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

TL;DR

This paper introduces an integral formulation for the 3-D Navier-Stokes equations using inverse Laplace transforms, establishing local and global existence results for smooth solutions based on Fourier transform norms and providing a numerical approach that extends existence times.

Contribution

It develops a novel integral equation approach for 3-D Navier-Stokes, linking solution existence to bounds on transformed variables and offering a new numerical method for extended existence times.

Findings

Proves local existence of smooth solutions using integral equations.

Establishes conditions for global existence based on bounds of transformed solutions.

Numerical evidence suggests longer existence times than classical estimates.

Abstract

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.