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

TL;DR

This paper establishes a new regularity criterion for the 3D MHD-Boussinesq system without thermal diffusion, and proves local well-posedness for various dissipative and inviscid cases, advancing understanding of non-dissipative hydrodynamic systems.

Contribution

It introduces the first Prodi-Serrin-type regularity criterion for a non-fully dissipative hydrodynamic system and provides constructive local well-posedness proofs for multiple variants.

Findings

Prodi-Serrin-type regularity criterion established

Local well-posedness proved for dissipative and inviscid cases

Results include special cases like 3D non-diffusive Boussinesq and MHD equations

Abstract

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.