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.

• Presentation Conference Type: Conference Paper (published)
• Conference Name: Proceedings of the 13th UK Conference on Boundary Integral Methods - ukBIM13
• 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