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Consider an exterior space-time domain where the incompressible Navier-Stokesequation and continuity equation hold with no bodies or force fields present,and smooth velocity at initial time. This is equivalent to the velocity beingimpulsively instantaneously started into motion and further assume that thisforce impulse is bounded. A smooth solution with a Stokeslet far-field decayfor all subsequent time is sought and found, demonstrating existence andsmoothness. This is given by a space-time boundary integral velocityrepresentation by a single layer potential linear distribution of Navier-Stokesfundamental solutions called NSlets. This is obtained by extending the theoryof hydrodynamic potentials to also include a non-linear potential thatsubsequently drops out of the formulation. Zero initial velocity gives the nullsolution and so there can be only one smooth solution demonstrating uniquenessin smooth space, but this is not to say that there are not other possiblesolutions in the wider class of non-smooth spaces.

Existence, Smoothness and Uniqueness (in smooth space) of the Navier-Stokes equation by using a new Boundary Integral representation