Wellposedness for the magnetohydrodynamics equation in critical space

Abstract

In this article, we study wellposedness of magnetohydrodynamics equation in Besov space in ℝ³ × [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 with (p, q, r) ∈ [1, ∞] × [2, ∞] × [1, ∞] such that by applying contraction argument directly. Moreover, we find that the bilinear operator ℬ seeing below is continuous from to for which improves the well-known result for r = ∞.

Keywords:

• magnetohydrodynamics equation
• Littlewood–Paley decomposition
• Besov space

AMS Subject Classifications: :

• 76W05, 35B65

Acknowledgement

The author would like to thank Professor Y. Chemin for sending his lecture.