Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings

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