Existence and smoothness of the Navier-Stokes equation using a Boundary Integral representation

Chadwick, Edmund

Abstract

In the current study we determine existence and smoothness of the Navier-Stokes equation as described in the Millennium problem [1]. This considers an exterior space-time domain where the incompressible Navier-Stokes equation and continuity equation hold with no bodies present, and smooth velocity at initial time. In particular, we also take the case of no force-fields. 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 spacetime boundary integral velocity representation by a single layer potential linear distribution of Navier-Stokes fundamental solutions called NSlets [2]. This is obtained by extending the theory of hydrodynamic potentials [3] to also include a non-linear potential that subsequently drops out of the formulation [2]. The proof relies on investigating boundary integrals in the near-field close to the NSlet which approximates to the Eulerlet [4] there, which has unstable singularities related to turbulence and finite-time blow up. In other approaches, turbulence and finite-time blow up are difficult to resolve [3] [5]. However, the advantage of using the Boundary Integral Method is all the required integral calculations in the formula are finite and bounded enabling us to demonstrate existence and uniqueness.

Citation

Chadwick, E. (in press). Existence and smoothness of the Navier-Stokes equation using a Boundary Integral representation.

• Conference Name: Proceedings of the 13th UK Conference on Boundary Integral Methods - ukBIM13
• Conference Location: University of Aberdeen
• Start Date: Jul 10, 2023
• End Date: Jul 11, 2023
• Acceptance Date: Apr 1, 2023
• Online Publication Date: Jul 10, 2023
• Deposit Date: Jul 3, 2023
• Publisher URL: https://abdn.pure.elsevier.com/en/activities/13th-uk-conference-on-boundary-integral-methods