Optimal cost for the null controllability of the Stokes system with controls having $n-1$ components and applications

In this work, we investigate the optimal cost of null controllability for the $n$-dimensional Stokes system when the control acts on $n-1$ scalar components. We establish a novel spectral estimate for low frequencies of the Stokes operator, involving solely $n-1$ components, and use it to show that the cost of controllability with controls having $n-1$ components remains of the same order in time as in the case of controls with $n$ components, namely $O(e^{C\mathrm{/}T})$, i.e. the cost of null controllability is not affected by the absence of one component of the control. We also give several applications of our results.