Global unique solutions for the incompressible MHD equations with variable density and electrical conductivity

We study the global unique solutions to the 2-D inhomogeneous incompressible MHD equations, with the initial data (u₀, B₀) being located in the critical Besov space \(\dot{B}_{p,1}^{{-1}+{{2}\over{p}}}(\mathbb{R}^{2})\) $${\dot{B}}_{p,1}^{-1+\frac{2}{p}}({\mathbb{R}}^2)$$ (1 < p < 2) and the initial density ρ₀ being close to a positive constant. By using weighted global estimates, maximal regularity estimates in the Lorentz space for the Stokes system, and the Lagrangian approach, we show that the 2-D MHD equations have a unique global solution.