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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 $$u_0$$u0 belongs to critical Besov spaces. A typical space is $$B=B^{-1+3/q}_{q,s}$$B=Bq,s-1+3/q with $$3<q<\infty $$3<q<∞, $$2<s<\infty $$2<s<∞ satisfying $$2/s+3/q \le 1$$2/s+3/q≤1 or $$B=\mathring{B}^{-1+3/q}_{q,\infty }$$B=B˚q,∞-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 $$u_0\mapsto u$$u0↦u is locally Lipschitz from B to $$C\left( [0,T];B\right) $$C[0,T];B. This implies that in the range $$3<q<\infty $$3<q<∞, $$2<s\le \infty $$2<s≤∞ with $$3/q +2/s \le 1$$3/q+2/s≤1 the problem is well posed which is in strong contrast to norm inflation phenomena in the space $$B^{-1}_{\infty ,s}$$B∞,s-1, $$1\le s <\infty $$1≤s<∞ proved in the last years for the whole space case.

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