J. math. fluid mech. 5 (2003) 70–91 1422-6928/03/010070-22 c  2003 Birkh¨auser Verlag, Basel Journal of Mathematical Fluid Mechanics Global Classical Solutions for MHD System Emanuela Casella, Paolo Secchi and Paola Trebeschi Communicated by H. Beir˜ ao da Veiga Abstract. In this paper we study the equations of magneto-hydrodynamics for a 2D incom- pressible ideal fluid in the exterior domain and in the half-plane. We prove the existence of a global classical solution in H¨older spaces, by applying Shauder fixed point theorem. Mathematics Subject Classification (2000). 35L50, 58J45, 46E35, 35Q35. Keywords. Incompressible MHD system, global existence, H¨older spaces. 1. Introduction Let us consider the equations of motion of ideal incompressible magneto-hydro- dynamics in unbounded domains. We study two different cases: the exterior domain and the half-plane cases, respectively. In the first problem, let O be an isolated rigid body in the plane, whose boundary Γ is sufficiently smooth, simple closed curve. Let Ω denote R 2 \ O and let Q T := Ω × (0,T ), T> 0. Let us denote by u = u(x, t)=(u 1 (x, t),u 2 (x, t)), B = B(x, t)=(B 1 (x, t),B 2 (x, t)) and π = π(x, t) the unknown velocity field, the magnetic field and the pressure of the fluid respectively. The system of partial differential equations we are going to consider is u t +(u ·∇)u + ∇π + 1 2 ∇|B| 2 − (B ·∇)B =0 in Q T , (1) B t +(u ·∇)B − (B ·∇)u − μΔB =0 in Q T , (2) div u =0 in Q T , (3) div B =0 in Q T , (4) u · ν =0 on Γ × (0,T ), (5) B · ν =0 on Γ × (0,T ), (6) rot B =0 on Γ × (0,T ), (7)