Global existence of solutions for Boussinesq system with energy dissipation

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, 0] for Boussinesq system coupled by bilinear energy dissipation phi(u) = 2 mu E(u) center dot E(u) on smooth bounded domain 12 C Rn or whole space Rn, n ,, 3, when the initial data [u0, 00] is an element of X0 = W sigma 1,n/2(12) x Ln/2(12) is sufficiently small and the external force F (0) = of0en has low regularity in the sense t1/2-b/2nf is an element of L infinity ((0,T) : Lb(12)) or f is an element of Ls((0,T) : Lb(12)), where 2s = 1 - nb and b is an element of [n, oo). 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 W1,n/2 sigma (Rn) that is scaling invariant. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper investigates the impact of energy dissipation on the global existence of solutions in the Boussinesq system. Specifically, it considers the case when the initial data belongs to scaling invariant function spaces. By introducing appropriate conditions, the paper demonstrates the existence of solutions.