The Navier-Stokes equations in the weak- $L^n$ space with time-dependent external force

Abstract. We consider the Navier-Stokes equations with time-dependent external force, either in the whole time or in positive time with initial data, with domain either the whole space, the half space or an exterior domain of dimension $n \ge 3$. We give conditions on the external force sufficient for the unique existence of small solutions in the weak-$L^n$ space bounded for all time. In particular, this result gives sufficient conditions for the unique existence and the stability of small time-periodic solutions or almost periodic solutions. This result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak-$L^n$ space with time-independent external force.