158 research outputs found
Global small solutions to the 3D compressible viscous non-resistive MHD system
Whether or not smooth solutions to the 3D compressible magnetohydrodynamic
(MHD) equations without magnetic diffusion are always global in time remains an
extremely challenging open problem. No global well-posedness or stability
result is currently available for this 3D MHD system in the whole space
or the periodic box even when the initial data is
small or near a steady-state solution. This paper presents a global existence
and stability result for smooth solutions to this 3D MHD system near any
background magnetic field satisfying a Diophantine condition.Comment: The final published version. To appear in Math. Models Methods Appl.
Sc
Analytical Study of Certain Magnetohydrodynamic-alpha Models
In this paper we present an analytical study of a subgrid scale turbulence
model of the three-dimensional magnetohydrodynamic (MHD) equations, inspired by
the Navier-Stokes-alpha (also known as the viscous Camassa-Holm equations or
the Lagrangian-averaged Navier-Stokes-alpha model). Specifically, we show the
global well-posedness and regularity of solutions of a certain MHD-alpha model
(which is a particular case of the Lagrangian averaged
magnetohydrodynamic-alpha model without enhancing the dissipation for the
magnetic field). We also introduce other subgrid scale turbulence models,
inspired by the Leray-alpha and the modified Leray-alpha models of turbulence.
Finally, we discuss the relation of the MHD-alpha model to the MHD equations by
proving a convergence theorem, that is, as the length scale alpha tends to
zero, a subsequence of solutions of the MHD-alpha equations converges to a
certain solution (a Leray-Hopf solution) of the three-dimensional MHD
equations.Comment: 26 pages, no figures, will appear in Journal of Math Physics;
corrected typos, updated reference
- …