Divergent Expansion, Borel Summability and 3-D Navier-Stokes Equation

We describe how Borel summability of divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for nonanalytic initial data, the present approach generates an integral equation applicable to much more general data. We apply these concepts to the 3-D Navier-Stokes system and show how the integral equation approach can give rise to local existence proofs. In this approach, the global existence problem in 3-D Navier-Stokes, for specific initial condition and viscosity, becomes a problem of asymptotics in the variable p (dual to 1/t or some positive power of 1/t). Furthermore, the errors in numerical computations in the associated integral equation can be controlled rigorously, which is very important for nonlinear PDEs such as Navier-Stokes when solutions are not known to exist globally. Moreover, computation of the solution of the integral equation over an interval [0, p0] provides sharper control of its p → ∞ behavior. Preliminary numerical computations give encouraging results.