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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 u on any time interval (0,T) (where 0<T⩽∞) is robust in the sense that small perturbations of the initial value in the norm of Lσ3(Ω) and the acting body force in the norm of L2(0,T; L3/2(Ω)) cause only a small perturbation of solution u in the norm of L3(Ω). This result particularly implies that the maximum length of the time interval, on which the solution starting from the initial value u0∈Lσ3(Ω) is regular, is a lower semi-continuous functional on Lσ3(Ω).