Well-posedness for the incompressible magneto-hydrodynamic system on modulation spaces

We study the Cauchy problem for an incompressible magneto-hydrodynamics (MHD) system in the modulation space Mq,σs(Rn) (n⩾2) with initial data (u0,b0)∈PMq,σs(Rn)×PMq,σs(Rn). We prove that this problem is locally well-posed in such a space when 1⩽q⩽∞, 1⩽σ<∞ and n(σ−1)σ−1⩽s, and globally well-posed when 1⩽q⩽n, nn−1⩽σ<∞ and max{n(σ−1)σ−1,n(σ−2)σ}