On partial regularity problem for 3D Boussinesq equations

In this paper, we study the partial regularity of the solutions for the three-dimensional Boussinesq equations. We first prove a criterion of local Hölder continuous of the suitable weak solutions of the Boussinesq equations, and show that one-dimensional Hausdorff measure of the singular point set is zero. Secondly, we present a local uniform gradient estimate on the suitable weak solutions and assert that the local behavior of the solution can be dominated by some scaled quantities, such as the scaled local L³-norm of the velocity. Besides, when the initial data v₀ and θ₀ decay sufficiently rapidly at ∞, the distribution of the regular point set of the suitable weak solutions is also considered. Based on it, one can find that MHD equations are more similar to Navier–Stokes equations than Boussinesq equations. Finally, we give a local regularity criterion of the suitable weak solutions near the boundary.