Style and standard: notation for partial derivative, $u_x$ or $\partial_x u$ or $\frac{\partial u}{\partial x}$? In the context of partial differential equations, what notation should one prefer to write partial derivatives, $u_x$ or $\partial_x u$ or $\frac{\partial u}{\partial x}$? The same question for higher order or mixed partial derivatives. 
I've seen both (even in the same paper), but I'd prefer to be consistent in my own writing. 

So I'd like to ask if there is a more widely accepted standard and what merits are there to the three kinds of notation.

 A: They're all popular in PDE. I'd say the full "fraction" notation is the most standard in the sense that everyone would know what it means, but $\partial_x u$ is also hard to misinterpret. 
The postfix notation is occasionally confusing (for example, in the context of evolution equations $u_t$ could mean either $\partial_t u$ or $u(\cdot, t)$). The advantage is compactness - if you have large, ugly expressions with higher derivatives in many different directions then it can help cut down on noise. If you're going to use this notation, I'd suggest always accompanying it with an explanation of exactly what you mean. 
A: Everything that is understandable is accepted, however the form $\frac{\partial u}{\partial x}$ is more popular and it shouls be used wherever you want to avoid misunderstandings.
The only form that you listed is not really acceptable $u_x$, and it is because it looks like a variable $u$ with subvariable (like $n_0, n_1,$ etc.
However if you like those notations that are less popular, you can still use them, just be sure to clearly point out which notation you are using to avoid confusion.
