Integral formulation of 3-D Navier-Stokes and longer time existence of smooth solutions
| dc.creator | Costin, O. | |
| dc.creator | Luo, G. | |
| dc.creator | Tanveer, S. | |
| dc.date | 2008-08-27 | |
| dc.date.accessioned | 2026-07-07T09:58:42Z | |
| dc.date.available | 2026-07-07T09:58:42Z | |
| dc.description | We consider the 3-D Navier-Stokes initial value problem, $$ v_t - νΔv = -\mathcal{P} [ v \cdot \nabla v ] + f , v(x, 0) = v_0 (x), x \in \mathbb{T}^3 (*) $$ 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, α, 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. | |
| dc.identifier | https://arxiv.org/abs/0808.3721 | |
| dc.identifier | http://arxiv.org/abs/0808.3721 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/167814 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Mathematical Physics | |
| dc.subject | 76D03, 76D05, 35Q3 | |
| dc.title | Integral formulation of 3-D Navier-Stokes and longer time existence of smooth solutions | |
| dc.type | text |