Well-Posedness and Ill-Posedness Problems of the Stationary Navier-Stokes Equations in Scaling Invariant Besov Spaces

We consider the stationary Navier-Stokes equations in Rn for n 3 in the scaling invariant Besov spaces. It is proved that if n < p 8 and 1 q 8, or p = n and 2 < q 8, then some sequence of external forces converging to zero in. B -3+ n p p, q can admit a sequence of solutions which never converges to zero in. B -1 8,8, especially in. B -1+ n p p, q. Our result may be regarded as showing the borderline case between ill-posedness and well-posedness, the latter of which Kaneko-KozonoShimizu proved when 1 p <= n and 1 q <= infinity.