Borel summability of Navier-Stokes equation in ${\mathbb{R}}^3$ and small time existence

Abstract

We consider the Navier-Stokes initial value problem, $$v_t - \nabla v = -\mathcal{P} [ v \cdot \nabla v \right ] + f, v(x, 0) = v_0 (x), x \in \mathbb{R}^3 $$ where $\mathcal{P}$ is the Hodge-Projection to divergence free vector fields in the assumption that $|f{|}_{\mu ,\beta }<\infty$ and $|v_0{|}_{\mu +2,\beta }<\infty$ for $\beta \ge 0,\mu >3$, where $$|\hat{f}(k)|=\underset{k\in {\mathbb{R}}^3}{\sup }{e}^{\beta |k|}(1+|k|{)}^{\mu }|\hat{f}(k)|$$ and $\hat{f}(k)=\mathcal{F}[f(\cdot )](k)$ is the Fourier transform in $x$. By Borel summation methods we show that there exists a classical solution in the form $$v(x,t)=v_0+{\int }_0^{\infty }{e}^{-p/t}U(x,p)dp$$ $t\in\CC$, $\Re \frac{1}{t}>\alpha$, and we estimate $\alpha$ in terms of $|{\hat{v}}_0{|}_{\mu +2,\beta }$ and $|\hat{f}{|}_{\mu ,\beta }$. We show that $|\hat{v}(\cdot ;t){|}_{\mu +2,\beta }<\infty$. Existence and $t$-analyticity results are analogous to Sobolev spaces ones. An important feature of the present approach is that continuation of $v$ beyond $t={\alpha }^{-1}$ becomes a growth rate question of $U(\cdot ,p)$ as $p\to \infty$, $U$ being is a known function. For now, our estimate is likely suboptimal. A second result is that we show Borel summability of $v$ for $v_0$ and $f$ analytic. In particular, we obtain Gevrey-1 asymptotics results: $v\sim v_0+{\sum }_{m=1}^{\infty }{v}_m{t}^m$, where $|v_m|\le m!A_0{B}_0^m$, with $A_0$ and $B_0$ are given in terms of to $v_0$ and $f$ and for small $t$, with $m(t)=\lfloor B_0^{-1}{t}^{-1}\rfloor$, $$|v(x,t)-v_0(x)-\sum _{m=1}^{m(t)}{v}_m(x)t^m|\le A_{0}m(t{)}^{1/2}{e}^{-m(t)}$$