On the continuity of the solutions to the Navier-Stokes equations with initial data in critical Besov spaces

It is well-known that there exists a unique local-in-time strong solution u of the initial-boundary value problem for the Navier-Stokes sytem in a three-dimensional smooth bounded domain when the initial velocity u0 belongs to critical Besov spaces. A typical space is B = B 1+3=q q;s with 3 < q < 1, 2 < s < 1 satisfying 2=s+3=q 1 or B = B 1+3=q q;1 . In this paper we show that the solution u is continuous in time up to initial time with values in B. Moreover, the solution map u0 7! u is locally Lip- schitz from B to C ([0; T];B). This implies that in the range 3 < q < 1, 2 < s 1 with 3=q + 2=s 1 the problem is well-posed which is in strong contrast to norm in ation phenomena for B 1 1;s, 1 s < 1