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

Abstract

We extend Borel summability methods to the analysis of the 3D Navier–Stokes initial value problem, ν_t-vΔν = -P[ν⋅∇ν]+f, ν(x, 0)=ν_0(x), x∈T^3 where P is the Hodge projection to divergence-free vector fields. We assume that the Fourier transform norms ∥f^∥l1(z^3) and ∥ν^_0∥l1(z^3) 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 R^+, 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 ∥ν^_0∥l1 and ∥f^∥l1. 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, q_0]. 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.

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