On the fine structure and hierarchy of gradient catastrophes for multidimensional homogeneous Euler equation

TL;DR

This paper analyzes the detailed structure and hierarchy of gradient blow-ups in multidimensional homogeneous Euler equations, revealing a layered hierarchy of singularities influenced by initial data and demonstrating superpositions of blow-up derivatives.

Contribution

It introduces a hierarchical classification of gradient catastrophes for multidimensional Euler equations based on the rank of matrices derived from initial data, and describes the structure of singularities.

Findings

Blow-ups form an $n+1$ level hierarchy.

Strongest singularity scales as $| abla x|^{-(n+1)/(n+2)}$.

Superpositions of blow-up derivatives are possible in multi-dimensional cases.

Abstract

Blow-ups of derivatives and gradient catastrophes for the $n$-dimensional homogeneous Euler equation are discussed. It is shown that, in the case of generic initial data, the blow-ups exhibit a fine structure in accordance of the admissible ranks of certain matrix generated by the initial data. Blow-ups form a hierarchy composed by $n+1$ levels with the strongest singularity of derivatives given by $\partial u_i/\partial x_k \sim |\delta \mathbf{x}|^{-(n+1)/(n+2)}$ along certain critical directions. It is demonstrated that in the multi-dimensional case there are certain bounded linear superposition of blow-up derivatives. Particular results for the potential motion are presented too. Hodograph equations are basic tools of the analysis.