Global well-posedness and long time behavior of 2D magnetohydrodynamic equations with partial dissipation in half space

Whether smooth solutions of partially dissipative 2D magnetohydrodynamic (MHD) equations develop singularities in finite time has been a long-standing open problem. Since the pioneering work by Cao and Wu [Adv. Math. 226, 1803–1822 (2011)], such problem has garnered much more attention. When the magnetic field is close to an equilibrium state, the study of stability and large-time behavior provides a significant example of the stabilizing effects of the magnetic field on electrically conducting fluids. In this paper, we consider the global well-posedness for 2D partially dissipation MHD equations around a uniform magnetic field in R+2 with no-slip boundary condition. An efficient Stokes estimate is developed, which can be used to obtain the global low order (H2) energy for the partially dissipative MHD system with physical boundary. In addition, we obtain an explicit decay rate for the linear system using spectral analysis methods. We also observe that the half-space shares some similarities with the Cauchy problem, but exhibits some differences, especially when considering spatial derivatives.