  We prove that the Navier-Stokes initial value problem is well-posed in the
logrithmically refined Besov spaces when the second index is not less than
certain critical value, and ill-posed in such spaces when the second index is
less than this critical value. The well-posedness result is proved by using
some sharp bilinear estimates obtained from some Hardy-Littlewood type
inequalities. The ill-posedness assertion is proved by refining the arguments
of Wang [18] and Yoneda [20].
