What does $ \vert \partial^{\alpha} v_o(x) \vert $ mean in the Navier-Stokes initial velocity condition?

Question

The initial condition $\displaystyle \mathbf{v}_0(x)$ is assumed to be a smooth and divergence-free function such that, for every multi-index $\displaystyle \alpha$ and any $\displaystyle K>0$, there exists a constant $\displaystyle C= c_{\alpha K}>0$ such that

$\displaystyle \vert \partial ^{\alpha }\mathbf {v_{0}} (x)\vert \leq {\frac {C}{(1+\vert x\vert )^{K}}\qquad }$ for all $ \displaystyle \qquad x\in \mathbb {R} ^{3}$

I just want to know what is $ \vert \partial ^{\alpha }\mathbf {v_{0}} (x)\vert$ and to express it in another way than it is. It seems a bit weird to me because there is no denominator. What would be another expression of it in cylindrical coordinates?

You may find precisions at the section:

"Statement of the problem in the whole space

Hypotheses and growth conditions"

on this page: https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness

Answer 1

It is standard muli-index notation: $\alpha = (\alpha_1,\ldots, \alpha_n) \in \mathbb N^n$ where $n$ is the dimension of $\mathbb R^n \ni x= (x_1,\ldots, x_n)$. With these definitions $$\partial^\alpha := \frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}}$$ where $|\alpha| := \alpha_1+\cdots + \alpha_n$.