On the regularity of weak solutions of the Boussinesq equations in Besov spaces

A. Barbagallo (1) , S. Gala , M A Ragusa (2) , Michel Thera (3, 4)

1 University of Naples Federico II = Università degli studi di Napoli Federico II
2 Unict - Università degli studi di Catania = University of Catania
3 UNILIM - Université de Limoges
4 Federation University Australia Ballarat

The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov spaceḂ −1 ∞,∞ (R 3), that, if the solution of the Boussinesq equation (1.1) below (starting with an initial data in H 2) is such that (∇u, ∇θ) ∈ L 2 0, T ;Ḃ −1 ∞,∞ (R 3) , then the solution remains smooth forever after T. In this contribution, we prove the same result for weak solutions just by assuming the condition on the velocity u and not on the temperature θ. 2 A. Barbagallo et al. Mathematics Subject Classification(2000): 35Q35, 35B65, 76D05.

Mots clés: Boussinesq equations; Besov space; weak solution; regularity criterion

Domaines: Equations aux dérivées partielles [math.AP]

A. Barbagallo, S. Gala, M A Ragusa, Michel Thera. On the regularity of weak solutions of the Boussinesq equations in Besov spaces. 2019. ⟨hal-02405555⟩