That first one doesn't compile in LaTeX (assuming \ce is from mhchem?). Perhaps the inline mode $'s shouldn't be in there? i.e.

$$ \ce{ \underset{global coordinate}{(x,y,z)} ->[\mathcal{R}][rotation] \underset{local coordinate}{(x',y',z')} -> \underset{descriptor}{\{\mathcal{D}_{ij}\} } -> E_i } $$

There does appear to be a bug though where the LaTeX expressions don't render if they're in markdown lists or tables.

I wonder if aligned multi line equations would work (I mean eqnarray or aligned environments in latex)

Kind of. See my other note here. The aligned environment work, but there should a curly braces around that are not shown, not sure the cause. https://github.com/chazeon/notes/blob/master/docs/notes/ml-p...

I would guess that the markdown parser does some overly eager escaping before handing off the input to the latex renderer. If I change the "\left\{" into "\left\\{" and the "\," into "\\," it renders like with LaTeX.