On the local well-posedness and a Prodi–Serrin-type regularity criterion of the three-dimensional MHD-Boussinesq system without thermal diffusion

• DOI: 10.1016/j.jde.2017.03.024
• Publication Date: 2017-03-22T05:45:46Z
• Authors: Adam Larios; Yuan Pei
• Keywords: Boussinesq approximation (buoyancy); Inviscid flow

We prove a Prodi-Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal diffusion, in terms of only two velocity and two magnetic components. This is the first Prodi-Serrin-type criterion for a hydrodynamic system which is not fully dissipative, and indicates that such an approach may be successful on other systems. In addition, we provide a constructive proof of the local well-posedness of solutions to the fully dissipative 3D MHD-Boussinesq system, and also the fully inviscid, irresistive, non-diffusive MHD-Boussinesq equations. We note that, as a special case, these results include the 3D non-diffusive Boussinesq system and the 3D MHD equations. Moreover, they can be extended without difficulty to include the case of a Coriolis rotational term.<br/>28 Pages<br/>