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We consider the 3-D Navier-Stokes initial value problem, $$ v_t - \nu \Deltav = -\mathcal{P} [ v \cdot \nabla v ] + f , v(x, 0) = v_0 (x), x \in\mathbb{T}^3 (*) $$ where $\mathcal{P}$ is the Hodge projection. We assume thatthe Fourier transform norms $ \| {\hat f} \|_{l^1 (\mathbb{Z}^3)}$ and $\|{\hat v}_0 \|_{l^{1} (\mathbb{Z}^3)}$ are finite. Using an inverse Laplacetransform approach, we prove that an integral equation equivalent to (*) has aunique solution ${\hat U} (k, q)$, exponentially bounded for $q$ in a sectorcentered on $\RR^+$, where $q$ is the inverse Laplace dual to $1/t^n$ for $n\ge 1$. This implies in particular local existence of a classical solution to(*) for $t \in (0, T)$, where $T$ depends on $\| {\hat v}_0 \|_{l^{1}}$ and $\|{\hat f} \|_{l^1}$. Global existence of the solution to NS follows if $\| {\hatU} (\cdot, q) \|_{l^1}$ has subexponential bounds as $q\to\infty$. If $f=0$,then the converse is also true: if NS has global solution, then there exists $n\ge 1 $ for which $\| {\hat U} (\cdot, q) \|$ necessarily decays. We show theexponential growth rate bound of U, \alpha, can be better estimated based onthe values of ${\hat U}$ on a finite interval $[0,q_0]$. We also show how theintegral equation can be solved numerically with controlled errors. Preliminarynumerical calculations suggest that this approach gives an existence time thatsubstantially exceeds classical estimate.

INTEGRAL FORMULATION OF 3D NAVIER–STOKES AND LONGER TIME EXISTENCE OF SMOOTH SOLUTIONS