Short Time Existence and Borel Summability in the Navier-stokes Equation in R

0 eU(x, p)dp t ∈ C, Re 1 t > α, and we estimate α in terms of ‖v̂0‖μ+2,β and ‖f̂‖μ,β . We show that ‖v̂(·; 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 v0 and f analytic. In particular, Borel summability implies a the Gevrey-1 asymptotics result: v ∼ v0 + P∞ m=1 vmt m, where |vm| ≤ m!A0B 0 , with A0 and B0 are given in terms of to v0 and f and for small t, with m(t) = ⌊B −1 0 t −1⌋,