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

Costin, O. and Luo, G. and Tanveer, S. (2011) Integral formulation of 3D Navier-Stokes and longer time existence of smooth solutions. Communications in Contemporary Mathematics, 13 (3). pp. 407-462. ISSN 0219-1997.

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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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Additional Information:© 2011 World Scientific Publishing Co. Received 27 October 2009; Revised 17 April 2010. This work was supported in part by the National Science Foundation (DMS-0406193, DMS-0601226, DMS-0600369 to OC and DMS-0405837, DMS-0733778 to S.T). We are grateful to P. Constantin for giving us useful references and to Alexey Cheskidov for pointing out that estimates of the time beyond which weak Leray solutions becomes classical are easy to obtain.
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Subject Keywords:Navier–Stokes; Borel summability; PDE
Issue or Number:3
Classification Code:AMSC numbers: 35Q30, 76D05, 76N10, 40G10
Record Number:CaltechAUTHORS:20110726-095821959
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:24542
Deposited By: Jason Perez
Deposited On:26 Jul 2011 20:12
Last Modified:03 Oct 2019 02:57

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