Integral formulation of 3-D Navier-Stokes and longer time existence of   smooth solutions
  We consider the 3-D Navier-Stokes initial value problem, $$ v_t - \nu \Delta v = -\mathcal{P} [ v \cdot \nabla v ] + f , v(x, 0) = v_0 (x), x \in \mathbb{T}^3 (*) $$ where $\mathcal{P}$ is the Hodge projection. We assume that the Fourier transform norms $ \| {\hat f} \|_{l^1 (\mathbb{Z}^3)}$ and $\| {\hat v}_0 \|_{l^{1} (\mathbb{Z}^3)}$ are finite. Using an inverse Laplace transform approach, we prove that an integral equation equivalent to (*) has a unique solution ${\hat U} (k, q)$, exponentially bounded for $q$ in a sector centered 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 $\| {\hat U} (\cdot, q) \|_{l^1}$ 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 the exponential growth rate bound of U, \alpha, can be better estimated based on the values of ${\hat U}$ on a finite interval $[0,q_0]$. We also show how the integral equation can be solved numerically with controlled errors. Preliminary numerical calculations suggest that this approach gives an existence time that substantially exceeds classical estimate. 
O. Costin, G. Luo and S. Tanveer