Integral formulation of 3D Navier-Stokes and longer time existence of smooth solutions

We extend Borel summability methods to the analysis of the 3D Navier-Stokes initial value problem, v_t - ν Δ v = -P[v · ∇ v] + f, v(x, 0) = v₀ (x), ∈ x T³, where P is the Hodge projection to divergence-free vector fields. We assume that the Fourier transform norms ∥f̂∥l¹ (ℤ³) and ∥v̂∥₀|l¹ (ℤ³) are finite. We prove that the integral equation obtained from (*) by Borel transform and Écalle acceleration, Û (k, q), is exponentially bounded for q in a sector centered on ⁺, where q is the inverse Laplace dual to 1/t ⁿ for n <1. This implies in particular local existence of a classical solution to (*) for t ∈ (0, T), where T depends on ∥v̂₀∥l¹ and ∥f̂l¹. Global existence of the solution to NS follows if ∥Û(·, q)∥ l¹ has subexponential bounds as q → ∞. If f = 0, then the converse is also true: if NS has global solution, then there exists n <1 for which ∥Û(·, q)∥ necessarily decays. More generally, if the exponential growth rate in q of Û is α, then a classical solution to NS exists for t ∈ (0, α^-1/n). We show that α can be better estimated based on the values of Û on a finite interval [0, q₀]. We also show how the integral equation can be solved numerically with controlled errors. Preliminary numerical calculations of the integral equation over a modest [0, 10], q-interval for n = 2 corresponding to Kida ([21]) initial conditions, though far from being optimized or rigorously controlled, suggest that this approach gives an existence time for 3D Navier-Stokes that substantially exceeds classical estimate. © 2011 World Scientific Publishing Company.