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

We consider the 3-D Navier-Stokes initial value problem, $$v_t-\nu \mathrm{\Delta }v=-\mathcal{P}[v\cdot \mathrm{\nabla }v]+f,v(x,0)=v_0(x),x\in {\mathbb{T}}^3(\ast )$$ where $\mathcal{P}$ is the Hodge projection. We assume that the Fourier transform norms $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{l^1({\mathbb{Z}}^3)}$ 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 $\widehat{U}(k,q)$, exponentially bounded for $q$ in a sector centered on ${\text{RR}}^{+}$, where $q$ is the inverse Laplace dual to $1\mathrm{/}{t}^n$ for $n \ge 1$. This implies in particular local existence of a classical solution to () for $t\in (0,T)$, where $T$ depends on $\mathrm{\parallel }{\widehat{v}}_0{\mathrm{\parallel }}_{l^1}$ and $| {\hat f} |{l^1}$\mathrm{.}GlobalexistenceofthesolutiontoNSfollowsif$| {\hat U} (\cdot, q) |{l^1}$hassubexponentialboundsas$q\to\infty$\mathrm{.}If$f=0$, then the converse is also true: if NS has global solution, then there exists $n \ge 1 $forwhich$| {\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 $\widehat{U}$ 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.