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

The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes (GCs) and dynamics of mappings of Rn → Rn 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 GC are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimension n. Several concrete examples in two- and three-dimensional cases are analysed. Properties of Rnu → Rnx mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.