Existence and smoothness of the Navier-Stokes equation #

Edmund Chadwick

12:10 Monday in 4Q08.

Part of the Analysis of continuum mechanics session.

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 set into motion and further assume this force impulse is bounded. A smooth solution with a Stokeslet far-field decay is sought and found. 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. Care must be taken in the analysis in treating instabilities in the NSlet in the near-field (in space-time) which tends to the singular Eulerlet and probably associated with turbulence.