Local and global solvability for the Boussinesq system in Besov spaces

Abstract

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.