  The paper is devoted to the analysis of the blow-ups of derivatives, gradient
catastrophes and dynamics of mappings of $\mathbb{R}^n \to \mathbb{R}^n$
associated with the $n$-dimensional homogeneous Euler equation. Several
characteristic features of the multi-dimensional case ($n>1$) are described.
Existence or nonexistence of blow-ups in different dimensions, boundness of
certain linear combinations of blow-up derivatives and the first occurrence of
the gradient catastrophe are among of them. It is shown that the potential
solutions of the Euler equations exhibit blow-up derivatives in any dimenson
$n$. Several concrete examples in two- and three-dimensional cases are
analysed. Properties of $\mathbb{R}^n_{\underline{u}} \to
\mathbb{R}^n_{\underline{x}}$ mappings defined by the hodograph equations are
studied, including appearance and disappearance of their singularities.
