Persistence of the incompressible Euler equations in a Besov space B d + , ( R d )

are considered. Here u(x, t) = (u,u, . . . ,ud) is the Eulerian velocity of a fluid flow and (u,∇)uk =∑di= u∂iu , k = , , . . . ,d with ∂i ≡ ∂ ∂xi . The best local existence and uniqueness results known for the Euler equations () in Besov spaces are a series of theorems on the space B p, (Rd) with  < p ≤ ∞ (see the introductions in [, ] for details and the references therein). The local existence for the limit case of p =  has not been reported yet possibly due to the lack of L-estimates. On the other hand, the ill-posedness of the Euler equations in [] for a range of Besov spaces has been recently studied, which signifies that it is worthwhile to clarify either the wellposedness or the ill-posedness of the solutions in some particular Besov spaces. This is why the existence problem in the space Bd+ , (Rd) is not trivial even though it is smaller than the space B∞,(R). This paper takes care of the local unique existence of the solution to the Euler equations () in a critical Besov space Bd+ , (Rd) and of the global existence for a two-dimensional case. Our main results are the following.