Well-posedness of the Viscous Boussinesq System in Besov Spaces of Negative Order Near Index $s=-1$

Abstract: 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 $({B}^{-1}{\infty,1}\cap{B^{-1,1}{\infty,\infty}})\times({B}^{-1}{\frac{n}{2},1}\cap{B^{-1,1}{\frac{n}{2},\infty}})$.