Local and global solvability for the Boussinesq system in Besov spaces

This article focuses on local and global existence and uniqueness for the strong solution to the Boussinesq system in R n {{\mathbb{R}}}^{n} ( n ≥ 3 n\ge 3 ) with full viscosity in Besov spaces. Under the hypotheses 1 < p < ∞ 1\lt p\lt \infty and − min { n ∕ p , 2 − n ∕ p } < s ≤ n ∕ p -\min \left\{n/p,2-n/p\right\}\lt s\le n/p , and the initial condition ( θ 0 , u 0 ) ∈ B ˙ p , 1 s − 1 × B ˙ p , 1 n ∕ p − 1 \left({\theta }_{0},{u}_{0})\in {\dot{B}}_{p,1}^{s-1}\times {\dot{B}}_{p,1}^{n/p-1} , the Boussinesq system is proved to have a unique local strong solution. Under the hypotheses n ≤ p < ∞ n\le p\lt \infty and − n ∕ p < s ≤ n ∕ p -n/p\lt s\le n/p , or especially n ≤ p < 2 n n\le p\lt 2n and − n ∕ p < s < n ∕ p − 1 -n/p\lt s\lt n/p-1 , and the initial condition ( θ 0 , u 0 ) ∈ ( B ˙ p , 1 s − 1 ∩ L n ∕ 3 ) × ( B ˙ p , 1 n ∕ p − 1 ∩ L n ) \left({\theta }_{0},{u}_{0})\in \left({\dot{B}}_{p,1}^{s-1}\cap {L}^{n/3})\times \left({\dot{B}}_{p,1}^{n/p-1}\cap {L}^{n}) with sufficiently small norms ‖ θ 0 ‖ L n ∕ 3 {\Vert {\theta }_{0}\Vert }_{{L}^{n/3}} and ‖ u 0 ‖ L n {\Vert {u}_{0}\Vert }_{{L}^{n}} , the Boussinesq system is proved to have a unique global strong solution.