Wellposedness for the magnetohydrodynamics equation in critical space

摘要

In this article, we study wellposedness of magnetohydrodynamics equation in Besov space in R{sup}3[0, T]. Comparing to Kato's space [T. Kato, Strong L{sup}p solutions of the Navier-Stokes equations in Rm 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 L{sup}([0,T]; (B{sub}(p,r)){sup}((3/p)+(2/p)-1)) with (p, q, r) ∈ [1,∞]×[2,∞]×[1,∞] such that (3/p)+(4/q)>2 by applying contraction argument directly. Moreover, we find that the bilinear operator B seeing below is continuous from L{sup}∞((B{sub}(p,r)){sup}((3/p)-1))×L{sup}∞((B{sub}(p,r)){sup}((3/p)-1)) to L{sup}∞((B{sub}(p,r)){sup}((3/p)-1) for 1≤p<3/2, 1≤r≤∞ which improves the well-known result for r=∞.