Asymptotics for the semi-dissipative 2D Boussinesq system

In this article, we consider the convergence from the semi-dissipative 2D Boussinesq equations to the 2D Navier–Stokes equations in some sense. When the temperature variable θ tends to 0, we prove that the component uθ of the solution of the semi-dissipative 2D Boussinesq equations is converging to the solution u of the 2D Navier–Stokes equations in the space H and uθ is asymptotically converging to u in the space H2(Ω) with the initial data (u0, θ0) ∈ H × L2(Ω). Moreover, we obtain the continuity of the local attractor Ar of the semi-dissipative 2D Boussinesq equations with respect to A×{0} in the space H2(Ω) × L2(Ω) at r = 0, here A×{0} is a natural embedding of the global attractor A of the 2D Navier–Stokes equations into the space H × L2(Ω).