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 M q , σ s ( R n ) ( n ⩾ 2 ) with initial data ( u 0 , b 0 ) ∈ P M q , σ s ( R n ) × P M q , σ s ( R n ) . 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 , n n − 1 ⩽ σ < ∞ and max { n ( σ − 1 ) σ − 1 , n ( σ − 2 ) σ } < s < n ( σ − 1 ) σ .