Here in the context of boundary value problems and the level set method for optimization the following notation
$$||\nabla_x u||$$
is presented as part of a PDE
$$F(t,x)||\nabla_x u||=1$$
Having weak background in differential calculus, what does
$$||\nabla_x u||$$
mean?
What I understand is that $u$ is some function, $\nabla_x$ is partial derivative with respect to $x$ and $||\cdot||$ is (some) norm. What I'm unsure about is what $\nabla_xu$ produces (I think it either produces a constant in the case of $ax$, a function of $t$ in the case of $a+t$, function of $t$ and a constant in the case of $ax+t$ or some other function of $t$, $x$ and constant) and what does $||\cdot||$ then do to it and what kind of norm is it.
You are right. Let $\Omega \subseteq \mathbb R^n$ and $I\subseteq \mathbb R$. Now if $u \colon \Omega \times I \to \mathbb R$ then $\nabla_x $ is given by: \begin{align*} \nabla_x u(x,t) = \begin{bmatrix} \frac{\partial u}{\partial x_1}(x,t)\\ \frac{\partial u}{\partial x_2}(x,t)\\ \vdots\\ \frac{\partial u}{\partial x_n}(x,t) \end{bmatrix} \end{align*} And the norm is just as you suspected the eucledian norm $||x|| = \sqrt{\sum_{i=1}^n x_i^2}$.
Hope this was what you had in mind.