,\epsilon}_{ p,q}$ ($s\in\mathbb{ R}$, $1\leq p,q\leq\infty$, $\epsilon>0$) is the logarithmically modified Besov space to the standard Besov space $B^{ s}_{ p,q}$. We also prove that this system is well-posed for small initial data in $({ B}^{ -1}_{ \infty,1}\cap{ B^{ -1,1}_{ \infty,\infty}})\times({ B}^{ -1}_{ \frac{ n}{ 2},1}\cap{ B^{ -1,1}_{ \frac{ n}{ 2},\infty}})$.
Categories: math.AP