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

TL;DR

This paper investigates the blow-up phenomena, gradient catastrophes, and singularity formation in solutions to the n-dimensional homogeneous Euler equations, analyzing their properties and mappings in multiple dimensions.

Contribution

It provides a detailed analysis of blow-up behavior, gradient catastrophes, and singularities in multi-dimensional Euler equations, including concrete examples and properties of associated mappings.

Findings

Blow-up derivatives occur in all dimensions n.

Existence of blow-ups depends on the dimension and conditions.

Properties of mappings defined by hodograph equations are characterized.

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