Global regularity for the 2D magnetic B\’{e}nard system
with fractional partial dissipation
Abstract
In this paper, we considered the global regularity for the 2D
incompressible anisotropic magnetic B\’{e}nard system
with fractional partial dissipation. More precisely, we established the
global existence and regularity for the 2D incompressible anisotropic
magnetic B\’{e}nard system with only vertical
hyperdiffusion
$\Lambda_{2}^{2\beta}b_1$ and
horizontal hyperdiffusion
$\Lambda_{1}^{2\beta}b_2$ and
$(-\Delta)^{\alpha}\theta$,
where $\Lambda_{1}$ and
$\Lambda_{2}$ are directional Fourier multiplier
operators with the symbols being
$|\xi_1|$ and
$|\xi_2|$, respectively. We prove
that, for $\beta>1$ and
$0<\alpha<1$, this system always
possesses a unique global-in-time classical solution when the initial
data is sufficiently smooth.