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

Abstract

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 ${\mathbb{R}}_{\underline{u}}^n\to {\mathbb{R}}_{\underline{x}}^n$ mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.

arXiv comment18 pages, 3 figures , typos corrected

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Cited by in corpus (1)

• Multi-dimensional Jordan chain and Navier-Stokes Equation