The two-dimensional Navier-Stokes system \(\begin{aligned} \left\{ \begin{array}{l} u_t + (u\cdot \nabla ) u =\Delta u+\nabla P + f(x,t), \\ \nabla \cdot u=0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}\) is considered along with homogeneous Dirichlet boundary conditions in a smoothly bounded planar domain \(\Omega \subset \mathbb {R}^2\) . By assuming that the initial fluid velocity \(u_0\) and the external force f are smooth and satisfy \(\Vert u_0\Vert _{L^{\infty }(\Omega )}+\Vert {\mathcal {P}}f\Vert _{L^q( ((t-1)_+,t);L^p(\Omega ))}\) \( \le M\) for all \(t \in (0,T)\) , and that (u, P) is a classical solution of ( \(\star \) ) on \(\Omega \times (0,T)\) , it is shown that \(\Vert u\Vert _{L^{\infty }(\Omega \times (0,T))}<\infty \) under the condition that \(p \in (1,\infty ]\) and \(q \in [1,\infty ]\) fulfill either \(\begin{aligned} q>1 \qquad \text{ and } \qquad \frac{1}{p} + \frac{1}{q} <1, \end{aligned}\) or \(\begin{aligned} q=1 \qquad \text{ and } \qquad p=\infty . \end{aligned}\) In addition, given any \(x_0 \in \Omega \) and \(T>0\) , certain smooth f and \(u=(u_1,u_2)\) are constructed which are such that \(\Vert f\Vert _{L^q((0,T);L^p(\Omega ))}<\infty \) for any \(p \in [1,\infty )\) and \(q \in (1,\infty ]\) fulfilling \(\begin{aligned} \frac{1}{p}+\frac{1}{q} \ge 1, \end{aligned}\) that u solves ( \(\star \) ) in \(\Omega \times (0,T)\) with some suitable P, but that \(\begin{aligned} u_1(x_0,t)\rightarrow + \infty \qquad \text{ as } t\nearrow T. \end{aligned}\)
