Existence, Smoothness and Uniqueness (in smooth space) of the Navier-Stokes equation by using a new Boundary Integral representation (2305.14219v1)

Abstract: Consider an exterior space-time domain where the incompressible Navier-Stokes equation and continuity equation hold with no bodies or force fields present, and smooth velocity at initial time. This is equivalent to the velocity being impulsively instantaneously started into motion and further assume that this force impulse is bounded. A smooth solution with a Stokeslet far-field decay for all subsequent time is sought and found, demonstrating existence and smoothness. This is given by a space-time boundary integral velocity representation by a single layer potential linear distribution of Navier-Stokes fundamental solutions called NSlets. This is obtained by extending the theory of hydrodynamic potentials to also include a non-linear potential that subsequently drops out of the formulation. Zero initial velocity gives the null solution and so there can be only one smooth solution demonstrating uniqueness in smooth space, but this is not to say that there are not other possible solutions in the wider class of non-smooth spaces.