Uniqueness result for the 3-D Navier-Stokes-Boussinesq equations with horizontal dissipation

Haroune Houamed (1) , Pierre Dreyfuss (1)

1 LJAD - Laboratoire Jean Alexandre Dieudonné

In this paper, for the 3-D Navier-Stokes-Boussinesq system with horizontal dissipation, where there is no smoothing effect on the vertical derivatives, we prove a uniqueness result of solutions (u, ρ) ∈ L ∞ T H 0,s × H 0,1−s with (h u, h ρ) ∈ L 2 T H 0,s × H 0,1−s and s ∈ [1/2, 1]. As a consequence, we improve the conditions stated in the paper [13] in order to obtain a global well-posedness result in the case of axisymmetric initial data.

Mots clés: Boussinesq sytem; horizontal dissipation; anisotropic inequalities; uniqueness; global well-posedness

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Haroune Houamed, Pierre Dreyfuss. Uniqueness result for the 3-D Navier-Stokes-Boussinesq equations with horizontal dissipation. 2019. ⟨hal-02088514⟩