On robustness of a strong solution to the Navier–Stokes equations with Navier's boundary conditions in the L³-norm

We recall or prove a series of results on solutions to the Navier–Stokes equation with Navier's slip boundary conditions. The main theorem says that a strong solution $\mathbf{u}$ on any time interval (0,T) (where $0<T\leqslant \infty$) is robust in the sense that small perturbations of the initial value in the norm of $\mathbf{L}_{\sigma}^{3}(\Omega )$ and the acting body force in the norm of ${{L}^{2}}\left(0,T;\,{{\mathbf{L}}^{3/2}}(\Omega )\right)$ cause only a small perturbation of solution $\mathbf{u}$ in the norm of ${{\mathbf{L}}^{3}}(\Omega )$. This result particularly implies that the maximum length of the time interval, on which the solution starting from the initial value ${{\mathbf{u}}_{0}}\in \mathbf{L}_{\sigma}^{3}(\Omega )$ is regular, is a lower semi-continuous functional on $\mathbf{L}_{\sigma}^{3}(\Omega )$.