Global existence of solutions for Boussinesq system with energy dissipation

Abstract

The Boussinesq system coupled by an energy dissipation brings new challenges in the study of global existence of solutions, for instance, this system does not have scale invariance which makes it difficult to show existence of mild solutions when initial data belongs to scaling invariant function spaces. In this paper we are interested to show global existence of solutions [u,θ] for Boussinesq system coupled by bilinear energy dissipation Φ(u)=2μE(u)⋅E(u) on smooth bounded domain Ω⊆Rn or whole space Rn, n⩾3, when the initial data [u0,θ0]∈X0=Wσ1,n/2(Ω)×Ln/2(Ω) is sufficiently small and the external force F(θ)=ϱfθen has low regularity in the sense t1/2−b/2nf∈L∞((0,T):Lb(Ω)) or f∈Ls((0,T):Lb(Ω)), where 2s=1−nb and b∈[n,∞). It seems that our results on the global and local well-posedness are the first to provide an Lp-approach when the initial velocity data u0 belongs to the space W˙σ1,n/2(Rn) that is scaling invariant.