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In this paper, we consider the 2D incompressible MHD equations withhorizontal dissipation and horizontal magnetic diffusion. We establish localexistence and uniqueness of solutions for initial data in $H^s$($s\geq 1$). Theproof relays heavily on new estimates of the nonlinear terms.

Local existence and uniqueness of solutions for the 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion