Existence and Smoothness of the Navier-Stokes equation using the Boundary Integral Method

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. A smooth solution with a stokeslet far-field decay for all subsequent time is sought and found, demonstrating existence and smoothness. A space-time boundary integral velocity representation is given by an integral distribution of fundamental solutions of the Navier-Stokes equation called nslets. These nslets approach eulerlets close to their origin which have a singularity line in the fluid that moves with the fluid to ensure that the velocity direction is defined. The boundary enclosing the fluid point is chosen to move with the fluid also and so in this reference frame the Lagrangian material derivative and Eulerian partial derivative become the same in the limit. Consequently, the contributions to the flux from the quadratic terms originating from the non-linearity vanish thereby enabling the boundary integral method standard theory of Oseen and Ladyzhenskaya to be used for this non-linear problem. It is then shown that the resulting representation exists and is smooth. Zero initial velocity gives the null solution. The non-linear interaction between the flow field and the fundamental solution alignment to it describes a dynamical system of two interacting linear systems incorporating chaos, and an example demonstrating reduction to the blinking vortex is given.