Remarks on pressure regularity criterion for the 3D Boussinesq equations

This paper establishes two logarithmically improved regularity criteria for the Cauchy problem of the 3D Boussinesq equations in termsof the pressure on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that if ∫0T‖‖‖P‖Lx1p‖Lx2q‖Lx3rβ1+ln(1+‖θ(⋅,t)‖L4)dt<∞,where2β+1p+1q+1r=2and2≤p,q,r≤∞,1−(1p+1q+1r)≥0,or ∫0T‖‖‖∇P‖Lx1p‖Lx2q‖Lx3rβ1+ln(1+‖θ(⋅,t)‖L4)dt<∞,where2β+1p+1q+1r=3and1≤p,q,r≤∞,1−(12p+12q+12r)≥0,for some T>0, then the corresponding weak solution (u,θ) to the 3D Boussinesq equations is regular on [0,T].