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

2016/09/20 by Adam Larios, Yuan Pei, Larios, Adam +1
Engineering · Mathematics · #35A01 #35K51 #35Q35 #35Q86 #76B03 #76D03 #76W05 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Navier-Stokes equation solutions #math.AP #msc:35A01 #msc:35K51 #msc:35Q35 #msc:35Q86 #msc:76B03 #msc:76D03 #msc:76W05

paper · pdf · doi:10.48550/arxiv.1609.06002

28 Pages

arxiv created 2016/09/20 · openalex publication_date 2016/09/20 · arxiv updated 2016/09/21 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/04

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.

Citations

• Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations
• Regularity criteria for the three-dimensional Navier-Stokes equations
• Global regularity for the 2D Boussinesq equations with partial viscosity terms

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