Short Time Existence and Borel Summability in the Navier–Stokes Equation in ℝ3

Abstract

We consider the Navier–Stokes initial value problem, where 𝒫 is the Hodge-Projection to divergence free vector fields in the assumption that ‖f‖_{μ, β} < ∞ and ‖ v ₀ ‖_{μ+2, β} < ∞ for β ≥ 0,μ > 3, where and [fcirc] (k) = ℱ [f (·)] (k) is the Fourier transform in x. By Borel summation methods we show that there exists a classical solution in the form t ∈ ℂ, , and we estimate α in terms of ‖ [vcirc]₀ ‖_{μ+2, β} and ‖ [fcirc] ‖_{μ, β}. We show that ‖ [vcirc] (·; t) ‖_{μ+2, β} < ∞. Existence and t-analyticity results are analogous to Sobolev spaces ones. An important feature of the present approach is that continuation of v beyond t = α⁻¹ becomes a growth rate question of U(·, p) as p → ∞, U being is a known function. For now, our estimate is likely suboptimal. A second result is that we show Borel summability of v for v ₀ and f analytic. In particular, Borel summability implies a the Gevrey-1 asymptotics result: , where , with A ₀ and B ₀ are given in terms of to v ₀ and f and for small t, with ,