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 system in a three-dimensional smooth bounded domain when the initial velocity u(0) belongs to critical Besov spaces. A typical space is B=(B) over circle (-1+3/q)(q,infinity) with 3 < q < infinity, 2 < s < infinity satisfying 2/s + 3/q <= 1 or B=B-q,infinity(-1+3/q). 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 u(0) -> u is locally Lipschitz from B to C ([0, T]; B). This implies that in the range 3 < q < infinity, 2 < s <= infinity with 3/q + 2/s <= 1 the problem is well posed which is in strong contrast to norm inflation phenomena in the space B-infinity,s(-1), 1 <= s < infinity proved in the last years for the whole space case.