A logarithmically improved regularity criterion for the Boussinesq equations in a bounded domain

The paper is concerned with the regularity of solutions of the Boussinesq equations for incompressible fluids without heat conductivity. The main goal is to prove a regularity criterion in terms of the vorticity for the initial boundary value problem in a bounded domain Ω of R3 with Navier-type boundary conditions and we prove that if ∫0T∥ω(·,t)∥BMO(Ω)log(e+∥ω(·,t)∥BMO(Ω))dt<∞,where ω: = curl u is the vorticity, then the unique local in time smooth solution of the 3D Boussinesq equations can be prolonged up to any finite but arbitrary time. © 2020, Springer Nature Switzerland AG.