Wellposedness for the magnetohydrodynamics equation in critical space

In this article, we study wellposedness of magnetohydrodynamics equation in Besov space in ℝ ³ × [0, T]. Comparing to Kato's space [T. Kato, Strong L ^p solutions of the Navier–Stokes equations in ℝ ^m with applications to weak solutions, Math. Z 187 (1984), pp. 471–480] for Navier–Stokes equation, we give existence and uniqueness of the solution of MHD in (Formula presented.) with (p, q, r) ∈ [1, ∞] × [2, ∞] × [1, ∞] such that (Formula presented.) by applying contraction argument directly. Moreover, we find that the bilinear operator ℬ seeing below is continuous from (Formula presented.) to (Formula presented.) for (Formula presented.) which improves the well-known result for r = ∞.