I am a bit lost. The Clay Mathematics Institute has written an $n$ dimensional Navier-Stokes problem description here.
On the first page, an initial condition $(4)$ is: $$\vert \partial_x^\alpha \, u°(x) \vert \le C_{\alpha K} (1+\vert x \vert)^{-K}\text{ on }\mathbb{R}^n\text{, for any }\alpha\text{ and }K.$$
May someone write it with quantifiers?
Does is it mean something like that:
$1.~\exists C_{\alpha K} \in \mathbb{R} ~~\forall x \in \mathbb{R}^n~~ \forall \alpha \in \mathbb{N}^n~~ \forall K \in \mathbb{R} ~~ \vert \partial_x^\alpha\, u°(x) \vert \le C_{\alpha K} (1+\vert x \vert)^{-K}$
but then $C_{\alpha K}$, depending on $\alpha$ and $K$, would be defined before it is the case for $\alpha$ and $K$.
Or does is it mean something like something which appears to always be the case, which so would not be that interesting?
$2.~\forall x \in \mathbb{R}^n~~ \forall \alpha \in \mathbb{N}^n~~ \forall K \in \mathbb{R} ~~ \exists C_{\alpha K} \in \mathbb{R} ~~\vert \partial_x^\alpha\, u°(x) \vert \le C_{\alpha K} (1+\vert x \vert)^{-K}$
or something else?
And also because, if we consider case $1$ a bit differently like:
$1'.~\exists C \in \mathbb{R} ~~\forall x \in \mathbb{R}^n~~ \forall \alpha \in \mathbb{N}^n~~ \forall K \in \mathbb{R} ~~ \vert \partial_x^\alpha\, u°(x) \vert \le C (1+\vert x \vert)^{-K}$
Then, if $~\exists x_0 \in \mathbb{R}^n ~ \vert u°(x_0) \vert \ne 0$, $~\exists C\in \mathbb{R} ~~\exists \epsilon >0~~\forall K \in \mathbb{R}~~0<\epsilon<\vert u°(x_0) \vert< C(1+\vert x_0 \vert)^{-K}$, which implies:
$~\exists C\in \mathbb{R} ~~\exists \epsilon >0~~\forall K \in \mathbb{R}~~(1+\vert x_0 \vert)^{K}\epsilon< C$, which is wrong, causing a null initial velocity for all $x$.
So, I may need some corrections, or precisions... Thanks for any help.
Generically when a constant is subscripted by some parameter $\theta$, it means that the quantifier for $\theta$ must have come before. It is also a good guess that if they bothered to subscript, then the constant cannot depend on variables that are not subscripted.
The correct reading is actually none of your guesses, it is:
$$\forall \alpha\in \mathbb N^n, \ \forall K>0,\ \exists C_{\alpha K}>0 \text{ s.t. } \forall x\in\mathbb R^n, \ |\partial_x^\alpha u°(x)| \le C_{\alpha K} (1+|x|)^{-K}.$$
(actually I didn't need to restrict $K>0$ but $K\le0$ is boring; the condition is supposed to indicate arbitrary polynomial decay at infinity, and gets stronger as $K\to\infty$)
PS: these functions are called Schwartz functions.