The 3D incompressible anisotropic magnetic Bénard system with fractional partial dissipation

Abstract In this paper, we consider the global strong solution to the 3D incompressible anisotropic magnetic Bénard system with fractional partial dissipation. It is well known that the global regularity or finite time singularity of the classical solution for the 3D Navier–Stokes and magnetohydrodynamic (MHD) equations remains an outstanding open problem. The same problem for the 3D incompressible anisotropic magnetic Bénard system with fractional partial dissipation is also difficult. For the initial data <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="bold">u</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi mathvariant="bold">b</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>θ</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mn>3</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> (\mathbf{u}_{0},\mathbf{b}_{0},\theta_{0})\in H^{1}(\mathbb{\mathbb{R}}^{3}) , we establish the global-in-time existence and uniqueness strong solutions to the 3D incompressible anisotropic magnetic Bénard system with fractional partial velocity dissipation and magnetic diffusion and thermal diffusion.